Enigma 106: Magic box
From New Scientist #1250, 23rd April 1981 [link]
Many interesting number patterns can be made with popular 15-puzzle in which the numbers from 1 to 15 can slide around in a four-by-four framework.
For example, it is possible to make the top left-hand three-by-three block into many different Magic Squares (i.e. each row of three, each column of three and each diagonal of three adds to the same sum). One such arrangement is shown.
It is also possible to do this in such a way that, again with the space in the bottom right-hand corner, the three numbers in the bottom row of the frame add up to the same total as the three numbers in the right-hand column of the frame, which also add up to the same total as the four numbers in the diagonal of the frame from the bottom left to the top right-hand corner. (And, incidentally, this total is more than each of the row sums of the Magic Square).
Give, in increasing order, the four numbers which must form that diagonal of this special arrangement.