Problems & Puzzles: Puzzles

 Puzzle 767. A second Follow up to Puzzle 764. Going up one more step in the direction that Dmitry Kamenetsky extended the Puzzle 764, now we ask for the 3D version of this puzzle.   "Place numbers in a NxNxN grid such that all gcds from 1 to 3*N*N*(N-1) are generated. A gcd is generated for every pair of neighboring numbers" Kamenetsky has produced one solution for the case 2x2x2 (He thinks that this solution has the minimal sum but...)   Sum = 350. 55 24  66 56  ------------------ 20 36  30 63  gcds from 1 to 12: Axis X: gcd(55,24)=1 Axis Y: gcd(55,66)=11 Axis X: gcd(56,66)=2 Axis Y: gcd(56,24)=8 Axis X: gcd(20,36)=4 Axis Y: gcd(20,30)=10 Axis X: gcd(63,30)=3 Axis Y gcd(63,36)=9 Axis Z: gcd(55,20)=5 Axis Z: gcd(24,36)=12 Axis Z: gcd(66,30)=6 Axis Z: gcd(56,63)=7 Q. Send your minimal sum solutions for N=2, 3, 4 & 5.

Contributions came from Dmitry Kamenetsky and Emmanuel Vantieghem.

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Dmitry wrote:

it turns out that my 2x2x2 was not optimal after all. I managed to find two better solutions, both with the same score:

2x2x2
score 338
66 12
63 36
------------------
55 10
56 40

2x2x2
score 338
42 45
77 36
------------------
30 40
44 24

Finding 3x3x3 (and above) was tricky, but I managed to do it:

3x3x3
score 64007
1833 1504 3128
4862 816 3927
6479 1488 792
------------------
1950 1440 1610
2200 1188 378
2014 1710 4995
------------------
2132 3567 989
520 4060 3612
1325 3675 1813

4x4x4
score 5852344
8804 3348 147963 237553
101990 13230 88800 169534
51606 137494 40425 466697
54229 196546 127765 36708
------------------
6106 4536 68112 101268
110121 20493 26208 133168
41922 44574 17955 130680
191389 35445 238165 36624
------------------
34314 8190 35178 317463
53580 3960 14625 58630
31152 27600 38500 41860
163812 25840 94468 48208
------------------
243853 44512 354497 11926
34238 11968 118400 36850
31270 4760 11648 117832
357481 29631 154245 202395

5x5x5
score 772650040
11766267 10165865 2593695 344565 901854
1337805 28196381 3849083 456940 819720
49245 7347756 1021496 8768424 76860
7473445 8754408 1451880 1034892 3632001
6707555 4740024 33759589 3608811 11592209
------------------
1802255 3486054 8159388 10480509 5001217
2419984 471120 1197840 2735192 4913168
6503133 2967990 13814710 23806158 516880
5629624 2905540 1210560 3288194 23541364
16636039 4116931 7355607 930495 3583195
------------------
15374515 640596 1157904 76464 5982836
573303 3738912 9165450 216972 10388658
20523327 3748024 14564750 3794310 1272240
7317280 5649490 3461568 8941842 700785
692718 3285711 21638525 9834415 813285
------------------
23038583 1034540 2369120 30888 8261163
2360915 3642912 266400 4635400 9686565
11615395 33343042 79500 1370880 1041675
1590624 3739428 7420636 764672 17687448
2708384 4195125 22089875 2152320 13169451
------------------
3055272 4727535 18180986 2050290 2527305
6754440 5042037 10185582 7330468 13302959
10591493 2373960 5160400 7543494 379575
4621617 4399564 3832752 2411200 4929925
54918 631350 506184 2210494 73502

***

Emmanuel wrote:

This is my minimal sum 338 solution for the  2x2x2 -cube :

10 12
55 66
------
40 36
63 56

I sure there is no smaller sum solution.

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