Mathematical_Recreations-Kraitchik-2e

50 Mathematical Recreations
to set up the page numbers of a 288-page book if he does not
use any piece twice? Answer: 756.
17. Under the conditions of problem 16, how many pages
are there in a book in which 15,321 pieces of type have been
used to number the pages? Answer: 4,107.
18. With the digits 1, 2, 3, 4, 5 we can form 5! = 120 numbers
of 5 digits each, no one of which has any digit repeated.
What is the sum of these numbers?
Solution: When the numbers have been arranged OIle below
the other in order to add them, we find that each column
contains the same digits as every other, though in a different
order, and each column contains each of the five digits equally
often, that is, -4-Q. = 24 times apiece. Thus the sum of the
digits in each column is (1 + 2 + 3 + 4 + 5)24 = 15 ·24 =
360. When these sums are added in their proper positions,
we get 11,111 X 360 = 3,999,960.
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19. PERFECT SQUARES. Set IOn - 1 = 9a = b - 1, and
c = 8a + 1. The following expressions are then perfect
squares:
(1) 4ab + c = 4a(9a + 1) + 8a + 1 = (6a + 1)2 = 66· .. 72 =
444·· ·888·· ·9.
(2) (a - l)b+ c = 9a 2 = (333·· ·333)2 = 111·· ·0888·· ·9.