functional-programming-emojis.tsx

/** @jsx jsx */
import { css, jsx, Global } from '@emotion/core'
import { useState, useContext } from 'react'
import Page from 'src/components/Page'
import Head from 'next/head'
import {
  ns,
  radii,
  fontSizes,
  colors,
  spaces,
  maxWidths,
  lineHeights
} from 'src/lib/theme'
import { lessonTitle } from 'src/lib/titles'
import Container from 'src/components/Container'
import ExpressionRunnerSeparator from 'src/components/ExpressionRunnerSeparator'
import EmojiSeparator from 'src/components/EmojiSeparator'
import ExpressionRunnerCaptionWrapper from 'src/components/ExpressionRunnerCaptionWrapper'
import BottomRightBadge from 'src/components/BottomRightBadge'
import Emoji from 'src/components/Emoji'
import H from 'src/components/H'
import EmojiForLetter from 'src/components/EmojiForLetter'
import EpisodeHero from 'src/components/EpisodeHero'
import InlinePrioritiesLabel from 'src/components/InlinePrioritiesLabel'
import EpisodePageFooter from 'src/components/EpisodePageFooter'
import EmojiNumber from 'src/components/EmojiNumber'
import CustomEmoji from 'src/components/CustomEmoji'
import ButtonWithTouchActiveStates from 'src/components/ButtonWithTouchActiveStates'
import * as R from 'src/components/Runners'
import { JimsTalk } from 'src/contents/0.en'
import {
  ExternalLink,
  P,
  Italic,
  Bold,
  InternalLink,
  Highlight,
  H3,
  Ul,
  Hr,
  UlLi
} from 'src/components/ContentTags'
import locale from 'src/lib/locale'
import { DateTime } from 'luxon'
import { enBaseUrl, githubRepo } from 'src/lib/meta'
import BaseAlert, { alertSpacing, AlertProps } from 'src/components/Alert'
import PrismHighlight, { defaultProps } from 'prism-react-renderer'
import theme from 'prism-react-renderer/themes/nightOwlLight'
import BubbleQuoteContext from 'src/components/BubbleQuoteContext'
import EmojiWithText from 'src/components/EmojiWithText'
import ExpressionRunnerConfigContext from 'src/components/ExpressionRunnerConfigContext'
import TwitterEmbed from 'src/components/TwitterEmbed'
import {
  StepOne,
  StepTwo,
  InstructionTwo,
  StepThree,
  InstructionThree,
  InstructionFour,
  StepFour
} from 'src/contents/4.en'
import VariableShadeContext from 'src/components/VariableShadeContext'
import TwoColGrid from 'src/components/TwoColGrid'

const numSteps = 10

const date = DateTime.fromISO('2019-11-08T12:00:00Z')
const dateString = date
  .setLocale('en')
  .setZone('America/Los_Angeles')
  .toFormat('LLL d, yyyy')
const dateSchemaString = date.setZone('America/Los_Angeles').toISO()

const title = 'You Can Explain Functional Programming Using Emojis'
const description =
  'A visual representation of Lambda calculus, Church encoding, and Y combinator'
const url = `${enBaseUrl}/functional-programming-emojis`
const ogImageUrl = `${enBaseUrl}/static/images/blog-og.png`

const tweetUrl = `https://twitter.com/intent/tweet?url=${encodeURIComponent(
  url
)}&via=chibicode&text=${encodeURIComponent(title)}`

const Ads = ({ shareId }: { shareId?: string }) => (
  <>
    <Ul>
      <UlLi>
        I’d love it if you could share this on Twitter.{' '}
        <ExternalLink href={tweetUrl}>
          <CustomEmoji type="twitter" /> Click here to Tweet this article.
        </ExternalLink>
      </UlLi>
      <UlLi>
        Also, the source code is{' '}
        <ExternalLink href={githubRepo}>available on GitHub</ExternalLink>:{' '}
        <iframe
          css={css`
            vertical-align: middle;
            transform: translateY(-0.1em);
          `}
          src="https://ghbtns.com/github-btn.html?user=chibicode&repo=Y-Combinator-for-Non-programmers&type=star&count=true&size=large"
          frameBorder="0"
          scrolling="0"
          width="160px"
          height="30px"
        ></iframe>
      </UlLi>
    </Ul>
    {shareId && (
      <>
        <P>Here’s my Twitter:</P>
        <TwitterEmbed showCard id={shareId} />
      </>
    )}
  </>
)

const Alert = (props: AlertProps) => (
  <BubbleQuoteContext.Provider value={{ inQuote: false }}>
    <div
      css={css`
        margin-left: ${spaces('-0.25')};
        margin-right: ${spaces('-0.25')};

        ${ns} {
          margin-left: 0;
          margin-right: 0;
        }
      `}
    >
      <BaseAlert {...props} />
    </div>
  </BubbleQuoteContext.Provider>
)

Alert.defaultProps = BaseAlert.defaultProps

const Subheading = ({
  step,
  noHrTop,
  children,
  coveredIn,
  ...props
}: JSX.IntrinsicElements['h3'] & {
  noHrTop?: boolean
  step: number | 'none'
  coveredIn: number | 'none'
}) => (
  <>
    {!noHrTop && (
      <Hr
        id={step !== 'none' ? `${step}` : undefined}
        css={css`
          margin-top: ${spaces(2.25)};
        `}
      />
    )}
    {step !== 'none' && coveredIn !== 'none' && (
      <div
        css={css`
          color: ${colors('indigo300')};
          margin-bottom: ${spaces(0.25)};
        `}
      >
        Step{' '}
        <Bold
          css={css`
            color: ${colors('indigo400')};
          `}
        >
          {step}
        </Bold>
        /{numSteps}
        <span
          css={css`
            font-size: ${fontSizes(0.75)};
            ${ns} {
              font-size: ${fontSizes(0.85)};
            }
          `}
        >
          {' '}
          &middot; From my course’s{' '}
          <InternalLink href={`/${coveredIn}`}>
            <H
              args={{
                name: 'titlePrefixColored'
              }}
              episodeNumberOverrides={coveredIn}
            />
          </InternalLink>{' '}
          level
        </span>
      </div>
    )}
    <H3
      {...props}
      css={css`
        margin: ${noHrTop ? spaces(2) : 0} 0 ${spaces(0.5)};
      `}
    >
      {children}
    </H3>
  </>
)

const codeFontFamily = `'Victor Mono', SFMono-Regular, Consolas,
Liberation Mono, Menlo, Courier, monospace`

const InlineCode = ({
  children,
  highlighted
}: {
  children: string
  highlighted?: boolean
}) => (
  <code
    css={css`
      font-weight: 400;
      font-family: ${codeFontFamily};
      background-color: ${highlighted ? colors('yellow200') : colors('codeBg')};
      display: inline-block;
      font-size: 0.85em;
      padding: 0.075em 0.2em;
      border-radius: ${radii(0.25)};
    `}
  >
    {children}
  </code>
)

const CodeBlock = ({
  children,
  shouldHighlight,
  result,
  pointToRunButton,
  defaultResultVisible,
  caption,
  noHighlight
}: {
  children: string
  shouldHighlight?: (lineNumber: number, tokenNumber: number) => boolean
  result?: string
  pointToRunButton?: boolean
  defaultResultVisible: boolean
  caption?: React.ReactNode
  noHighlight?: boolean
}) => {
  const [resultVisible, setResultVisible] = useState(defaultResultVisible)
  const buttonOnClick = () => setResultVisible(true)
  return (
    <div
      css={css`
        margin-left: auto;
        margin-right: auto;
        max-width: ${maxWidths('xs')};
      `}
    >
      {caption && (
        <ExpressionRunnerCaptionWrapper
          css={css`
            margin-top: ${spaces(1.75)};
          `}
        >
          {caption}
        </ExpressionRunnerCaptionWrapper>
      )}
      <PrismHighlight
        {...defaultProps}
        code={children}
        theme={theme}
        language={noHighlight ? 'diff' : 'javascript'}
      >
        {({ tokens, getLineProps, getTokenProps }) => (
          <pre
            css={[
              alertSpacing,
              css`
                background-color: ${colors('codeBg')};
                font-weight: 400;
                font-family: ${codeFontFamily};
                margin-top: ${caption ? 0 : spaces(1.75)};
                margin-left: ${spaces('-0.25')};
                margin-right: ${spaces('-0.25')};
                margin-bottom: ${result ? 0 : spaces(1.75)};
                font-size: ${fontSizes(0.8)};

                ${ns} {
                  margin-left: 0;
                  margin-right: 0;
                  font-size: ${fontSizes(0.85)};
                }
              `,
              result
                ? css`
                    border-top-left-radius: ${radii(0.5)};
                    border-top-right-radius: ${radii(0.5)};
                  `
                : css`
                    border-radius: ${radii(0.5)};
                  `,
              (!(result && resultVisible) || !result) &&
                css`
                  border-bottom-right-radius: ${radii(0.5)};
                `
            ]}
          >
            <div
              css={css`
                overflow-x: auto;
              `}
            >
              {tokens.map((line, i) => (
                <div {...getLineProps({ line, key: i })}>
                  {line.map((token, key) => (
                    <span
                      {...getTokenProps({ token, key })}
                      css={[
                        css`
                          font-style: normal !important;
                        `,
                        !!shouldHighlight &&
                          shouldHighlight(i, key) &&
                          css`
                            background: ${colors('yellow300')};
                            border-bottom: 2px solid ${colors('deepOrange400')};
                          `
                      ]}
                    />
                  ))}
                </div>
              ))}
            </div>
          </pre>
        )}
      </PrismHighlight>
      {result && (
        <>
          <div
            css={css`
              max-width: ${maxWidths('xs')};
              margin-bottom: ${spaces(1.75)};
              margin-left: ${spaces('-0.25')};
              margin-right: ${spaces('-0.25')};
              ${ns} {
                margin-left: 0;
                margin-right: 0;
              }
            `}
          >
            {resultVisible ? (
              <div
                css={[
                  alertSpacing,
                  css`
                    border-top-left-radius: 0;
                    border-top-right-radius: 0;
                    border-bottom-left-radius: ${radii(0.5)};
                    border-bottom-right-radius: ${radii(0.5)};
                    background: #fff;
                    border-left: 0.25rem solid ${colors('codeBg')};
                    border-bottom: 0.25rem solid ${colors('codeBg')};
                    border-right: 0.25rem solid ${colors('codeBg')};
                    padding-top: 0.425rem;
                    padding-bottom: 0.425rem;
                    margin-top: 0;

