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.

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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