Mathematical_Recreations-Kraitchik-2e

66 Mathematical Recreations
10. How TO GUESS A NUMBER OF THREE DIGITS (Rose­
Innes). Denote the number by ABC, where A, B, and Care
the digits. Ask for the remainders from division by 11 of the
numbers formed by the digits arranged ABC, BCA, CAB in
turn. Let the respective remainders be a, b, c.
Form the
sums a + b, b + c, c + a. If any of these is odd, increase or
decrease it by 11 so as to get a positive number less than 20.
Then divide each of these numbers by 2, and the resulting
numbers are the digits A, B, C in order.
4. FIGURATE NUMBERS
Figurate numbers are of very ancient origin. Pascal's
"Treatise on Figurate Numbers" was published in 1665;
but Diophantus (about the second century A.D.) had already
solved many problems concerning them.
1. Consider the arithmetical progression beginning with
1 and with common difference i. Its terms are
1, 1 + i, 1 + 2i, 1 + 3i, ....
The terms of this progression are called the linear figurate
numbers, or the figurate numbers of the first dimension.
Here are the first few terms in these progressions:
i Linear figurate numbers General term
1 1 2 3 4 5 n
2 1 3 5 7 9 2n -1
3 1 4 7 10 13 3n - 2
4 1 5 9 13 17 4n - 3
5 1 6 11 16 21 5n - 4
i 1 1 + i 1 + 2i 1 + 3i 1 + 4i in - (i - 1)
2. Consider now the successive sums of the terms of these