                    ${ns} {
                      padding-top: 0.65rem;
                      padding-bottom: 0.65rem;
                    }
                  `
                ]}
              >
                <Bold
                  css={css`
                    color: ${colors('indigo400')};
                    font-size: ${fontSizes(0.85)};
                    margin-left: -0.25rem;
                  `}
                >
                  Result:{' '}
                </Bold>
                <span
                  css={css`
                    color: ${colors('indigo500')};
                  `}
                >
                  <InlineCode highlighted>{result}</InlineCode>
                </span>
              </div>
            ) : (
              <>
                <ButtonWithTouchActiveStates
                  onClick={buttonOnClick}
                  activeBackgroundColor={colors('indigo50')}
                  css={[
                    alertSpacing,
                    css`
                      border-top-left-radius: 0;
                      border-top-right-radius: 0;
                      border-bottom-left-radius: ${radii(0.5)};
                      border-bottom-right-radius: ${radii(0.5)};
                      line-height: 1.1rem;
                      border: none;
                      margin-top: 0;
                      margin-bottom: 0;
                      font-weight: bold;
                      font-size: ${fontSizes(0.85)};
                      background: ${colors('codeButtonBg')};
                      color: ${colors('indigo500')};
                      padding-left: ${spaces(1.25)};
                      padding-right: ${spaces(1.25)};

                      &:enabled {
                        cursor: pointer;
                      }

                      @media (hover: hover) {
                        &:hover:enabled {
                          background: ${colors('indigo50')};
                        }
                        &:focus {
                          box-shadow: inset 0 0 0 1px ${colors('codeBg')};
                          outline: none;
                        }
                      }
                      &:active:enabled {
                        background: ${colors('indigo50')};
                      }
                    `
                  ]}
                >
                  Run <Emoji>▶️</Emoji>
                </ButtonWithTouchActiveStates>
                {pointToRunButton && (
                  <span
                    css={[
                      alertSpacing,
                      css`
                        font-size: ${fontSizes(0.85)};
                        animation: pointToCodeRunButton 1s infinite;
                        color: ${colors('grey700')};

                        @keyframes pointToCodeRunButton {
                          0% {
                            margin-left: 0px;
                          }
                          50% {
                            margin-left: -5px;
                          }
                          100% {
                            margin-left: 0px;
                          }
                        }
                      `
                    ]}
                  >
                    ← <H args={{ name: 'pointToRunButton' }} />
                  </span>
                )}
              </>
            )}
          </div>
        </>
      )}
    </div>
  )
}

CodeBlock.defaultProps = {
  defaultResultVisible: false
}

const PointToRunButton = ({ children }: { children: React.ReactNode }) => {
  const config = useContext(ExpressionRunnerConfigContext)
  return (
    <ExpressionRunnerConfigContext.Provider
      value={{
        ...config,
        pointToRunButton: true
      }}
    >
      {children}
    </ExpressionRunnerConfigContext.Provider>
  )
}

const DimUnhighlighted = ({ children }: { children: React.ReactNode }) => (
  <VariableShadeContext.Provider
    value={{
      shadeNonHighlighted: true
    }}
  >
    {children}
  </VariableShadeContext.Provider>
)

const guessTheResult = (
  <>
    Guess what the result would be
    <br />
    before pressing the <H args={{ name: 'run' }} /> button.
  </>
)

export default () => {
  let step = 1
  return locale === 'en' ? (
    <Page>
      <Global
        styles={[
          css`
            @font-face {
              font-family: 'Victor Mono';
              src: url('/static/fonts/VictorMono-SemiBold.woff2')
                  format('woff2'),
                url('/static/fonts/woff/SemiBold.woff') format('woff');
              font-weight: 400;
              font-display: fallback;
              font-style: normal;
            }
          `
        ]}
      />
      <Head>
        <title key="title">{title}</title>
        <meta property="og:title" content={title} />
        <meta property="og:site_name" content={lessonTitle} />
        <meta property="og:url" content={url} />
        <link rel="canonical" href={url} />
        <meta property="og:description" content={description} />
        <meta name="description" content={description} />
        <meta property="og:image" content={ogImageUrl} />
        <meta property="article:published_time" content={dateSchemaString} />
      </Head>
      <Container
        Component="article"
        size="smsm"
        horizontalPadding={0.875}
        cssOverrides={css`
          padding-top: ${spaces(1.5)};
          padding-bottom: ${spaces(6)};
        `}
      >
        <EpisodeHero mainTitle={title} threeLineTitle />
        <h2
          css={css`
            text-align: center;
            color: ${colors('indigo400')};
            font-size: ${fontSizes(1)};
            font-style: italic;
            font-weight: normal;
            margin: 0 0 ${spaces(0.5)};

