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.

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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 

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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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