99 lines
6.8 KiB
HTML
99 lines
6.8 KiB
HTML
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<h2>Searching a triangular array for a sub-triangle having minimum-sum.</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 150</h3><span style="width:300px;color:#666;">Published on Friday, 13th April 2007, 10:00 pm; Solved by 1902</span></div>
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<p>In a triangular array of positive and negative integers, we wish to find a sub-triangle such that the sum of the numbers it contains is the smallest possible.</p>
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<p>In the example below, it can be easily verified that the marked triangle satisfies this condition having a sum of <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />42.</p>
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<p>We wish to make such a triangular array with one thousand rows, so we generate 500500 pseudo-random numbers <span style="font-style: italic">s<sub>k</sub></span> in the range <img src='images/symbol_plusmn.gif' width='11' height='11' alt='±' border='0' style='vertical-align:middle;' />2<sup>19</sup>, using a type of random number generator (known as a Linear Congruential Generator) as follows:</p>
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<p style="margin-left:50px;"><span style="font-style: italic">t</span> := 0
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<br />
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for k = 1 up to k = 500500:
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<br />
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<span style="font-style: italic">t</span> := (615949*<span style="font-style: italic">t</span> + 797807) modulo 2<sup>20</sup>
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<br />
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<span style="font-style: italic">s<sub>k</sub></span> := <span style="font-style: italic">t</span><img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />2<sup>19</sup></p>
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<p>Thus: <span style="font-style: italic">s<sub>1</sub></span> = 273519, <span style="font-style: italic">s<sub>2</sub></span> = <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' />153582, <span style="font-style: italic">s<sub>3</sub></span> = 450905 etc</p>
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<p>Our triangular array is then formed using the pseudo-random numbers thus:</p>
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<div style="text-align:center;font-style: italic;">
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s<sub>1</sub>
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<br />
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s<sub>2</sub> s<sub>3</sub>
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<br />
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s<sub>4</sub> s<sub>5</sub> s<sub>6</sub>
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<br />
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s<sub>7</sub> s<sub>8</sub> s<sub>9</sub> s<sub>10</sub>
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<br />
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...
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</div>
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<p>Sub-triangles can start at any element of the array and extend down as far as we like (taking-in the two elements directly below it from the next row, the three elements directly below from the row after that, and so on).
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<br />
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The "sum of a sub-triangle" is defined as the sum of all the elements it contains.
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<br />
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Find the smallest possible sub-triangle sum.</p>
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