            ${ns} {
              font-size: ${fontSizes(1.2)};
            }
          `}
        >
          {description}
        </h2>
        <div
          css={css`
            font-size: ${fontSizes(0.8)};
            color: ${colors('grey600')};
            text-align: center;
            margin-bottom: ${spaces(2)};
          `}
        >
          Shu Uesugi (
          <ExternalLink href="https://twitter.com/chibicode">
            @chibicode
          </ExternalLink>
          )
        </div>
        <img
          src="/static/images/animated@2x.gif"
          alt={title}
          css={css`
            width: 6rem;
            margin: ${spaces(2)} auto ${spaces(2.25)};
            display: block;
            ${ns} {
              width: 7rem;
            }
          `}
        />
        <ExpressionRunnerConfigContext.Provider
          value={{ churchNumerals: true }}
        >
          <BubbleQuoteContext.Provider value={{ inQuote: true }}>
            <P>
              Last month, I published a free online course called “
              <InternalLink href="/">
                <Bold>Y Combinator for Non-programmers</Bold>
              </InternalLink>
              ”. In this 17-page course, I explain functional programming
              concepts such as lambda calculus, Church encoding, and Y
              combinator in a way such that people{' '}
              <Italic>who have zero programming knowledge</Italic> can
              understand.
            </P>
            <P>
              I didn’t use any code to explain these concepts. Instead, I
              created something called “<Bold>emoji puzzles</Bold>” that can{' '}
              <Italic>visualize</Italic> functional code. In this article, I’ll
              explain how my emoji puzzles can represent and execute functional
              code visually.
            </P>
            <Subheading noHrTop step="none" coveredIn="none">
              Quick Demo
            </Subheading>
            <P>
              First, here’s simple, functional JS code.{' '}
              <Highlight>
                Try pressing <H args={{ name: 'run' }} /> below
              </Highlight>{' '}
              to see the result:
            </P>
            <CodeBlock
              result={`'sandwich'`}
              pointToRunButton
              caption={<>Functional JS code:</>}
            >{`(sushi => sushi)('sandwich')`}</CodeBlock>
            <P>
              The above code can be expressed visually using my{' '}
              <Bold>emoji puzzle</Bold> below.{' '}
              <Highlight>
                Try pressing <H args={{ name: 'run' }} />
              </Highlight>
              :
            </P>
            <PointToRunButton>
              <R.Itbm>Equivalent emoji puzzle:</R.Itbm>
            </PointToRunButton>
            <P>
              I’ll explain how it works shortly. If you like to teach
              programming, or if you like functional programming, I think you’ll
              enjoy this article.
            </P>
            <Subheading noHrTop step="none" coveredIn="none">
              Overview
            </Subheading>
            <P>
              This article has <Bold>10 steps:</Bold>
            </P>
            <Ul>
              <UlLi>
                <Bold>In the first half (steps 1 - 5):</Bold> I’ll show you how
                simple JavaScript code can be represented visually using my
                emoji puzzles. You’ll be able to understand it even if you’re
                not familiar with JS.
              </UlLi>
              <UlLi>
                <Bold>In the second half (steps 6 - 10):</Bold> I’ll talk about
                how I used my emoji puzzles to explain functional programming
                concepts such as lambda calculus, Church encoding, and Y
                combinator.
              </UlLi>
            </Ul>
            <Alert>
              <div
                css={css`
                  font-size: ${fontSizes(0.85)};
                `}
              >
                <Emoji>⚠️</Emoji> This article is for programmers. If you’re a
                non-programmer, check out my course instead: “
                <InternalLink href="/">
                  <Bold>Y Combinator for Non-programmers</Bold>
                </InternalLink>
                ”.
              </div>
            </Alert>
            <Subheading step={step++} coveredIn={3}>
              Identity function in JS
            </Subheading>
            <P>
              Take a look at the following code. It’s an{' '}
              <Italic>identity function</Italic> in JavaScript that returns the
              argument.
            </P>
            <CodeBlock>{`// Identity function in JS
sushi => sushi`}</CodeBlock>
            <P>
              If you apply the above function on a string{' '}
              <InlineCode>'sandwich'</InlineCode>, the result will be{' '}
              <InlineCode>'sandwich'</InlineCode>.
            </P>
            <CodeBlock>{`// The result will be 'sandwich'
(sushi => sushi)('sandwich')`}</CodeBlock>
            <P>
              One day, I realized that the above JS code can be{' '}
              <Italic>described visually</Italic> using emojis like below. I
              called this an “<Bold>emoji puzzle</Bold>”.
            </P>
            <R.Ilpo>
              An “<Bold>emoji puzzle</Bold>” that visually describes
              <br />
              <InlineCode>(sushi => sushi)('sandwich')</InlineCode>
            </R.Ilpo>
            <P>
              The above emoji puzzle is equivalent to the earlier JS code.
              First, the identity function{' '}
              <InlineCode>sushi => sushi</InlineCode>…
            </P>
            <CodeBlock
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 0 && tokenNumber > 0 && tokenNumber < 5
              }
            >{`(sushi => sushi)('sandwich')`}</CodeBlock>
            <P>
              …is represented by <Italic>the bottom two items</Italic>, which
              are both <EmojiWithText letter="a" />.
            </P>
            <DimUnhighlighted>
              <R.Elku>
                <InlineCode>sushi => sushi</InlineCode> is represented
                <br />
                by the bottom two items
              </R.Elku>
            </DimUnhighlighted>
            <P>
              Second, the argument <InlineCode>'sandwich'</InlineCode> to the
              identity function…
            </P>
            <CodeBlock
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 0 && tokenNumber > 6 && tokenNumber < 8
              }
            >{`(sushi => sushi)('sandwich')`}</CodeBlock>
            <P>
              …is represented by <Italic>the top item</Italic>, which is a{' '}
              <EmojiWithText letter="b" />.
            </P>
            <DimUnhighlighted>
              <R.Vowa>
                The argument <InlineCode>'sandwich'</InlineCode> is represented
                <br />
                by the top item
              </R.Vowa>
            </DimUnhighlighted>
            <P>
              That’s how my emoji puzzles can visually describe a simple JS
              expression. Next, let’s talk about how we can{' '}
              <H args={{ name: 'run', lowerCase: true }} /> it.
            </P>
            <Alert backgroundColor="brown">
              <P>
                <Bold>Note:</Bold> To keep things simple, emoji puzzles don’t
                distinguish between variable names (e.g.{' '}
                <InlineCode>sushi</InlineCode>) and strings (e.g.{' '}
                <InlineCode>'sushi'</InlineCode>). Therefore, both{' '}
                <InlineCode>sushi</InlineCode> and{' '}
                <InlineCode>'sushi'</InlineCode> will be represented as{' '}
                <EmojiForLetter letter="a" size="semilg" />.
              </P>
              <P
                css={css`
                  margin-bottom: 0;
                `}
              >
                <Bold>Why emojis?</Bold> I used emojis because they are not
                scary-looking for non-programers. I used food emojis because…I
                like food. <Emoji>😉</Emoji>
              </P>
            </Alert>
            <Subheading step={step++} coveredIn={4}>
              Running the function
            </Subheading>
            <P>
              I’ve added the <H args={{ name: 'run' }} /> button to the JS code
              snippet below. If you press it, you’ll see that the result is{' '}
              <InlineCode>'sandwich'</InlineCode>.
            </P>
            <CodeBlock
              result={`'sandwich'`}
              pointToRunButton
            >{`(sushi => sushi)('sandwich')`}</CodeBlock>
            <P>
              We can also “run” the equivalent emoji puzzle and get the same
              result.{' '}
              <Highlight>
                Try pressing the <H args={{ name: 'run' }} /> button below.
              </Highlight>
            </P>
            <PointToRunButton>
              <R.Itbm>
                An emoji puzzle that’s equivalent to
                <br />
                <InlineCode>(sushi => sushi)('sandwich')</InlineCode>
              </R.Itbm>
            </PointToRunButton>
            <P>
              The result is a <EmojiWithText letter="b" />, which is the same as
              what happens when you run the equivalent JS code.
            </P>
            <P>
              So, you can <H args={{ name: 'run', lowerCase: true }} /> an emoji
              puzzle just as you can run a piece of JS code. This is how I
              taught functional programming to non-programmers in my course (
              <InternalLink href="/">
                Y Combinator for Non-programmers
              </InternalLink>
              )—without showing any code.
            </P>
            <P>
              <Bold>Let’s take a look at another example.</Bold> Here’s a piece
              of JS code that’s slightly different from before. It’s a function
              that <Italic>ignores the input</Italic> and always returns{' '}
              <InlineCode>'pizza'</InlineCode>.
            </P>
            <CodeBlock>{`// A function that ignores the input
// and always returns 'pizza'
sushi => 'pizza'`}</CodeBlock>
            <P>
              Let’s run the above code with <InlineCode>'sandwich'</InlineCode>{' '}
              as the argument. <H args={{ name: 'pressRun' }} />
            </P>
            <CodeBlock
              result={`'pizza'`}
            >{`(sushi => 'pizza')('sandwich')`}</CodeBlock>
            <P>
              As expected, the result is <InlineCode>'pizza'</InlineCode>. Now,
              this code can be represented using an emoji puzzle as follows.{' '}
              <H args={{ name: 'pressRun' }} />
            </P>
            <R.Qcme>
              An emoji puzzle that’s equivalent to
              <br />
              <InlineCode>(sushi => 'pizza')('sandwich')</InlineCode>
            </R.Qcme>
            <P>
              Just like the JS code, the emoji puzzle ended up with a{' '}
              <EmojiWithText letter="f" /> after running it.
            </P>
            <Alert backgroundColor="pink">
              <P>
                <Bold>What we have learned so far:</Bold> Simple JS code like
                below can be represented using emoji puzzles.
              </P>
              <CodeBlock
                defaultResultVisible
                result={`'sandwich'`}
                caption={<>Functional JS code:</>}
              >{`(sushi => sushi)('sandwich')`}</CodeBlock>
              <R.Ilpo>Equivalent emoji puzzle:</R.Ilpo>
              <ExpressionRunnerSeparator />
              <R.Lngo></R.Lngo>
