Chapter 10: Brainteasers 377
Ones and zeros
Show that any natural number has a multiple whose
decimal representation only contains the digits 0 and 1.
For example, if the number is 13, we get 13=1001.
(Thanks to idgregorio.)
Solution
Consider then+1 numbers 1, 11, 111, 1111, etc. Two of
them will be congruent modulon. Subtract the smaller
one from the bigger one. You will get a number contain-
ing only 0s and 1s.
Bookworm
There is a two-volume book set on a shelf, the volumes
being side by side, first then second. The pages of each
volume are two centimeters thick and each cover is two
millimeters thick. A worm has nibbled the set, perpen-
dicularly to the pages, from the first page of the first
volume to the last page of the second one. What is the
length of the path he has nibbled?
(Thanks to Vito.)
Solution
Just four millimeters. Think about where the first page
of the first volume and the last page of the second
volume will be relative to each other.
Compensation
A number of quants are at dinner, and start dis-
cussing compensation. They want to calculate the