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copied!<p><strong>EDIT</strong>: for those who do not know what R is, here is a <a href="http://www.r-project.org/" rel="noreferrer">link</a>.</p> <p>My C being a bit rusty... I will write a solution in R, it should not be difficult to adapt. The solution runs very fast in R, so it should be even faster in C.</p> <pre><code># Create an hash table of cubes from 1 to 100 numbers <- 1:100 cubes <- numbers ^ 3 # The possible pairs of numbers pairs <- combn(numbers, 2) # Now sum the cubes of the combinations # This takes every couple and sums the values of the cubes # with the appropriate index sums <- apply(pairs, 2, function(x){sum(cubes[x])}) </code></pre> <p>So:</p> <p><code>numbers</code> will be: <code>1, 2, 3, 4, ..., 98, 99, 100</code><br> <code>cubes</code> will be: <code>1, 8, 27, 64, ..., 941192, 970299, 1000000</code><br> <code>pairs</code> will contain:</p> <pre><code> [,1] [,2] [,3] [,4] [,5] ... [,4949] [,4950] [1,] 1 1 1 1 1 ... 98 99 [2,] 2 3 4 5 6 ... 100 100 </code></pre> <p><code>sums</code> will be: <code>9 28 65 126 217 344 ... 1911491 1941192 1970299</code></p> <p>A quick check that we are on the right track...</p> <pre><code>> which(sums == 1729) [1] 11 765 <--- the ids of the couples summing to 1729 > pairs[,11] [1] 1 12 > pairs[,765] [1] 9 10 </code></pre> <p>Now, let's find which are the couples with the same sums.</p> <p><code>table(sums)</code> gives us a neat summary like</p> <pre><code>> 9 28 35 65 72 91 ... 1674 1729 1736 1 1 1 1 1 1 .... <lots of 1s here> ... 1 2 1 </code></pre> <p>So let's just find which elements of <code>table(sums)</code> are == 2</p> <pre><code>doubles <- which(table(sums) == 2) taxi.numbers <- as.integer(names(doubles)) [1] 1729 4104 13832 20683 32832 39312 40033 46683 64232 65728 [11] 110656 110808 134379 149389 165464 171288 195841 216027 216125 262656 [21] 314496 320264 327763 373464 402597 439101 443889 513000 513856 515375 [31] 525824 558441 593047 684019 704977 805688 842751 885248 886464 920673 [41] 955016 984067 994688 1009736 1016496 </code></pre> <p>And finally (to be read two-by-two), the corresponding integer pairs</p> <pre><code>> pairs[,doubles] [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14] [,15] [1,] 1 9 2 9 2 18 10 19 4 18 2 15 9 16 3 [2,] 12 10 16 15 24 20 27 24 32 30 34 33 34 33 36 [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25] [,26] [,27] [,28] [,29] [1,] 27 17 26 12 31 4 36 6 27 12 38 8 29 20 [2,] 30 39 36 40 33 48 40 48 45 51 43 53 50 54 [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43] [1,] 38 17 24 9 22 3 22 5 45 8 36 4 30 18 [2,] 48 55 54 58 57 60 59 60 50 64 60 68 66 68 [,44] [,45] [,46] [,47] [,48] [,49] [,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57] [1,] 32 30 51 6 54 42 56 5 48 17 38 10 45 34 [2,] 66 67 58 72 60 69 61 76 69 76 73 80 75 78 [,58] [,59] [,60] [,61] [,62] [,63] [,64] [,65] [,66] [,67] [,68] [,69] [,70] [,71] [1,] 52 15 54 24 62 30 57 7 63 51 64 2 41 11 [2,] 72 80 71 80 66 81 72 84 70 82 75 89 86 93 [,72] [,73] [,74] [,75] [,76] [,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84] [,85] [1,] 30 23 63 8 72 12 54 20 33 24 63 35 59 29 [2,] 92 94 84 96 80 96 90 97 96 98 89 98 92 99 [,86] [,87] [,88] [,89] [,90] [1,] 60 50 59 47 66 [2,] 92 96 93 97 90 </code></pre> <p>So:</p> <p>1,12 and 9,10 give 1729<br> 2,16 and 9,15 give 4104<br> 2,24 and 18,20 give 13832<br> and so on!</p>

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