              <Hr />
              <CodeBlock
                defaultResultVisible
                result={`'pizza'`}
                caption={<>Functional JS code:</>}
              >{`(sushi => 'pizza')('sandwich')`}</CodeBlock>
              <R.Bjny>Equivalent emoji puzzle:</R.Bjny>
              <ExpressionRunnerSeparator />
              <R.Ukzq></R.Ukzq>
            </Alert>
            <Subheading step={step++} coveredIn={4}>
              Visualizing evaluation rules
            </Subheading>
            <P>
              If you know how to code, you have a <Italic>mental model</Italic>{' '}
              of how function evaluation works:
            </P>
            <Ul>
              <UlLi>
                If you see <InlineCode>(sushi => sushi)('sandwich')</InlineCode>
                , you can quickly figure out that the result would be{' '}
                <InlineCode>'sandwich'</InlineCode>.
              </UlLi>
              <UlLi>
                If you see{' '}
                <InlineCode>(sushi => 'pizza')('sandwich')</InlineCode>, you
                know that the result would be <InlineCode>'pizza'</InlineCode>.
              </UlLi>
              <UlLi>
                You know what free variables and bound variables mean.
              </UlLi>
            </Ul>
            <P>
              On the other hand, most non-programmers don’t have a mental model
              of how function evaluation works. To help them develop the mental
              model, I created a <Italic>step-by-step visualization</Italic> of
              function evaluation rules using emoji puzzles.
            </P>
            <P>Let’s reuse the earlier example again:</P>
            <CodeBlock
              caption={<>Functional JS code:</>}
            >{`(sushi => sushi)('sandwich')`}</CodeBlock>
            <R.Ilpo>Equivalent emoji puzzle:</R.Ilpo>
            <P>
              On the puzzle below,{' '}
              <Highlight>
                try pressing the <H args={{ name: 'run' }} /> button
              </Highlight>
              . This button is a bit different from the last time—it shows{' '}
              <Italic>every step</Italic> that happens in between.
            </P>
            <PointToRunButton>
              <R.Wunw></R.Wunw>
            </PointToRunButton>
            <P>Here are the four steps it displayed:</P>
            <P>
              <Bold>Step 1.</Bold> <StepOne />
            </P>
            <P>
              First, we label each emoji. The top item is labeled as{' '}
              <BottomRightBadge inline bottomRightBadgeType="callArg" /> (for “
              <Bold>T</Bold>op”), the left item is labeled as{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcArg" /> (for “
              <Bold>L</Bold>eft”), and the right item is labeled as{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcBound" /> (for
              “<Bold>R</Bold>ight”).
            </P>
            <R.Zzxj>
              <StepOne />
            </R.Zzxj>
            <P>
              <Bold>Step 2.</Bold> <StepTwo />
            </P>
            <P>
              Second, <InstructionTwo lowerCase /> In this case, both the
              bottom-left and the bottom-right are <EmojiWithText letter="a" />,
              so there’s a match.
            </P>
            <R.Keck>
              <StepTwo />
            </R.Keck>
            <P>
              <Bold>Step 3.</Bold> <StepThree />
            </P>
            <P>
              Third, <InstructionThree lowerCase /> In this case, we copy the{' '}
              <EmojiWithText letter="b" /> on the top to the bottom-right.
            </P>
            <R.Qoms>
              <StepThree />
            </R.Qoms>
            <P>
              <Bold>Step 4.</Bold> <StepFour />
            </P>
            <P>
              Finally, <InstructionFour lowerCase /> We’re left with just the{' '}
              <EmojiWithText letter="b" /> at the end.
            </P>
            <R.Mhgm>
              <StepFour />
            </R.Mhgm>
            <ExpressionRunnerSeparator />
            <R.Lngo></R.Lngo>
            <P>
              The above steps are a visual representation of how functions are
              evaluated. They are equivalent to how JavaScript evaluates{' '}
              <InlineCode>(sushi => sushi)('sandwich')</InlineCode>.
            </P>
            <P>
              By learning these rules, non-programmers will be able to evaluate
              functions intuitively.
            </P>
            <Alert>
              <P>
                <Bold>Note:</Bold> The above steps would look{' '}
                <Italic>very</Italic> confusing to most programmers. As
                programmers, we already know how to evaluate functional
                expressions, and we instinctively try to map the above diagrams
                to code in our head.
              </P>
              <P>
                But in my testing,{' '}
                <Italic>
                  it seems to be less confusing to non-programmers
                </Italic>{' '}
                who don’t already have a mental model of function evaluation.
                Furthermore,{' '}
                <InternalLink href="/">in the course itself</InternalLink>, I
                slow down and spend a lot more time covering the above rules—the
                explanations are spread out in a full page (
                <InternalLink href="/4">here</InternalLink>
                ). And I don’t show any code in my course—I only show emoji
                puzzles, so one less source of confusion compared to this
                article.
              </P>
              <P>
                One thing I learned is that the feedback on my course from
                programmers and non-programmers are complete opposites—
                <Italic>generally negative from programmers</Italic> but{' '}
                <Italic>generally positive from non-programmers</Italic>. I
                believe that’s because{' '}
                <Bold>
                  many programmers find that using code is simpler than using
                  the above diagrams, but many non-programmers think the
                  opposite.
                </Bold>
              </P>
              <P
                css={css`
                  margin-bottom: 0;
                `}
              >
                Some people have incredible difficulty understanding code or
                math symbols, and sometimes visual alternatives help them. I
                believe the more alternative teaching methods are offered in CS
                education, the better.
              </P>
            </Alert>
            <Subheading step={step++} coveredIn={4}>
              If there’s no match
            </Subheading>
            <P>Let’s take a look at the other example from earlier:</P>
            <CodeBlock
              caption={<>Functional JS code:</>}
            >{`(sushi => 'pizza')('sandwich')`}</CodeBlock>
            <R.Bjny>Equivalent emoji puzzle:</R.Bjny>
            <P>
              <Highlight>
                Try pressing the <H args={{ name: 'run' }} /> button and see
                what happens:
              </Highlight>
            </P>
            <R.Gthd></R.Gthd>
            <P>Here are the three steps it displayed:</P>
            <P>
              <Bold>Step 1.</Bold> <StepOne />
              —this is the same as before.
            </P>
            <R.Gogn>
              <StepOne />
            </R.Gogn>
            <P>
              <Bold>Step 2.</Bold> This time, there’s <Bold>no match</Bold>{' '}
              between <BottomRightBadge inline bottomRightBadgeType="funcArg" />{' '}
              and <BottomRightBadge inline bottomRightBadgeType="funcBound" />.
              The bottom-left is <EmojiWithText letter="a" />, but the
              bottom-right is <EmojiWithText letter="f" />.
            </P>
            <R.Zvpc>
              <Bold>No match:</Bold>{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcArg" />{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcBound" />{' '}
              <Emoji>❌</Emoji>
            </R.Zvpc>
            <P>
              If there’s no match, we skip step 3 (<StepThree />) and go
              directly to step 4.
            </P>
            <P>
              <Bold>Step 4.</Bold> <StepFour />
            </P>
            <R.Nsds></R.Nsds>
            <ExpressionRunnerSeparator />
            <R.Ukzq></R.Ukzq>
            <P>
              So we’re left with a <EmojiWithText letter="f" />. That’s how the
              emoji puzzle visualizes the evaluation of a function when there’s
              no matching variable in the function body.
            </P>
            <Alert backgroundColor="blue">
              <P>
                <Bold>More examples (optional read):</Bold> Here are more
                examples that might be helpful for your understanding. You can{' '}
                <Highlight>
                  press <H args={{ name: 'run' }} />
                </Highlight>{' '}
                on each example to see the evaluation visualization.{' '}
                <H args={{ name: 'mentionRightArrow' }} />
              </P>
              <CodeBlock
                defaultResultVisible
                result={`(spaghetti => 'bread')`}
                caption={<>Functional JS code:</>}
              >{`(pizza => pizza)(spaghetti => 'bread')`}</CodeBlock>
              <R.Hluq>Equivalent emoji puzzle:</R.Hluq>
              <Hr />
              <CodeBlock
                defaultResultVisible
                result={`'hotDog'`}
                caption={<>Functional JS code:</>}
              >{`(salad => 'hotDog')(curry => 'tacos')`}</CodeBlock>
              <R.Zuus>Equivalent emoji puzzle:</R.Zuus>
            </Alert>
            <Subheading step={step++} coveredIn={5}>
              More complicated expressions
            </Subheading>
            <P>
              Let’s take a look at more complicated functional JS expressions
              and see if they can be represented using an emoji puzzle.
            </P>
            <P>
              Check out the following JS expression, and try to guess what the
              result would be before{' '}
              <Highlight>
                pressing the <H args={{ name: 'run' }} /> button.
              </Highlight>
            </P>
            <CodeBlock
              result={`'hamburger'`}
              caption={
                <>
                  Guess what the result would be
                  <br />
                  before pressing the <H args={{ name: 'run' }} /> button.
                </>
              }
            >{`(sushi => sandwich => sushi)(
  'hamburger'
)('chicken')`}</CodeBlock>
            <P>
              The result was <InlineCode>{`'hamburger'`}</InlineCode>. It’s
              because:
            </P>
            <Ul>
              <UlLi>
                <InlineCode>{'sushi'}</InlineCode> is bound to{' '}
                <InlineCode>{`'hamburger'`}</InlineCode>,
              </UlLi>
              <UlLi>
                <InlineCode>{'sandwich'}</InlineCode> is bound to{' '}
                <InlineCode>{`'chicken'`}</InlineCode>,
              </UlLi>
              <UlLi>
                And it returns the value of <InlineCode>{'sushi'}</InlineCode>,
                which is <InlineCode>{`'hamburger'`}</InlineCode>.
              </UlLi>
            </Ul>
            <P>Now, here’s the equivalent emoji puzzle:</P>
            <R.Cvtc></R.Cvtc>
            <P>Let’s break it down. First, the function expression…</P>
            <CodeBlock
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 0 && tokenNumber > 0 && tokenNumber < 9
