Problems & Puzzles: Puzzles

 Puzzle 342. Primes on a dice. Q1.  "It is possible to write a distinct prime on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a distinct prime? (in total, 14 distinct primes)". If so, please send the solution with the minimal sum of the six primes. Perhaps some of you will recognize that this is a stolen statement from one of the always interesting puzzles from one the pages of my friend Frank Rubin. But his puzzle asks for (14) squares not for primes, which makes his problem a lot harder than the mine!... Q2.  It is possible to write a distinct prime on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a distinct square?. If so, please send the solution with the minimal sum of the six primes.Q3.  It is possible to write a distinct square on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a distinct prime?. If so, please send the solution with the minimal sum of the six squares. _____________ Note:  Please send your solutions just as: p5 p1, p2, p3, p4 p6 In this format (p5, p6), (p1, p3) & (p2, p4) are in opposite sides of the dice.

Contributions came from Adam, Stinchcombe, Jacques Tramu & David Terr.

I was told to report p6;p1,p2,p3,p4;p5 which I assumed meant prime on face 5, prime on 1, etc.  So then p1=3, p2=7 and the die I am looking at in my hand has a vertex with 1,2, and 3 bordering each other (3+7+13=23 is prime) and 1,2 and 4 bordering each other (3+7+19=29 is prime). 3+7+11=21 (is not prime) but represents faces 1,2 and 6.  Face 6 is directly opposite face 1.   Am I misunderstanding something in the puzzle?

The other solutions that add 90 and had 8 additional primes used faces of (out of order, my programming did not retain the order of the faces, permuting them around until they found all prime sums) A) 3, 7, 11, 13, 19, 37, B) 3, 7, 13, 17, 19, 31, C) 5, 7, 11, 13, 17, 37, and D) 5, 7, 11, 17, 19, 31.

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Jacques wrote for Q3:

36
1,4 ,361 ,784
144

sum     1330

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David wrote for Q1:

Here's a solution:

3
5   11   17    53
13

Sum = 102

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It happened that the question Q1 this puzzle was posed in a more general way in Puzzle 290. So the only new questions are Q2 & Q3 (sorry).

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Christian Boyer found the minimal solution for Q2:

Eight solutions with the smallest possible sum of 6 prime numbers = 32045.
One of them:

2
61, 2053, 12829, 7333
9767

But 8 possible dice have this same sum, and always 2 and 9767 on their opposite faces (the first solution being the one sent yesterday):

2 61 2053 12829 7333 9767
2 421 1693 5701 14461 9767
2 571 1543 13339 6823 9767
2 631 1483 13399 6763 9767
2 811 1303 6091 14071 9767
2 823 1291 13591 6571 9767
2 883 1231 6163 13999 9767
2 1021 1093 13789 6373 9767

It is interesting to note that these 8 solutions have:
- exactly the same sum of 6 faces,
- the same couple of opposite faces 2/9767,
- but also the same set of 8 distinct squares sums (the 8 sums of the faces surrounding each vertex):
46², 86², 109², 122², 131², 142², 157², 173²

After 32045, the next two smallest sums of faces are 77285 and 89570.
But if we accept solutions of any set of 8 squares sums (some of them allowed to be equal), then 50690 is another possible sum of faces.

Regarding the original Frank's puzzle, Boyer thinks this way:

I do not have the solution, and I do not know if a solution is possible...
...but if you are interested by answers very close to the solution, I have some examples, one of them:

132² 176² 468² 351²
1200²

The sum of the 6 faces is 1830625. And 7 on the 8 sums of the faces surrounding each vertex are squares, meaning that only ONE is not a square.
The 8 sums of this example are: 220², 375², 500², 585², 1220², 1580625, 1300², 1335² with unfortunately 1580625 = 75² x 281 = (1257.229...)²

Perhaps that your problem is as difficult as the "magic squares of squares"
problem: www.multimagie.com/English/SquaresOfSquares.htm

...

There are several and great similarities between the 2 problems.

1) Left side of equations
The two problems are very close:
-The Frank's puzzle is a system of 8 equations, the left parts of the 8 equations summing 3 squares.
-The 3x3 magic square of squares problem is ALSO a system of 8 equations (3 rows + 3 columns + 2 diagonals), the left parts of the 8 equations ALSO summing 3 squares

2) Right side of equations
-In the Frank's puzzle, the right parts of the 8 equations are squares.
-In the 3x3 magic squares of squares problem, the right parts of the 8 equations are not asked to be squares... BUT in the family of near-solutions of Edouard Lucas, the right parts are ALSO squares!

3) Near-solutions
-Near-solutions of the 3x3 magic squares of squares are known, solving 7 equations on 8 (and we know a great number of near-solutions).
-My near-solution of the Frank's puzzle, sent below, solves ALSO 7 equations on 8 (and I have a great number of other near-solutions with 7 correct equations).

4) Impossible?
I have not deeply analyzed the Frank's puzzle, but at a first look, I do not see evident logical impossibility, as it is already for the 3x3 magic squares of squares problem. And for both problems, a solution is not easy to get... and is perhaps impossible to get.

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Wilfrid Whiteside wrote (March, 2006):

I had fun trying the squares on a cube problem but did not succeed, probably not even close.  I ran a few cpu days doing systematic searches.  I tried all dice with faces less than 23230^2 and I tried all dice with faces less than 20000^2 but allowing the face opposite to the smallest face go up to 32000^2.  All I got was 40 cases (after weeding out multiples) where 7 of the 8 corner triplets added up to squares (see below).

Unfortunately finding 7 of 8 is a far cry from getting 8 of 8.  So I did not find glory.

Cases found with 7 of 8 sums forming squares:

0 1056 1980 4131 2808 1600
0 1232 3276 11232 4224 17325
0 1232 3276 11232 4224 2145
0 1287 1716 3520 2640 11700
0 1287 1716 9360 7020 4400
0 132 176 468 351 1200
0 364 1248 8064 2352 11781
0 4620 4851 16632 15840 4640
0 5500 5775 13860 13200 13104
0 704 1872 7371 2772 15600
0 935 2244 12240 5100 20592
2 55 110 1511 1514 1510
6 165 330 4533 4542 1490
8 220 440 6044 6056 505
55 880 1276 6680 7040 2240
81 882 2558 3078 3294 2901
99 144 148 924 512 384
112 516 597 14184 6960 30512
138 690 2645 6366 6859 6330
143 364 368 4576 1144 6688
216 552 612 1329 7308 9416
216 7035 7812 16065 10800 9716
231 252 896 8712 3888 3024
288 401 492 13536 4188 33264
336 648 1232 2772 2727 9264
342 361 834 693 1254 2178
396 1008 5376 19103 16272 7224
432 576 1092 2704 2652 1521
440 756 792 4620 3240 1485
456 532 3192 11172 12264 19551
504 1792 2913 7776 6048 24984
594 1323 2394 12243 7722 18326
828 1712 5961 8928 10944 12864
1584 2772 2928 5292 7392 2849
1840 2240 3520 2944 7820 4117
1881 3686 4218 9306 6822 10143
1920 3491 9612 19584 14400 15888
1980 2385 3840 7040 7920 8448
2268 8793 11556 10674 12676 24972
2660 3080 3640 9880 8645 10868

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