Enigma 1740

Spoiler for New Scientist Enigma 1740: “Sudoprime” (Follow the link to see the puzzle.)

Label the squares:

A - - - B
C D - E F
- G H I -
J K - L M
N - - - O

A is 5 and H is 9. C, D, F, I, K, L, M, N, and O must be odd and not 5.

AC must be 53 or 59. Try 59 first:

Then CD must be 97. K cannot be 7 or 9 so must be 1 or 3. Then DGK is 743, 751, or 761. L cannot be 7 or 9 so must be 1 or 3. If DGK is 743 then GHI is 491 and L cannot be 1 or 3. Dead end. If DGK is 751 then GHI is 593 and again L cannot be 1 or 3. Dead end. So DGK is 761, GHI is 691, and L is 3. EIL must be 613. This means O must be 1. F cannot be 1 or 7 or 9 so must be 3; then EF can only be 63 which is not prime. Dead end.

So AC must be 53. CD can be 31 or 37. Try 37:

Same argument as above except we conclude F must be 9, so EF is 69 which is not prime. Dead end.

Then CD is 31. N cannot be 3 or 9 so must be 1 or 7. K, L, and O must be 3 or 7. So LM and MO are 31 and 17 or 71 and 13. Either way M is 1. Try 31 and 17:

Then F is 9. I is 1 or 7. The 3-digit primes ending in 13 or 73 are 113, 173, 313, 373, 613, 673, 773. None is legal for EIL. 

So LM and MO are 71 and 13. Then I is 1 or 3. The 3-digit primes ending in 17 or 37 are 137, 317, 337, 617, 937. Only 617 is legal. Then EF is 67. K is 3. GHI is 491 or 691. But 143 is not prime so DGK is 163. B is 4. The only possibilities for JN are 41 and 47, but we already have BF = 47 so JN is 41.

The grid is:

5 - - - 4
3 1 - 6 7
- 6 9 1 -
4 3 - 7 1
1 - - - 3

and the answers required are 617 and 41.

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