              }
            >{`(sushi => sandwich => sushi)(
  'hamburger'
)('chicken')`}</CodeBlock>
            <P>…is represented by the bottom row:</P>
            <DimUnhighlighted>
              <R.Lxzn>
                <InlineCode>sushi => sandwich => sushi</InlineCode>
              </R.Lxzn>
            </DimUnhighlighted>
            <P>And the two arguments…</P>
            <CodeBlock
              shouldHighlight={(lineNumber, tokenNumber) =>
                (lineNumber === 1 && tokenNumber > 0 && tokenNumber < 2) ||
                (lineNumber === 2 && tokenNumber > 2 && tokenNumber < 4)
              }
            >{`(sushi => sandwich => sushi)(
  'hamburger'
)('chicken')`}</CodeBlock>
            <P>…are represented by the top and the middle rows:</P>
            <DimUnhighlighted>
              <R.Ujuq>
                <InlineCode>('hamburger')('chicken')</InlineCode>
              </R.Ujuq>
            </DimUnhighlighted>
            <P>
              How do we evaluate this function? Well, in JS, we first evaluate
              the function call with the argument{' '}
              <InlineCode>'hamburger'</InlineCode>.
            </P>
            <CodeBlock
              caption={<>In JS, we evaluate this part first</>}
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 0 ||
                (lineNumber === 1 && tokenNumber > 0 && tokenNumber < 2) ||
                (lineNumber === 2 && tokenNumber < 2)
              }
            >{`(sushi => sandwich => sushi)(
  'hamburger'
)('chicken')`}</CodeBlock>
            <P>
              Equivalently, in an emoji puzzle, we evaluate the bottom two rows
              first. We ignore the top row initially, which is shaded in blue.
            </P>
            <R.Bcgp>
              <Bold>Focus</Bold> on <Italic>the bottom two rows</Italic>, and
              <br />
              <Bold>Ignore</Bold> the <Italic>top row</Italic> for now
            </R.Bcgp>
            <P>
              Also, if you look at the left edge of the puzzle above, the bottom
              two rows correspond to the pair of{' '}
              <InlinePrioritiesLabel revert>1</InlinePrioritiesLabel>
              ’s. That’s how you know that you must start with the bottom two
              rows. In an emoji puzzle with 3 rows, you start with the pair of{' '}
              <InlinePrioritiesLabel revert>1</InlinePrioritiesLabel>’s.
            </P>
            <P>
              Next, let’s label each item on the bottom two rows. This time, in
              addition to{' '}
              <BottomRightBadge inline bottomRightBadgeType="callArg" />{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcArg" />{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcBound" />, we
              now have a new label{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcUnbound" /> for
              the <EmojiWithText letter="b" />.
            </P>
            <R.Pbhg>
              There’s a new label{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcUnbound" />
              <br />
              for the <EmojiWithText letter="b" />
            </R.Pbhg>
            <P>
              <Bold>Here’s the rule:</Bold> The middle item on the bottom row is
              labeled as{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcUnbound" />{' '}
              (for “<Bold>M</Bold>iddle”), and you can <Bold>ignore it</Bold>.
              You can pretend that{' '}
              <BottomRightBadge inline bottomRightBadgeType="funcUnbound" />
              ’s don’t exist.
            </P>
            <P>
              Let’s take a look at the next few steps in this iteration, which
              is just like what we saw earlier:
            </P>
            <R.Itvv></R.Itvv>
            <ExpressionRunnerSeparator />
            <R.Nefh></R.Nefh>
            <ExpressionRunnerSeparator />
            <R.Qwtn></R.Qwtn>
            <ExpressionRunnerSeparator />
            <R.Hbbv></R.Hbbv>
            <P>This is exactly like how the earlier JS code is evaluated.</P>
            <CodeBlock
              caption={<>After evaluating this part…</>}
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 0 ||
                (lineNumber === 1 && tokenNumber > 0 && tokenNumber < 2) ||
                (lineNumber === 2 && tokenNumber < 2)
              }
            >{`(sushi => sandwich => sushi)(
  'hamburger'
)('chicken')`}</CodeBlock>
            <ExpressionRunnerSeparator halfMargin />
            <CodeBlock
              caption={
                <>
                  …it becomes this, which is
                  <br />
                  equivalent to the above emoji puzzle
                  <br />
                  (but it’s not done yet!)
                </>
              }
            >{`(sandwich => 'hamburger')('chicken')`}</CodeBlock>
            <P>
              <Bold>But we’re not done yet!</Bold> Let’s continue to the end by{' '}
              <Highlight>
                pressing the <H args={{ name: 'run' }} /> button below:
              </Highlight>
            </P>
            <R.Iwxm>Let’s continue to the end!</R.Iwxm>
            <P>
              We ended up with a <EmojiWithText letter="c" />, which is exactly
              the same as what the JS code evaluated to.
            </P>
            <Alert backgroundColor="pink">
              <P>
                <Bold>What we have learned so far:</Bold> Complex functional JS
                expressions can be represented and evaluated using an emoji
                puzzle.
              </P>
              <CodeBlock
                defaultResultVisible
                result={`'hamburger'`}
                caption={<>Functional JS code:</>}
              >{`(sushi => sandwich => sushi)(
  'hamburger'
)('chicken')`}</CodeBlock>
              <R.Cvtc>
                Equivalent emoji puzzle:
                <br />
                (Start with the pair of{' '}
                <InlinePrioritiesLabel>1</InlinePrioritiesLabel>’s)
              </R.Cvtc>
              <ExpressionRunnerSeparator />
              <R.Yxel></R.Yxel>
            </Alert>
            <Alert backgroundColor="blue">
              <P>
                <Bold>More examples (optional read):</Bold> Here’s another
                example that might be helpful for your understanding. Check out
                the following JS expression, and try to guess what the result
                would be before{' '}
                <Highlight>
                  pressing the <H args={{ name: 'run' }} /> button.
                </Highlight>
              </P>
              <CodeBlock
                result={`'salad'`}
                caption={guessTheResult}
              >{`(friedPotatoes => pizza => pizza)(
  spaghetti => spaghetti
)('salad')`}</CodeBlock>
              <P>
                The result was <InlineCode>'salad'</InlineCode>. Now, here’s the
                equivalent emoji puzzle below.{' '}
                <Highlight>
                  Try pressing <H args={{ name: 'run' }} />.
                </Highlight>{' '}
                Again, remember that:
              </P>
              <Ul>
                <UlLi>
                  We start with the pair of{' '}
                  <InlinePrioritiesLabel>1</InlinePrioritiesLabel>’s.
                </UlLi>
                <UlLi>
                  The middle item on the bottom row is labeled as{' '}
                  <BottomRightBadge inline bottomRightBadgeType="funcUnbound" />
                  , and you can <Bold>ignore it</Bold>.
                </UlLi>
              </Ul>
              <R.Xhje>Equivalent emoji puzzle:</R.Xhje>
              <Hr />
              <P>
                Next, here’s slightly different JS code. Compared to the last
                time, there’s an extra pair of parentheses around{' '}
                <InlineCode>(spaghetti => spaghetti)('salad')</InlineCode>,
                which changes the result.
              </P>
              <CodeBlock
                result={`pizza => pizza`}
                caption={guessTheResult}
              >{`// There’s an extra pair of parens around
// (spaghetti => spaghetti)("salad")
(friedPotatoes => pizza => pizza)(
  (spaghetti => spaghetti)("salad")
)`}</CodeBlock>
              <P>
                Let’s take a look at the equivalent emoji puzzle. This time, the
                pair of <InlinePrioritiesLabel>1</InlinePrioritiesLabel>’s is on
                the top two rows instead. So we start with the top two rows this
                time.{' '}
                <Highlight>
                  Try pressing <H args={{ name: 'run' }} />
                </Highlight>{' '}
                and see what happens.
              </P>
              <R.Hyvj>
                This time, the pair of{' '}
                <InlinePrioritiesLabel>1</InlinePrioritiesLabel>’s is
                <br />
                on the top two rows instead
              </R.Hyvj>
              <P>
                <Bold>Summary:</Bold>
              </P>
              <Ul>
                <UlLi>
                  In <Bold>JavaScript</Bold>, you can change the evaluation
                  order of an expression by placing parentheses on different
                  locations.
                </UlLi>
                <UlLi>
                  In <Bold>emoji puzzles</Bold>, you can change the evaluation
                  order by placing the numbers (e.g.{' '}
                  <InlinePrioritiesLabel>1</InlinePrioritiesLabel>’s and{' '}
                  <InlinePrioritiesLabel>2</InlinePrioritiesLabel>’s) on
                  different locations.
                </UlLi>
              </Ul>
            </Alert>
            <Subheading step={step++} coveredIn={6}>
              Church numerals
            </Subheading>
            <Alert backgroundColor="yellow">
              <div
                css={css`
                  font-size: ${fontSizes(0.85)};
                `}
              >
                <P>
                  <Bold>You’re halfway there:</Bold> 5 steps down, 5 more to go!
                </P>
                <P
                  css={css`
                    margin-bottom: 0;
                  `}
                >
                  If you’re thinking of taking a break, I’d appreciate it if you
                  could{' '}
                  <ExternalLink href={tweetUrl}>
                    <CustomEmoji type="twitter" />{' '}
                    <Bold>Share this article on Twitter</Bold>
                  </ExternalLink>{' '}
                  before you close this page.
                </P>
              </div>
            </Alert>
            <P>
              From now on, we’ll apply what we’ve learned so far to solve more
              interesting problems.
            </P>
            <P>
              Here’s a function called <InlineCode>convert</InlineCode> that
              takes a function <InlineCode>f</InlineCode> as an argument. It
              then calls <InlineCode>f</InlineCode> with{' '}
              <InlineCode>(n => n + 1)(0)</InlineCode>.
            </P>
            <CodeBlock>{`function convert(f) {
  return f(n => n + 1)(0)
}`}</CodeBlock>
            <P>
              Now, suppose that we apply <InlineCode>convert</InlineCode> on
              this function:{' '}
              <InlineCode>sushi => sandwich => sandwich</InlineCode>. What would
              the result be? Try to guess before{' '}
              <Highlight>
                pressing the <H args={{ name: 'run' }} /> button.
              </Highlight>
            </P>
            <CodeBlock
              result="0"
              caption={guessTheResult}
            >{`function convert(f) {
  return f(n => n + 1)(0)
}

