Mathematical_Recreations-Kraitchik-2e

Magic Squares 185
must take. You can then tell him that the sum of the numbers
selected is 34.
Symmetric magic series are those that contain only pairs
of complementary numbers. (Two numbers are complementary
if their sum is n 2 + 1, in this case 17.) All possible
symmetric magic series of order 4 will be formed by taking
all possible pairs of numbers not greater than 8, together
with their complements. Hence there are 8 ~ 7 = 28 distinct
symmetric magic series, of which 12 are also algebraic, and
so have already been listed. They are marked in the preceding
table with an asterisk. The other 16 are:
1 2 15 16
3 14
5 12
8 9
2 4 13 15
611
7 10
3 4 13 14
3 6 11 14
710
4 5 12 13
8 9
5 6 11 12
7 10
6 8 9 11
7 8 9 10
The remaining magic series, which are neither algebraic
nor symmetric, we shall call arithmetic. They are due merely
to the chance of arithmetical combinations and contain many
anomalies. There are the following 34:
1 5 13 15(0) 2 9 10 13(l) 4 7 8 15(8)
6 13 14 11 12(l) 11 12
7 11 15(0) 3 4 11 16 8 10 12(e)
9 10 14(l) 12 15 9 10 11(l)
11 13(lo) 5 11 15(0) 5 6 7 16(8)
10 11 12(l) 7 8 16(8) 8 15(8)
2 4 12 16(e) 9 15(0) 9 14
5 13 14 7 11 13(0) 10 13
6 10 16(e) 8 11 12 7 8 14(8)
6 12 14(e) 9 10 12(l) 9 13(0)
8 10 14(e) 4 6 8 16(8e) 6 7 8 13(8)
10 14(e)
Every magic series (of order 4) contains two large numbers
(> 8) and two small « 9), except those arithmetic series
marked (l) or (8) respectively. Likewise each magic series