Problems & Puzzles: Puzzles

Puzzle 833. A special set of integers. This puzzle is based in the entry 1445 of the always interesting Claudio Meller's site. Here we ask for a set of distinct integers allocated in a Rx3 matrix such that the three integers a, b & c in any row has the same smallest sum S=a+b+c and the same product P=a*b*c. Here is an example for R=9 and S=smallest sum, that I computed and submitted to Claudio.   S a b c P 1287 99 588 600 34927200 100 539 648 105 462 720 112 405 770 126 336 825 132 315 840 162 245 880 165 240 882 196 200 891 By my side I started solving the same puzzle but adding the following condition: S=smallest prime sum. My best solution for R=7 and S=smallest prime sum is S=991 and P=10054800 Q. Send your set of integers with the smallest prime S for R=7, 8,...,15.

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Contributions came from Carlos Rivera and Emmanuel Vantieghem

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Rivera wrote:

Here are my solutionns for R=9 & 10

R=9, S=minimal and prime=991
A B C Suma Prod
45 456 490 991 10054800
49 342 600 991 10054800
56 270 665 991 10054800
63 228 700 991 10054800
76 180 735 991 10054800
95 140 756 991 10054800
105 126 760 991 10054800
 

R=10, S=minimal and prime=2111
A B C Suma Prod
255 920 936 2111 219585600
260 816 1035 2111 219585600
270 736 1105 2111 219585600
288 650 1173 2111 219585600
299 612 1200 2111 219585600
312 575 1224 2111 219585600
325 544 1242 2111 219585600
368 468 1275 2111 219585600

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Emmanuel wrote:

I could not find a solution for  R > 13.  Either there was not enough memory or there was lack of time. But here are the results for the other asked values of  R :

R = 7;  S = 991; P = 10054800

  45, 456, 490

  49, 342, 600

  56, 270, 665

  63, 228, 700

  76, 180, 735

  95, 140, 756

  105, 126, 760

R = 8; S = 2111; P = 219585600

  255, 920, 936

  260, 816, 1035

  270, 736, 1105

  288, 650, 1173

  299, 612, 1200

  312, 575, 1224

  325, 544, 1242

  368, 468, 1275

R = 9; S = 4259; P = 908107200

  224, 1890, 2145

  245, 1440, 2574

  275, 1176, 2808

  280, 1144, 2835

  297, 1050, 2912

  360, 819, 3080

  416, 693, 3150

  441, 650, 3168

  480, 594, 3185

R = 10; S = 4703; P = 1917115200

  418, 2080, 2205

  420, 1995, 2288

  429, 1824, 2450

  475, 1456, 2772

  480, 1430, 2793

  528, 1235, 2940

  539, 1200, 2964

  560, 1140, 3003

  585, 1078, 3040

  728, 840, 3135

R = 11; S = 5419; P = 2464862400

  390, 2464, 2565

  396, 2223, 2800

  399, 2160, 2860

  416, 1925, 3078

  455, 1620, 3344

  475, 1512, 3432

  528, 1300, 3591

  532, 1287, 3600

  594, 1120, 3705

  630, 1045, 3744

  728, 891, 3800

R = 13; S = 11777; P = 5751345600

  171, 5600, 6006

  175, 4914, 6688

  198, 3675, 7904

  224, 3003, 8550

  228, 2925, 8624

  240, 2717, 8820

  245, 2640, 8892

  385, 1512, 9880

  416, 1386, 9975

  462, 1235, 10080

  540, 1045, 10192

  600, 936, 10241

  637, 880, 10260

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