convert(sushi => sandwich =>
  sandwich
)`}</CodeBlock>
            <P>
              The result is <InlineCode>0</InlineCode> because:
            </P>
            <Ul>
              <UlLi>
                <InlineCode>sushi</InlineCode> is bound to{' '}
                <InlineCode>n => n + 1</InlineCode>
              </UlLi>
              <UlLi>
                <InlineCode>sandwich</InlineCode> is bound to{' '}
                <InlineCode>0</InlineCode>
              </UlLi>
              <UlLi>
                And it returns <InlineCode>sandwich</InlineCode>, which is{' '}
                <InlineCode>0</InlineCode>.
              </UlLi>
            </Ul>
            <P>
              <Bold>Next:</Bold> What if the input to{' '}
              <InlineCode>convert</InlineCode> changes as follows?{' '}
              <Highlight>
                Try pressing <H args={{ name: 'run' }} /> on each example.
              </Highlight>
            </P>
            <CodeBlock result="1">{`convert(sushi => sandwich =>
  sushi(sandwich)
)`}</CodeBlock>
            <CodeBlock result="2">{`convert(sushi => sandwich =>
  sushi(sushi(sandwich))
)`}</CodeBlock>
            <CodeBlock result="3">{`convert(sushi => sandwich =>
  sushi(sushi(sushi(sandwich)))
)`}</CodeBlock>
            <P>
              The results are <InlineCode>1</InlineCode>,{' '}
              <InlineCode>2</InlineCode>, and <InlineCode>3</InlineCode>{' '}
              respectively because:
            </P>
            <Ul>
              <UlLi>
                <InlineCode>sushi</InlineCode> is bound to{' '}
                <InlineCode>n => n + 1</InlineCode>
              </UlLi>
              <UlLi>
                <InlineCode>sandwich</InlineCode> is bound to{' '}
                <InlineCode>0</InlineCode>
              </UlLi>
              <UlLi>
                So it applies <InlineCode>n => n + 1</InlineCode> to{' '}
                <InlineCode>0</InlineCode> for the number of times{' '}
                <InlineCode>sushi</InlineCode> is used in the function body.
              </UlLi>
            </Ul>
            <Alert backgroundColor="pink">
              <P>
                <Bold>What we have learned so far:</Bold> If we have a function
                that has one of the following patterns:
              </P>
              <CodeBlock>{`// a & b can be any variable name
a => b => b
a => b => a(b)
a => b => a(a(b))
a => b => a(a(a(b)))
// ...and so on...`}</CodeBlock>
              <P>
                Then, when the function is passed to{' '}
                <InlineCode>convert()</InlineCode>, it returns the number of
                times <InlineCode>a</InlineCode> is applied to{' '}
                <InlineCode>b</InlineCode>
                in the function body.
              </P>
              <CodeBlock>{`convert(a => b => b)          // 0
convert(a => b => a(b))       // 1
convert(a => b => a(a(b)))    // 2
convert(a => b => a(a(a(b)))) // 3
// ...and so on...`}</CodeBlock>
            </Alert>
            <P>
              <Bold>Important:</Bold> These functions that can be converted to a
              number using <InlineCode>convert()</InlineCode> have a{' '}
              <Italic>special name.</Italic> They are called{' '}
              <ExternalLink href="https://en.wikipedia.org/wiki/Church_encoding">
                <Bold>Church numerals</Bold>
              </ExternalLink>
              . Each function represents a Church numeral, like this:
            </P>
            <CodeBlock>{`// Church numeral 0
a => b => b

