How many triangles?

exercise.png

helper.png

result.png

triangles.js

const { equal } = require('assert')

const UNIT_SIDE = 28

const getNumberOfTrianglesOfLengthOne = (lineLength = UNIT_SIDE) => lineLength ** 2

equal(1, getNumberOfTrianglesOfLengthOne(1))
equal(4, getNumberOfTrianglesOfLengthOne(2))
equal(9, getNumberOfTrianglesOfLengthOne(3))

const getAllTriangles = (fullLineLength = UNIT_SIDE, log = console.log) => {
  let sum = getNumberOfTrianglesOfLengthOne(fullLineLength)

  if (typeof log !== 'function') log = () => {}

  for (let lengthOfTriangle = 2; lengthOfTriangle <= fullLineLength; lengthOfTriangle++) {
    log('check for', lengthOfTriangle)

    let sumForLengthOfTriangle = 0
    let sumForLengthOfTriangleReversed = 0

    // triangles pointing up
    let startLine = 0
    let stopLine = fullLineLength - lengthOfTriangle
    for (; startLine <= stopLine; startLine++) {
      const lengthOfThisLine = fullLineLength - startLine;
      sumForLengthOfTriangle += lengthOfThisLine - (lengthOfTriangle - 1)

      log('    →  I can put', lengthOfThisLine - (lengthOfTriangle - 1), 'of size', lengthOfTriangle, 'on a line sized', lengthOfThisLine)
    }

    log('  →  I can put', sumForLengthOfTriangle, 'of length', lengthOfTriangle)
    sum += sumForLengthOfTriangle

    // triangles pointing downx
    startLine = lengthOfTriangle - 1
    stopLine = fullLineLength - lengthOfTriangle - 1

    for (; startLine <= stopLine; startLine++) {
      const lengthOfThisLine = fullLineLength - startLine;

      sumForLengthOfTriangleReversed += lengthOfThisLine - lengthOfTriangle

      log('    →  I can put', lengthOfThisLine - lengthOfTriangle, 'reversed of size', lengthOfTriangle, 'on a line sized', lengthOfThisLine)
    }

    log('  →  I can put', sumForLengthOfTriangleReversed, 'reversed of length', lengthOfTriangle)
    sum += sumForLengthOfTriangleReversed
  }

  return sum
}

// checked by hand (cf. helper.png)
equal(1, getAllTriangles(1, null))
equal(5, getAllTriangles(2, null))
equal(13, getAllTriangles(3, null))
equal(27, getAllTriangles(4, null))
equal(48, getAllTriangles(5, null))
equal(78, getAllTriangles(6, null))

// more checks I found online after solution
equal(118, getAllTriangles(7, null))
equal(170, getAllTriangles(8, null))

console.log('getAllTriangles(1) =', getAllTriangles(1, null))
console.log('getAllTriangles(2) =', getAllTriangles(2, null))
console.log('getAllTriangles(3) =', getAllTriangles(3, null))
console.log('getAllTriangles(4) =', getAllTriangles(4, null))
console.log('getAllTriangles(5) =', getAllTriangles(5, null))
console.log('getAllTriangles(6) =', getAllTriangles(6, null))
console.log('getAllTriangles(28) =', getAllTriangles(28, null))