110 lines
2.4 KiB
Markdown
110 lines
2.4 KiB
Markdown
---
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id: 5900f3c11000cf542c50fed4
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title: 'Problem 85: Counting rectangles'
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challengeType: 5
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forumTopicId: 302199
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dashedName: problem-85-counting-rectangles
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---
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# --description--
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By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles:
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<img class="img-responsive center-block" alt="a diagram of the different rectangles found within a 3 by 2 rectangular grid" src="https://cdn-media-1.freecodecamp.org/project-euler/counting-rectangles.png" style="background-color: white; padding: 10px;" />
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Although there may not exists a rectangular grid that contains exactly `n` rectangles, find the area of the grid with the nearest solution.
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# --hints--
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`countingRectangles(18)` should return a number.
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```js
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assert(typeof countingRectangles(18) === 'number');
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```
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`countingRectangles(18)` should return `6`.
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```js
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assert.strictEqual(countingRectangles(18), 6);
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```
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`countingRectangles(250)` should return `22`.
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```js
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assert.strictEqual(countingRectangles(250), 22);
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```
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`countingRectangles(50000)` should return `364`.
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```js
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assert.strictEqual(countingRectangles(50000), 364);
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```
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`countingRectangles(1000000)` should return `1632`.
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```js
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assert.strictEqual(countingRectangles(1000000), 1632);
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```
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`countingRectangles(2000000)` should return `2772`.
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```js
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assert.strictEqual(countingRectangles(2000000), 2772);
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```
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# --seed--
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## --seed-contents--
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```js
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function countingRectangles(n) {
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return true;
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}
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countingRectangles(18);
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```
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# --solutions--
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```js
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function countingRectangles(n) {
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function numberOfRectangles(h, w) {
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return (h * (h + 1) * w * (w + 1)) / 4;
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}
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function rectangleArea(h, w) {
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return h * w;
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}
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let rectanglesCount = 1;
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let maxSide = 1;
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while (rectanglesCount < n) {
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maxSide++;
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rectanglesCount = numberOfRectangles(maxSide, 1);
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}
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let bestDiff = Math.abs(rectanglesCount - n);
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let bestSize = [maxSide, 1];
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let curHeight = maxSide - 1;
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let curWidth = 1;
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for (curWidth; curWidth < curHeight; curWidth++) {
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for (curHeight; curHeight > curWidth; curHeight--) {
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rectanglesCount = numberOfRectangles(curHeight, curWidth);
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const curDiff = Math.abs(rectanglesCount - n);
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if (curDiff < bestDiff) {
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bestDiff = curDiff;
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bestSize = [curHeight, curWidth];
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}
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if (rectanglesCount < n) {
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break;
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}
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}
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}
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return rectangleArea(...bestSize);
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}
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```
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