SOLUTIONS
Of course, at a circular table, A will be next to the man at the
end of the line.
I first gave this problem for six persons on ten days, in the
Daily Mail
for the 13th and 16th October, 1905, and it has
since been discussed in various periodicals by mathematicians. Of
course, it is easily seen that the maximum number of sittings for
n
persons is ^
^
^ ways. The comparatively easy method
for solving all cases where
n
is a prime + 1 was first discovered by
Ernest Bergholt. I then pointed out the form and construction of
a solution that I had obtained for 10 persons, from which E. D.
Bewley found a general method for all even numbers. The odd
numbers, however, are extremely difficult, and for a long time
no progress could be made with their solution, the only numbers
that could be worked being 7 (given above) and 5, 9, 17, and 33,
these last four being all powers of 2 + 1. Recently, however,
(though not without much difficulty), I discovered a subtle method
for solving all cases, and have written out schedules for every
number up to 25 inclusive. The method is far too complex for me
to explain here, and the arrangements alone would take up too
much space. The case of 11 has been solved also by W. Nash.
Perhaps the reader will like to try his hand at 13. He will find
it an extraordinarily hard nut.
91.—
The Five Tea-Tins.
There are twelve ways of arranging the boxes without consider-
ing the pictures. If the thirty pictures were all different the answer
would be 93,312. But the necessary deductions for cases where
changes of boxes may be made without affecting the order of
pictures amount to 1,728, and the boxes may therefore be arranged,
in accordance with the conditions, in 91,584 different ways. I will
leave my readers to discover for themselves how the figures are to be
arrived at.
92.—
The Four Porkers.
The number of ways in which the four pigs may be placed in the
thirty-six sties in accordance with the conditions is seventeen, includ-
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