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    <p>Notice that there must be an integer number of squares which fill the width and height. Therefore, the aspect ratio must be a rational number.</p> <p>Input: <code>width</code>(float or int), <code>height</code>(float or int)</p> <p>Algorithm:</p> <pre><code>aspectRatio = RationalNumber(width/height).lowestTerms #must be rational number # it must be the case that our return values # numHorizontalSqaures/numVerticalSquares = aspectRatio return { numHorizontalSquares = aspectRatio.numerator, numVerticalSquares = aspectRatio.denominator, squareLength = width/aspectRatio.numerator } </code></pre> <p>If the width/height is a rational number, your answer is merely any multiple of the aspect ratio! (e.g. if your aspect ratio was 4/3, you could fill it with 4x3 squares of length <code>width/4</code>=<code>height/3</code>, or 8x6 squares of half that size, or 12x9 squares of a third that size...) If it is not a rational number, your task is impossible.</p> <p>You convert a fraction to lowest terms by factoring the numerator and denominator, and removing all duplicate factor pairs; this is equivalent to just using the greatest common divisor algorithm <code>GCD(numer,denom)</code> , and dividing both numerator and denominator by that. </p> <p>Here is an example implementation in python3:</p> <pre><code>from fractions import Fraction def largestSquareTiling(width, height, maxVerticalSquares=10**6): """ Returns the minimum number (corresponding to largest size) of square which will perfectly tile a widthxheight rectangle. Return format: (numHorizontalTiles, numVerticalTiles), tileLength """ if isinstance(width,int) and isinstance(height,int): aspectRatio = Fraction(width, height) else: aspectRatio = Fraction.from_float(width/height) aspectRatio2 = aspectRatio.limit_denominator(max_denominator=maxVerticalSquares) if aspectRatio != aspectRatio2: raise Exception('too many squares') #optional aspectRatio = aspectRatio2 squareLength = width/aspectRatio.numerator return (aspectRatio.numerator, aspectRatio.denominator), squareLength </code></pre> <p>e.g.</p> <pre><code>&gt;&gt;&gt; largestSquareTiling(2.25, 11.75) ((9, 47), 0.25) </code></pre> <p>You can tune the optional parameter <code>maxVerticalSquares</code> to give yourself more robustness versus floating-point imprecision (but the downside is the operation may fail), or to avoid a larger number of vertical squares (e.g. if this is architecture code and you are tiling a floor); depending on the range of numbers you are working with, a default value of <code>maxVerticalSquares=500</code> might be reasonable or something (possibly not even including the exception code).</p> <p>Once you have this, and a range of desired square lengths <code>(minLength, maxLength)</code>, you just multiply:</p> <pre><code># inputs desiredTileSizeRange = (0.9, 0.13) (minHTiles, minVTiles), maxTileSize = largestSquareTiling(2.25, 11.75) # calculate integral shrinkFactor shrinkFactorMin = maxTileSize/desiredTileSizeRange[0] shrinkFactorMax = maxTileSize/desiredTileSizeRange[1] shrinkFactor = int(scaleFactorMax) if shrinkFactor&lt;shrinkFactorMin: raise Exception('desired tile size range too restrictive; no tiling found') </code></pre> <p>If <code>shrinkFactor</code> is now <code>2</code> for example, the new output value in the example would be <code>((9*2,47*2), 0.25/2)</code>.</p>
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