// Church numeral 1
a => b => a(b)

// Church numeral 2
a => b => a(a(b)))

// Church numeral 3
a => b => a(a(a(b))))

// ...and so on...`}</CodeBlock>
            <P>
              <Bold>You might be wondering:</Bold> “What’s the point of this?”
              Don’t worry—I’ll tell you why Church numerals are interesting
              shortly. But before I do, let me explain how emoji puzzles can
              express Church numerals.
            </P>
            <P>
              Here’s an emoji puzzle that represents{' '}
              <InlineCode>sushi => sandwich => sandwich</InlineCode>, which is{' '}
              Church numeral <Bold>0</Bold>.
            </P>
            <R.Jarm>
              <Bold>Represents:</Bold>
              <br />
              <InlineCode>sushi => sandwich => sandwich</InlineCode>
              <br />
              (Church numeral <Bold>0</Bold>)
            </R.Jarm>
            <P>
              <Bold>Here’s what’s new:</Bold> We now have the “
              <H args={{ name: 'convertToChurchNumeral' }} />” button below the
              puzzle which converts it to the corresponding Church number. Try
              pressing it:
            </P>
            <PointToRunButton>
              <R.Jjjh></R.Jjjh>
            </PointToRunButton>
            <P>
              Other Church numeral functions can also be represented using emoji
              puzzles, and they can be converted to a number. And we can use
              emojis other than <EmojiForLetter letter="a" /> and{' '}
              <EmojiForLetter letter="b" />
              —as long as they follow the same pattern.
            </P>
            <R.Mifg>
              <Bold>Represents:</Bold>
              <br />
              <InlineCode>
                hamburger => chicken => hamburger(chicken)
              </InlineCode>
              <br />
              (Church numeral <Bold>1</Bold>)
            </R.Mifg>
            <R.Fgen>
              Church numeral <Bold>2</Bold>
            </R.Fgen>
            <R.Pebp>
              Church numeral <Bold>3</Bold>
            </R.Pebp>
            <P>
              Now that we’ve covered the basics of Church numerals, we’ll talk
              next about why Church numerals are interesting.
            </P>
            <Subheading step={step++} coveredIn={8}>
              Arithmetic with functions
            </Subheading>
            <P>
              Church numerals are interesting because{' '}
              <Italic>they let you do arithmetic with functions.</Italic> Take a
              look at this function:
            </P>
            <CodeBlock>{`sushi => sandwich => pizza =>
  sandwich(
    sushi(sandwich)(pizza)
  )`}</CodeBlock>
            <P>
              It looks pretty complicated, but don’t worry too much. Now,
              suppose that:
            </P>
            <Ul>
              <UlLi>
                We apply the above function to the Church numeral zero (has the
                pattern <InlineCode>a => b => b</InlineCode>), and
              </UlLi>
              <UlLi>
                Run <InlineCode>convert()</InlineCode> (from earlier) on it.
              </UlLi>
              <UlLi>What would the result be?</UlLi>
            </Ul>
            <P>
              Let’s take a look. Try to guess before{' '}
              <Highlight>
                pressing the <H args={{ name: 'run' }} /> button.
              </Highlight>
            </P>
            <CodeBlock result="1">{`// Function from the above
const f = sushi => sandwich => pizza =>
  sandwich(
    sushi(sandwich)(pizza)
  )

// Church numeral zero
const zero = chicken => salad => salad

// What happens when you apply f to zero,
// and convert it?
convert(f(zero))`}</CodeBlock>
            <P>
              The result is <InlineCode>1</InlineCode>, which is 1 greater than
              the input Church numeral, which was <InlineCode>0</InlineCode>. In
              other words, <InlineCode>0</InlineCode> became{' '}
              <InlineCode>1</InlineCode>.
            </P>
            <EmojiSeparator
              nodes={[
                <EmojiNumber number={0} />,
                <CustomEmoji type="singleArrow" />,
                <EmojiNumber number={1} />
              ]}
              description={
                <>
                  <InlineCode>0</InlineCode> became <InlineCode>1</InlineCode>
                </>
              }
            />
            <P>
              <Bold>Here’s the secret:</Bold> This function{' '}
              <InlineCode>f</InlineCode> we used actually <Bold>adds 1</Bold> to
              the input Church numeral. When you apply{' '}
              <InlineCode>f</InlineCode> to a Church numeral, it returns a{' '}
              <Italic>new Church numeral</Italic> that’s 1 greater than before.
            </P>
            <CodeBlock>{`// If we apply this function to a
// Church numeral, it returns a new
// Church numeral that’s greater by 1.
const f = sushi => sandwich => pizza =>
  sandwich(
    sushi(sandwich)(pizza)
  )`}</CodeBlock>
            <P>You can try out more examples to verify this:</P>
            <CodeBlock result="2">{`// Church numeral one
const one = chicken => salad =>
  chicken(salad)

convert(f(one))`}</CodeBlock>
            <CodeBlock result="3">{`// Church numeral two
const two = chicken => salad =>
  chicken(chicken(salad))

convert(f(two))`}</CodeBlock>
            <P>
              This is what I mean by “
              <Italic>doing arithmetic with functions</Italic>”:
            </P>
            <Ul>
              <UlLi>
                By using a Church numeral function that can be converted to{' '}
                <InlineCode>someNumber</InlineCode>…
              </UlLi>
              <UlLi>
                …and the function <InlineCode>f</InlineCode> we saw earlier,
              </UlLi>
              <UlLi>
                You can calculate <InlineCode>someNumber + 1</InlineCode>.
              </UlLi>
            </Ul>
            <EmojiSeparator
              nodes={[
                <CustomEmoji type="blankNumber" />,
                <Emoji>➕</Emoji>,
                <EmojiNumber number={1} />
              ]}
              description={
                <>
                  You can calculate <InlineCode>someNumber + 1</InlineCode>
                  <br />
                  using just functions
                </>
              }
            />
            <P>
              Now, let’s see if we can explain this to non-programmers using
              emoji puzzles.
            </P>
            <P>
              First, here’s an emoji puzzle representation of the Church numeral{' '}
              <Bold>0</Bold>. You can confirm that it can be converted to{' '}
              <EmojiNumber number={0} /> by pressing the button below.
            </P>
            <R.Hfgh>
              <Bold>Represents:</Bold>
              <br />
              <InlineCode>chicken => salad => salad</InlineCode>
              <br />
              (Church numeral <Bold>0</Bold>)
            </R.Hfgh>
            <P>
              And here’s the function <InlineCode>f</InlineCode> we saw earlier,
              represented as an emoji puzzle:
            </P>
            <CodeBlock>{`const f = sushi => sandwich => pizza =>
  sandwich(
    sushi(sandwich)(pizza)
  )`}</CodeBlock>
            <ExpressionRunnerSeparator halfMarginTop />
            <R.Qyhn>Equivalent emoji puzzle:</R.Qyhn>
            <P>
              Let’s apply <InlineCode>f</InlineCode> to the Church numeral{' '}
              <Bold>0</Bold>. To do that, we just combine the above two emoji
              puzzles and run it. <H args={{ name: 'pressRun' }} />
            </P>
            <R.Afxj>Combined the above two emoji puzzles</R.Afxj>
            <P>
              The result is equivalent to{' '}
              <InlineCode>sandwich => pizza => sandwich(pizza)</InlineCode>,
              which is Church numeral <Bold>1</Bold>.
            </P>
            <R.Uxno>
              <Bold>Represents:</Bold>
              <br />
              <InlineCode>sandwich => pizza => sandwich(pizza)</InlineCode>
              <br />
              (Church numeral <Bold>1</Bold>)
            </R.Uxno>
            <Alert backgroundColor="pink">
              <P>
                <Bold>What just happened:</Bold> An emoji puzzle that can be
                converted to <EmojiNumber number={0} /> became a puzzle that can
                be converted to <EmojiNumber number={1} />.
              </P>
              <R.Pjih></R.Pjih>
              <ExpressionRunnerSeparator />
              <R.Jzwg></R.Jzwg>
              <P
                css={css`
                  margin-bottom: 0;
                `}
              >
                <Bold>In other words:</Bold> We can use an emoji puzzle to
                calculate <InlineCode>someNumber + 1</InlineCode>.
              </P>
            </Alert>
            <Alert backgroundColor="blue">
              <P>
                <Bold>More examples (optional read):</Bold> Let’s see if we can
                calculate <InlineCode>1 + 1 = 2</InlineCode> using the same
                method. Here’s an emoji puzzle that can be converted to{' '}
                <EmojiNumber number={1} />:
              </P>
              <R.Yfwq>
                <Bold>Represents:</Bold>
                <br />
                <InlineCode>chicken => salad => chicken(salad)</InlineCode>
                <br />
                (Church numeral <Bold>1</Bold>)
              </R.Yfwq>
              <P>
                And let’s combine it with the earlier emoji puzzle and run it.{' '}
                <H args={{ name: 'pressRun' }} />
              </P>
              <R.Egka></R.Egka>
              <P>
                The result is equivalent to Church numeral <Bold>2</Bold> and
                can be converted to <EmojiNumber number={2} />.
              </P>
              <R.Dvio>
                Church numeral <Bold>2</Bold>
              </R.Dvio>
              <P>
                So, it successfully calculated{' '}
                <InlineCode>1 + 1 = 2</InlineCode>. Again, this is what just
                happened:
              </P>
              <R.Kqtz></R.Kqtz>
              <ExpressionRunnerSeparator />
              <R.Cawl></R.Cawl>
            </Alert>
            <Subheading step={step++} coveredIn={15}>
              Multiplications
            </Subheading>
            <P>
              You can do pretty much any computation with Church numerals.
              Consider <Bold>multiplication.</Bold> Here’s a JS function that{' '}
              <Italic>multiplies two Church numerals</Italic>:
            </P>
            <CodeBlock>{`// Multiplies two Church numerals
const mul = sushi => sandwich => pizza =>
  sushi(sandwich(pizza))`}</CodeBlock>
            <P>
              Let’s compute <InlineCode>2 x 3</InlineCode> using the above{' '}
              <InlineCode>mul</InlineCode> function. We use two Church numeral
              functions—one for <InlineCode>2</InlineCode> and the other for{' '}
              <InlineCode>3</InlineCode>, and feed them into{' '}
              <InlineCode>mul</InlineCode>. Take a look at the code below and{' '}
              <Highlight>
                press <H args={{ name: 'run' }} />
              </Highlight>
              :
            </P>
            <CodeBlock result="6">{`// Church numeral two
const two = chicken => salad =>
  chicken(chicken(salad))

