Mathematical_Recreations-Kraitchik-2e

184 Mathematical Recreations
1 2 3 4 1 2 5 6 1 3 5 7
5 6 7 8 3 4 7 8 2 4 6 8
9 10 11 12 9 10 13 14 9 11 13 15
13 14 15 16 11 12 15 16 10 12 14 16
FIGURE 60.
ment from each orthogonal of an addition table. This can
be done in 3·4! = 72 ways; but each series appears twice,
so there are only 36 different series. They are:
1 4 13 16* 2 3 13 16 3 5 10 16 4 5 11 14
14 15 14 15* 12 14* 6 9 15
6 11 16* 5 11 16 6 9 16 11 13*
12 15 12 15* 10 15 7 9 14
7 10 16* 7 9 16 12 13 10 13*
12 14 11 14 8 9 14* 5 8 9 12*
8 10 15 12 13 10 13 10 11
11 14 8 9 15'" 4 5 9 16 6 7 9 12
12 13 11 13 10 15 10 11*
Two of these, 2, 8, 9, 15 and 3, 5, 12, 14, are not only magic
but bimagic and trimagic as well.
Here is a very beautiful recreation based on the fact that
any four numbers selected from an addition table so that one
is taken from each orthogonal will always add up to 34.
Write the numbers from 1 to 16 on 16 like cards and arrange
them in the form of an addition table, but face downwards.
Ask a person to pick up anyone of the cards without showing
you the number written on it. When he has done so, gather
up the cards remaining in the two orthogonals in which the
selected card lay. Then let him choose another card. Again
remove the remaining cards in its orthogonals. When this
has been done once more there is only one card left, which he
• These series are symmetric; see discussion below.