Enigma 1730

Spoiler for New Scientist Enigma 1730: “Prime position” (Follow the link to see the puzzle.)

This one goes quickly, once you realize the 1-digit horizontal and vertical numbers do not count among the 11 answers. Label the blank squares like this:

2---A
BC-DE
-FGH-
IJ-KL
M---N

The required primes are BC, DE, FGH, IJ, KL, 2B, AE, CFJ, DHK, IM, and LN (but not 2, A, M, N, or G). All columns have the same sum, namely G, and since CFJ and DHK are prime their digits must not sum to a multiple of 3; therefore G≤8. Prime 2B must be 23 or 29 but 29 gives too large a column sum, therefore B=3. Now I+M≤3 so IM=11 and G=7.

BC=3C=31 or 37, but the latter ruins the column sum. IJ=1J=11 or 13 (not 17). If IJ=13 then CFJ=133 which is not prime, so IJ=11 and CFJ=151, which is prime.

FGH=57H and H<7 so H=1 to give the prime 571. Then D+K=6 and K is odd and not 5, so DHK=313 or 511; only 313 is prime. E and L can only be 1. Now A+N=5 and N is odd. A=2, N=3 doesn’t work (AE=21 isn’t prime) so A=4, N=1. The completed grid is

2---4
31-31
-571-
11-31
1---1

and the shaded numbers are 41 and 571.

 

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