// Church numeral three
const three = curry => hamburger =>
  curry(curry(curry(hamburger)))

const result = mul(two)(three)

convert(result)`}</CodeBlock>
            <P>
              The result was <InlineCode>6</InlineCode>, so it successfully
              calculated <InlineCode>2 x 3</InlineCode>.
            </P>
            <P>
              Now, let’s see if we can do the same using emoji puzzles. First,
              here’s an emoji puzzle that’s equivalent to the{' '}
              <InlineCode>mul</InlineCode> function.
            </P>
            <CodeBlock>{`// Multiplies two Church numerals
const mul = sushi => sandwich => pizza =>
  sushi(sandwich(pizza))`}</CodeBlock>
            <ExpressionRunnerSeparator halfMarginTop />
            <R.Nnzx>Equivalent emoji puzzle:</R.Nnzx>
            <P>
              Let’s combine it with emoji puzzles that can be converted to{' '}
              <EmojiNumber number={2} /> and <EmojiNumber number={3} />:
            </P>
            <TwoColGrid
              noBottomNegativeMargin
              noTopNegativeMargin
              maxVariableSize="sm"
              left={
                <>
                  <R.Njqi>
                    Converts to <EmojiNumber number={2} />
                  </R.Njqi>
                </>
              }
              right={
                <>
                  <R.Kdxf>
                    Converts to <EmojiNumber number={3} />
                  </R.Kdxf>
                </>
              }
            />
            <P>
              Here’s the combined puzzle.{' '}
              <Highlight>
                Press <H args={{ name: 'run' }} />
              </Highlight>{' '}
              and see what happens. (Because it takes time, we’ll fast-forward
              it at 4x speed.)
            </P>
            <R.Anfx></R.Anfx>
            <P>
              The result is equivalent to Church numeral <Bold>6</Bold> and can
              be converted to <EmojiNumber number={6} />.
            </P>
            <R.Mili>Church numeral 6</R.Mili>
            <P>
              So emoji puzzles can calculate multiplications too.{' '}
              <InternalLink href="/10">In my course</InternalLink>, I also show
              how we can do <Bold>subtractions</Bold> using emoji puzzles.
              Divisions are very complicated but possible.
            </P>
            <Subheading step={step++} coveredIn={12}>
              Conditionals
            </Subheading>
            <P>
              In addition to arithmetic, we can also implement{' '}
              <Bold>conditionals</Bold> such as <InlineCode>if/else</InlineCode>{' '}
              statements using emoji puzzles.
            </P>
            <P>
              Consider the following JS code. This is a simple{' '}
              <InlineCode>if/else</InlineCode> statement that does different
              things based on what <InlineCode>x</InlineCode> is.
            </P>
            <CodeBlock>{`if (x === 0) {
  // is Zero
} else {
  // is NOT Zero
}`}</CodeBlock>
            <P>
              It turns out that <InlineCode>if/else</InlineCode> statements like
              the above can also be expressed using Church numerals. To save
              time, I won’t show JS code this time and will only show the emoji
              puzzle. Check out the following:
            </P>
            <R.Pmss>
              An emoji puzzle that represents
              <br />
              <InlineCode>{`if (x === 0) { ... } else { ... }`}</InlineCode>
            </R.Pmss>
            <P>The above emoji puzzle will become:</P>
            <Ul>
              <UlLi>
                <CustomEmoji type="isZero" size="mdlg" /> if you put an emoji
                puzzle that can be converted to <EmojiNumber number={0} /> on
                the bottom
              </UlLi>
              <UlLi>
                <CustomEmoji type="isNotZero" size="mdlg" /> otherwise
              </UlLi>
            </Ul>
            <P>
              Let’s try it out. First, we’ll put an emoji puzzle that can be
              converted to <EmojiNumber number={0} /> on the bottom.{' '}
              <H args={{ name: 'pressRun' }} />
            </P>
            <R.Vdcr>
              The bottom emoji puzzle (yellow background)
              <br />
              can be converted to <EmojiNumber number={0} />
            </R.Vdcr>
            <P>
              <Bold>What just happened:</Bold> We started out with{' '}
              <EmojiNumber number={0} />
              , and it ended up with <CustomEmoji type="isZero" size="mdlg" />.
            </P>
            <P>
              Now, what if we started out with an emoji puzzle that can be
              converted to <EmojiNumber number={1} />?{' '}
              <H args={{ name: 'pressRun' }} />
            </P>
            <R.Vyic>
              The bottom emoji puzzle (yellow background)
              <br />
              can be converted to <EmojiNumber number={1} />
            </R.Vyic>
            <P>
              <Bold>What just happened:</Bold> We started out with{' '}
              <EmojiNumber number={1} />, and it ended up with{' '}
              <CustomEmoji type="isNotZero" size="mdlg" />.
            </P>
            <P>
              As you just saw, in addition to math expressions, we can also
              represent <Bold>conditional</Bold> statements using emoji puzzles.
            </P>
            <Alert backgroundColor="brown">
              <P>
                <Bold>Column: Lambda calculus and Church encoding:</Bold>
              </P>
              <P>
                There’s a programming language called{' '}
                <Bold>Lambda calculus</Bold> (
                <ExternalLink href="https://en.wikipedia.org/wiki/Lambda_calculus">
                  Wikipedia
                </ExternalLink>
                ), created by a mathematician Alonzo Church in 1930s. It only
                has two features: variables and anonymous functions. Here’s a
                piece of lambda calculus code:
              </P>
              <CodeBlock
                noHighlight
                caption={
                  <>
                    <InlineCode>λ</InlineCode> is the greek alphabet “lambda”
                  </>
                }
              >{`λsushi.sushi sandwich`}</CodeBlock>
              <P>
                The above code is equivalent to the following JS code. There are
                no strings in lambda calculus—everything is either a variable or
                an anonymous function.
              </P>
              <CodeBlock>{`(sushi => sushi)(sandwich)`}</CodeBlock>
              <P>
                You might have realized that all the functional JS code we
                represented using emoji puzzles can be expressed in lambda
                calculus. For example, the emoji puzzle that multiplies two
                Church numerals…
              </P>
              <R.Nnzx>
                Emoji puzzle that multiplies
                <br />
                two Church numerals
              </R.Nnzx>
              <P>…is equivalent to the following lambda calculus code:</P>
              <CodeBlock
                noHighlight
                caption={<>Equivalent lambda calculus code</>}
              >{`λsushi.λsandwich.λpizza
  sushi (sandwich pizza)`}</CodeBlock>
              <P>
                <Bold>So, here’s the secret:</Bold> My emoji puzzles are
                actually a visual representation of lambda calculus. And by
                using emoji puzzles, lambda calculus can be explained visually
                to non-programmers!
              </P>
              <EmojiSeparator
                nodes={[
                  <Emoji>🍣</Emoji>,
                  <CustomEmoji type="doubleArrow" />,
                  <CustomEmoji type="lambda" />
                ]}
                description={
                  <>
                    My emoji puzzles are actually
                    <br />a visual representation of lambda calculus
                  </>
                }
              />
              <P>
                <Bold>Finally:</Bold> We saw that emoji puzzles, or lambda
                calculus, can express not only numbers and arithmetic but also
                conditionals. In fact, lambda calculus can express pretty much
                anything any programming language can do—it’s Turing complete.
                And this method of encoding data and operators using lambda
                calculus is called <Bold>Church encoding</Bold> (
                <ExternalLink href="https://en.wikipedia.org/wiki/Church_encoding">
                  Wikipedia
                </ExternalLink>
                ).
              </P>
            </Alert>
            <Subheading step={step++} coveredIn={14}>
              Control flow and Y combinator
            </Subheading>
            <Alert backgroundColor="yellow">
              <div
                css={css`
                  font-size: ${fontSizes(0.85)};
                `}
              >
                <Bold>You’re almost done:</Bold> This is the final step in this
                article.
              </div>
            </Alert>
            <P>
              <Bold>Question:</Bold> If we can express conditionals (e.g.{' '}
              <InlineCode>if</InlineCode>) using functions/emoji puzzles, can we
              express <Bold>control flow</Bold> (e.g. loops) as well?
            </P>
            <P>
              The answer is yes. We can express control flow using{' '}
              <Bold>Y combinator</Bold>.
            </P>
            <P>
              <Bold>Y combinator</Bold> is a function that allows you to create
              a recursive function without using named functions.
            </P>
            <P>
              Y combinator is complex, so if we go into detail we’ll need
              another article. In fact, I used two full pages (
              <InternalLink href="/14">here</InternalLink> and{' '}
              <InternalLink href="/15">here</InternalLink>) in my course to
              explain Y combinator using emoji puzzles. So here I’ll briefly
              explain what Y combinator is.
            </P>
            <P>
              Take a look at this JS code. It calculates the{' '}
              <Bold>factorial</Bold> of a number using recursion.
            </P>
            <CodeBlock
              caption={
                <>
                  Calculates the factorial of a number:
                  <br />
                  <InlineCode>n * n-1 * ... * 1</InlineCode>
                </>
              }
            >{`function fact(n) {
  if (n === 0) {
    return 1
  }
  else {
    return n * fact(n - 1)
  }
}`}</CodeBlock>
            <P>
              If you run it on <InlineCode>5</InlineCode>, it calculates{' '}
              <InlineCode>5 * 4 * 3 * 2 * 1</InlineCode>.{' '}
              <H args={{ name: 'pressRun' }} />
            </P>
            <CodeBlock
              caption={
                <>
                  <InlineCode>5 * 4 * 3 * 2 * 1</InlineCode>
                </>
              }
              result="120"
            >{`fact(5)`}</CodeBlock>
            <P>
              Note that the above recursive function was a{' '}
              <Italic>named</Italic> function. It had the name{' '}
              <InlineCode>fact</InlineCode>, which was called from the function
              body to do recursion.
            </P>
            <CodeBlock
              shouldHighlight={(lineNumber, tokenNumber) =>
                (lineNumber === 0 && tokenNumber > 1 && tokenNumber < 3) ||
                (lineNumber === 5 && tokenNumber > 4 && tokenNumber < 6)
              }
              caption={
                <>
                  This is a <Italic>named</Italic> function called{' '}
                  <InlineCode>fact</InlineCode>
                </>
              }
            >{`function fact(n) {
  if (n === 0) {
    return 1
  }
  else {
    return n * fact(n - 1)
  }
}`}</CodeBlock>
            <P>
              You’d usually use a named function to do recursion. However, if
              you use Y combinator, you can do recursion without using a named
              function.
            </P>
            <P>
              Let me show you how. Here’s the Y combinator function{' '}
              <InlineCode>yc</InlineCode>:
            </P>
            <CodeBlock
              caption={<>The Y combinator function</>}
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 0 && tokenNumber > 1 && tokenNumber < 3
              }
            >{`const yCombinator = sushi =>
  (pizza =>
    sushi(sandwich =>
      pizza(pizza)(sandwich)
    ))(pizza =>
    sushi(sandwich =>
      pizza(pizza)(sandwich)
    )
  )`}</CodeBlock>
            <P>
              Now, we’ll apply <InlineCode>yCombinator</InlineCode> on another
              anonymous function. This time, <InlineCode>fact</InlineCode> is
              NOT a function name, but it’s a <Italic>parameter</Italic> name.
              We haven’t used any named function yet—we’ve only used anonymous
              functions.
            </P>
            <CodeBlock
              caption={
                <>
                  <InlineCode>fact</InlineCode> is now a parameter
                </>
              }
              shouldHighlight={(lineNumber, tokenNumber) =>
                (lineNumber === 0 && tokenNumber > 1 && tokenNumber < 3) ||
                (lineNumber === 4 && tokenNumber > 4 && tokenNumber < 6)
              }
            >{`yCombinator(fact => n => {
  if (n === 0) {
    return 1
  } else {
    return n * fact(n - 1)
  }
})`}</CodeBlock>
            <P>
              Finally, let’s run the above function on{' '}
              <InlineCode>5</InlineCode> and see what happens:
            </P>
            <CodeBlock
              caption={
                <>
                  Run it on <InlineCode>5</InlineCode>
                </>
              }
              result="120"
              shouldHighlight={(lineNumber, tokenNumber) =>
                lineNumber === 6 && tokenNumber > 3 && tokenNumber < 5
              }
            >{`yCombinator(fact => n => {
  if (n === 0) {
    return 1
  } else {
    return n * fact(n - 1)
  }
})(5) // ← run it on 5`}</CodeBlock>
            <P>
              The result was <InlineCode>120</InlineCode>, so it successfully
              calculated the factorial{' '}
              <InlineCode>5 * 4 * 3 * 2 * 1</InlineCode>.
            </P>
            <P>
              <Bold>So:</Bold> By using the <InlineCode>yCombinator</InlineCode>{' '}
              function, you can create a recursive function without using named
              functions. It allows you to implement control flow (loops) using
              anonymous functions.
            </P>
            <P>
              Of course, Y combinator can be represented using an emoji puzzle:
            </P>
            <CodeBlock
              caption={<>The Y combinator function</>}
            >{`const yCombinator = sushi =>
  (pizza =>
    sushi(sandwich =>
      pizza(pizza)(sandwich)
    ))(pizza =>
    sushi(sandwich =>
      pizza(pizza)(sandwich)
    )
  )`}</CodeBlock>
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            <R.Weow>Y combinator using emojis</R.Weow>
            <P>
              In{' '}
              <InternalLink href="/15">
                the final lesson of my course
              </InternalLink>
              , I show how to use the Y combinator emoji puzzle to calculate
              factorials (I won’t show it here because it’s pretty complex).
            </P>
            <P>
              In any case, you can teach Y combinator to non-programmers using
              emoji puzzles.
            </P>
            <Alert backgroundColor="blue">
              <P>
                The Y Combinator I showed above is for{' '}
                <ExternalLink href="https://en.wikipedia.org/wiki/Evaluation_strategy#Applicative_order">
                  applicative order
                </ExternalLink>{' '}
                languages like JavaScript. For{' '}
                <ExternalLink href="https://en.wikipedia.org/wiki/Evaluation_strategy#Normal_order">
                  normal order
                </ExternalLink>{' '}
                languages like Haskell, Y Combinator would look visually
                simpler, like this:
              </P>
              <R.Xjzx>
                Y Combinator for
                <br />
                normal order languages
              </R.Xjzx>
              <P>
                The emoji puzzles on this page run in the applicative order
                because I tried to compare it with JavaScript. However, in my
                course, the emoji puzzles actually run in the normal order
                instead, simply because Y Combinator would be visually less
                complex.
              </P>
              <P>
                The bottom line is that my emoji puzzles support both
                applicative and normal orders of evaluation, which was
                nontrivial to implement.
              </P>
            </Alert>
            <Alert backgroundColor="brown">
              <JimsTalk onBlog />
            </Alert>
            <Subheading step="none" coveredIn="none">
              Conclusion
            </Subheading>
            <P>
              <Bold>One-line summary:</Bold> By using emoji puzzles, you explain
              functional programming concepts such as lambda calculus, Church
              encoding, and Y combinator to non-programmers—without using any
              code.
            </P>
            <P>
              <Bold>Why did I make emoji puzzles?</Bold> As someone who is
              passionate about teaching, I wanted to challenge myself to explain
              a <Italic>difficult computer science concept</Italic> (like Y
              combinator) to non-programmers while satisfying the following
              constraints:
            </P>
            <Ul>
              <UlLi>Don’t use any code</UlLi>
              <UlLi>Don’t sacrifice rigor</UlLi>
              <UlLi>Must be doable on a smartphone in under 2-3 hours</UlLi>
            </Ul>
            <P>
              That’s how I came up with emoji puzzles (they’re
              smartphone-friendly too). In the future, I plan to develop
              something similar with other CS topics. In the meantime, you can
              take a look at my course,{' '}
              <InternalLink href="/">
                Y Combinator for Non-programmers
              </InternalLink>
              , here:
            </P>
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                  View My Course
                </span>
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                    display: block;
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                  Y Combinator for Non-programmers
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              Thank you for reading!
            </Subheading>
            <Ads shareId="1192843729565962241" />
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              This article was published on{' '}
              <time dateTime={dateSchemaString}>{dateString}</time>.
            </P>
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