Problems & Puzzles: Puzzles

Puzzle 1144 Queen attacking primes over a Knight tour 8x8 Matrix. This new puzzle is related to the puzzles 30, 310 and 1141 Now, Honaker, Jr. poses the following puzzle, on Set 12, 2023: We know since Nov 1998, by Mike Keith, that exists at least one simple (not magic neither semimagic*) Knight Tour attacks all the l8 the primes between 1 & 64 in a chess square (8x8). See it here http://www.cadaeic.net/primeq.htm or here https://www.instagram.com/p/CJ4F-iXjJmi/ * as a matter of fact in 5) from PUzzle 1141 we learnt that "In 2003, Hugues Mackay (Canada), Jean-Charles Meyrignac (Francia) and Günther Stertenbrink (Alemania) produced ALL the possible Semi magic Knight 8x8 tours (140, 63 closed and 77 open) with the surprising result that NONE of them resulted to be "magic". All of these were semi magic." The queen is at the square 35 and the 18 attacked primes are: 2, 3, 5, ..., 59, 61. The rule for this puzzle, is that the queen does not attack the square where it is positioned, no matter if this square is 1, prime or composite. The new question posed by Honaker is the following one: a) Discarding rotations, reflection, etc, how many distinct solutions are for this puzzle? b) Disregarding the condition of 18 primes being attacked by the queen, is there a solution such that seven primes in one of the diagonals could be attacked by this bellicose queen?

---

From Set 23-30 contributions came from Alessandro Casini

***

Alessandro wrote:

Q1) I don't know if there's a easy way to just count the solutions without generating them. Let me point out some simple symmetries that hold for general knight tour aren't valid for this puzzle (for instance, inverting the path). For the moment, I've found 11936 distinct solutions, of which 1585 with a closed knight tour. However, it isn't an exhaustive search, so the solutions do not claim to be all possible ones. I attach here all the solutions I found in the Frenicle form, with the relative Queen positions indicated.

 

Q2) Actually, it's very easy to find a knight tour of this kind, so I took the liberty of increasing the conditions. I report here one of the re-entrant Knight Tours with both diagonals of the Queen composed entirely of primes (hence a configuration of maximal primes attack for a Bishop, and thus 7 primes on one diagonal and 6 on the other one) and furthermore the Queen attacks all the 18 primes.

 

11 14 39 60 9 64 7 62

38 41 10 13 54 61 58 1

15 12 53 40 59 8 63 6

42 37 50 55 52 5 2 57

49 16 43 36 29 56 25 4

44 19 46 51 24 3 30 33

17 48 21 28 35 32 23 26

20 45 18 47 22 27 34 31

Great Work! (CR)

***