Mathematical_Recreations-Kraitchik-2e

302 Mathematical Recreations
numbers with different dominoes having the same sum.
Thus the two squares of order 3 in Figure 163 are ordinary
35 03 06 22 51
26 12 13 03 11 32 61 45 40
26 01 15 16 00 05 14 02 36 11 62 46 00 21 24
12 14 16 02 04 06 05 15 01 06 01 31 52 63 33
13 36 11 03 26 01 00 25 04 16 44 41 34 02 05
FIGURE 163. Domino Magic Squares.
magic squares, while those of order 4 and order 5 have repetitions
such as 1-3 and 0-4, and 3-5, 4-4, and 6-2.
Paillot gives the following arrangement in which the last
•.. I . "'1 ... . ....... • '1" . 1
:·:1 . : :1--. -·.1-·. :.: -
:::1:·: . 1 • ".1:::' .1
• I: : : : 1 ::: : : I:: • 1
.. 1:-: . .I::: :·:1
:::1::: . .1"'••• 1' . ".1
'1'. :':1:: .. 1:: :::1
FIGURE 164.
column can be omitted, leaving a
magic square of order 7 (Figure
164).
Lucas uses the term quadrille to
describe a certain arrangement of
the whole set of dominoes in which
the pairs of rows consist of sets of
squares composed of four equal
Domino half-dominoes. We give four distinct
examples, each of which gives
Magic Square with One
Border of Blanks.
rise to many variations by permuting
the numbers 0, 1,2,3,4,5,6. (Figure 165). Delannoy
found three other distinct forms (Figure 166).
2. PERMUTATIONAL GAMES
1. THE 15 PUZZLE. Rather than attempt a detailed
description of this familiar puzzle, we shall consider it to be
equivalent to a set of 15 equal cubes, numbered from 1 to 15,
placed in a shallow box that just holds 16 such cubes. The
empty space may be in any of the 16 possible positions, but