8
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Can you fill a 4x4 grid with every number from 1 to 16, such that every row, every column and every 2x2 sub-grid of numbers sum to the same value?

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8
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Solution:

     1 15  2 16
    14  4 13  3
     7  9  8 10
    12  6 11  5
 

Notes:

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

Must have
A+B = G+H = I+J = O+P
C+D = E+F = K+L = M+N
A+E = J+N = C+G = L+P
I+M = B+F = K+O = D+H
B+C = J+K = E+H = M+P
F+G = N+O = A+D = I+L
E+I = G+K = B+N = D+P
F+J = H+L = A+M = C+O
which are enough constraints to make it manageable by trial and error.

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1
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This one has a kind of rhythm I find cool:

1 16 2 15
14 3 13 4
12 5 11 6
7 10 8 9
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    $\begingroup$ The central 2x2 appears to sum to 32 only. $\endgroup$ – loopy walt Apr 6 at 17:22
  • $\begingroup$ @loopywalt thats correct, my mistake, I forgot to consider this one. It appears that exchanging rows 3 and 4 fixes it. Then it is interesting to note that it gives a solution similar to yours but with columns 2 and 4 exchanged. So it seems there are not plenty of solutions... $\endgroup$ – Fred Apr 12 at 15:05
1
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1 16 5 12
8 9 4 13
3 14 7 10
15 2 11 6

Each 2x2 square should be (largest + smallest) * 4 / 2. In this case that would be 34.

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    $\begingroup$ You forgot the columns, they don't add up properly. $\endgroup$ – Christoph Apr 6 at 14:22
  • $\begingroup$ Ah yes, I didn't notice that. $\endgroup$ – Dmitry Kamenetsky Apr 7 at 2:25

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