Problems & Puzzles: Puzzles

Puzzle 338. Domino-Prime-Pyramid The following puzzle is just a way of driving your attention to the nice puzzles in the page of Leonid Mochalov, a very recent site added to my Link's page, on his personal request. As a matter of fact, me and my friends of the Ramanujan's seminars celebrated in the Bars of our town have enjoyed the Mochalov's puzzles a lot, specially the related to the dominoes game, by obvious reasons ;-) Due to the fact that Mochalov, unfortunately, provides the solution of every puzzle, I will have to switch some things of his original puzzle in order to make interesting to you the puzzle of this week in my pages. Question 1 (about his puzzle # 3, "The pyramid"): Regarding the following arrangement of the domino pieces: Find an arrangement such that the sum of the points in every row is a prime number, such that all the primes gotten are distinct. Keep in mind that "the tiles in the horizontal rows are positioned according to the rules of dominoes". Question 2 (about his puzzle # 10, "Frameworks"): Regarding the following arrangement of the domino pieces, If you are asked to get an arrangement such that the sum of each side of each square is the same integer: a) Can you explain why this sum must be necessarily 13, or it can be some other numbers? While the solution given by Mochalov is such that the pieces do not follow the rules of dominoes, b) Can you get one solution that follow the rules of dominoes? c) Can you get another solution simply distinct to the Mochalov's one?

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Jacques Tramu wrote:

Hi Carlos. There seems to be a lot of solutions for this puzzle.

So I added the constraint : "doubles must be first or last in row", and found the following:

0|2   (2)
0|0  0|3   (3)
1|1  1|0  0|4   (7)
2|2  2|1  1|3  3|5   (19)
3|3  3|2  2|4  4|1  1|6   (29)
4|4  4|3  3|6  6|0  0|5  5|1   (41)
5|5  5|2  2|6  6|4  4|5  5|6  6|6   (67)

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For question 2, Carlos Rivera has proved that you can get solutions with sum for side other than 13 (from 12 to 15) and has found one particular solution for each sum value in the valid range, just to verify the proof.

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