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Problem 85: Counting rectangles

curriculum/challenges/english/blocks/project-euler-problems-1-to-100/5900f3c11000cf542c50fed4.md

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--description--

By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles:

Although there may not exists a rectangular grid that contains exactly n rectangles, find the area of the grid with the nearest solution.

--hints--

countingRectangles(18) should return a number.

js
assert(typeof countingRectangles(18) === 'number');

countingRectangles(18) should return 6.

js
assert.strictEqual(countingRectangles(18), 6);

countingRectangles(250) should return 22.

js
assert.strictEqual(countingRectangles(250), 22);

countingRectangles(50000) should return 364.

js
assert.strictEqual(countingRectangles(50000), 364);

countingRectangles(1000000) should return 1632.

js
assert.strictEqual(countingRectangles(1000000), 1632);

countingRectangles(2000000) should return 2772.

js
assert.strictEqual(countingRectangles(2000000), 2772);

--seed--

--seed-contents--

js
function countingRectangles(n) {

  return true;
}

countingRectangles(18);

--solutions--

js
function countingRectangles(n) {
  function numberOfRectangles(h, w) {
    return (h * (h + 1) * w * (w + 1)) / 4;
  }

  function rectangleArea(h, w) {
    return h * w;
  }

  let rectanglesCount = 1;
  let maxSide = 1;
  while (rectanglesCount < n) {
    maxSide++;
    rectanglesCount = numberOfRectangles(maxSide, 1);
  }

  let bestDiff = Math.abs(rectanglesCount - n);
  let bestSize = [maxSide, 1];

  let curHeight = maxSide - 1;
  let curWidth = 1;

  for (curWidth; curWidth < curHeight; curWidth++) {
    for (curHeight; curHeight > curWidth; curHeight--) {
      rectanglesCount = numberOfRectangles(curHeight, curWidth);
      const curDiff = Math.abs(rectanglesCount - n);
      if (curDiff < bestDiff) {
        bestDiff = curDiff;
        bestSize = [curHeight, curWidth];
      }

      if (rectanglesCount < n) {
        break;
      }
    }
  }
  return rectangleArea(...bestSize);
}