Sunday Times Teaser 3239 – Box Set
by Victor Bryant
Published Sunday October 20 2024 (link)
I have a shallow box whose base is 20 centimetres wide and a certain whole number of centimetres long. In this box there are some thin plain tiles. Most (but not all) of the tiles are rectangles, ten centimetres by 20, and the rest are squares of side 20 centimetres. They just fit snugly, jigsaw fashion, in the bottom of the box. With the box in a fixed position, I have calculated the number of different jigsaw layouts possible using all the tiles to fill the box. The answer is a three-figure perfect square.
How long is the box, and how many of the tiles are square?
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BRG permalink12345678910111213141516171819202122232425262728293031323334353637383940from itertools import countfrom functools import cache# count the number of ways in which <n12> 1x2 tiles and <n22># 2x2 tiles can be arranged in a 2x(n12 + 2 x n22) rectangle@cachedef count_ways(n12, n22):if not (n12 or n22):return 1cnt = 0if n12 > 1:# place two vertical 1x2 tilescnt += count_ways(n12 - 2, n22)if n12:# place a horizontal 1x2 tilecnt += count_ways(n12 - 1, n22)if n22:# place a 2x2 tilecnt += count_ways(n12, n22 - 1)return cnt# consider the total number of tiles ...for nt in count(3):# ... and the number of 2x2 tilesfor n22 in range(1, (nt + 1) // 2):n12 = nt - n22if n12 > n22:# count the number of ways to place the tilesways = count_ways(n12, n22)# ways is a three digit squareif ways < 1000:if ways >= 100 and round(ways ** 0.5) ** 2 == ways:length = 10 * n12 + 20 * n22print(f"{length = }cm, {n12 = }, {n22 = }, {ways = }.")breakelse:continuebreak
