Mathematical_Recreations-Kraitchik-2e

178 Mathematical Recreations
I: (the identity)
P: (01) (23) (45) (67)
Q: (02) (13) (46) (57)
R: (03) (12) (47) (56)
S: (04) (15) (26) (37)
T: (05) (14) (27) (36)
U: (06) (17) (24) (35)
V: (07) (16) (25) (34)
I' :
pI:
Q':
R':
S':
T':
U ' :
V':
(the identity)
(aD) (bC) (cB)
(aC) (bD) (cA)
(ab) (cd) (AB)
(aB) (bA) (cD)
(ae) (bd) (AC)
(ad) (be) (AD)
(aA) (bB) (cC)
(dA)
(dB)
(CD)
(dC)
(BD)
(BC)
(dD)
For the first column of our square we use the magic line
of 8:
Oa 2A 4b 6B Ie 3C 5d 7D.
For the downward main diagonal we use
Oa 3B 6d 5C 5b 6A 3c OD.
We form the other columns of the square by applying to the
first column the substitution on the digits and the substitution
on the letters that will give the correct element in the
main diagonal.
11. LATIN SQUARES
A Latin square is formed with n sets of the numbers 1 to n
arranged in the cells of a square of order n in such a way that
1 2 3 4 5 1 2 3 4 5
2 3 4 5 1 3 4 5 1 2
3 4 5 1 2 5 1 2 3 4
4 5 1 2 3 2 3 4 5 1
5 1 2 3 4 4 5 1 2 3
FIGURE 54.
no orthogonal contains the same two numbers. A diagonal
Latin square is one in which the main diagonals are also without
repetitions. The squares in Figure 54 are Latin squares,
of which the second is diagonal.