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Since the square is known to be magic, we can assume from basic
"magic theory" that the missing numbers are those absent at the
moment from the series 1 to 16. The main insight then required is
to spot that the third column from the left must contain both 2 and
3, in order to achieve the 34 constant (because 1 and 4 are already
positioned).
Next, examination of the possibilities for the 15, ?, ?, 12
diagonal reveals that the 2 will need to be next to the 12, thus
making the complete diagonal 15, 5, 2, 12. The rest is easy.
Ivor Smith will receive £20 for this neat little
contribution.
Here is the completed array:
| 1 | 8 | 13 | 12 |
|---|
14  | 11 | 2 | 7 |
4 | 5 | 16 | 9 |
15 | 10 | 3 | 6 |