COUNTER CROSSES.
(
Combination and Group Problems)
All that we need for this puzzle is nine counters, numbered 1, 2, 3, 4,
5, 6, 7, 8, and 9. It will be seen that in the illustration A these are
arranged so as to form a Greek cross, while in the case of B they form a
Latin cross. In both cases the reader will find that the sum of the
numbers in the upright of the cross is the same as the sum of the
numbers in the horizontal arm. It is quite easy to hit on such an
arrangement by trial, but the problem is to discover in exactly how many
different ways it may be done in each case. Remember that reversals and
reflections do not count as different. That is to say, if you turn this
page round you get four arrangements of the Greek cross, and if you turn
it round again in front of a mirror you will get four more. But these
eight are all regarded as one and the same. Now, how many different ways
are there in each case?
[Illustration:
(1) (2)
(2) (4) (5) (1) (6) (7)
(3) (4) (9) (5) (6) (3)
(7) (8)
A (8) B (9)
]
Answer:
Let us first deal with the Greek Cross. There are just eighteen forms in
which the numbers may be paired for the two arms. Here they are:--
12978 13968 14958
34956 24957 23967
23958 13769 14759
14967 24758 23768
12589 23759 13579
34567 14768 24568
14569 23569 14379
23578 14578 25368
15369 24369 23189
24378 15378 45167
24179 25169 34169
35168 34178 25178
Of course, the number in the middle is common to both arms. The first
pair is the one I gave as an example. I will suppose that we have
written out all these crosses, always placing the first row of a pair in
the upright and the second row in the horizontal arm. Now, if we leave
the central figure fixed, there are 24 ways in which the numbers in the
upright may be varied, for the four counters may be changed in 1 x 2 x 3
x 4 = 24 ways. And as the four in the horizontal may also be changed in
24 ways for every arrangement on the other arm, we find that there are
24 x 24 = 576 variations for every form; therefore, as there are 18
forms, we get 18 x 576 = 10,368 ways. But this will include half the
four reversals and half the four reflections that we barred, so we must
divide this by 4 to obtain the correct answer to the Greek Cross, which
is thus 2,592 different ways. The division is by 4 and not by 8, because
we provided against half the reversals and reflections by always
reserving one number for the upright and the other for the horizontal.
In the case of the Latin Cross, it is obvious that we have to deal with
the same 18 forms of pairing. The total number of different ways in this
case is the full number, 18 x 576. Owing to the fact that the upper and
lower arms are unequal in length, permutations will repeat by
reflection, but not by reversal, for we cannot reverse. Therefore this
fact only entails division by 2. But in every pair we may exchange the
figures in the upright with those in the horizontal (which we could not
do in the case of the Greek Cross, as the arms are there all alike);
consequently we must multiply by 2. This multiplication by 2 and
division by 2 cancel one another. Hence 10,368 is here the correct
answer.
