000RM.dvi

724 Figurate numbers

Exercise

1.Prove that every hexagonal number is a triangular number.

2.Find two triangular numbers (apart from 1) that are squares.

3.Find a pentagonal numbers (apart from 1) that is also a square.

4.It is known that there is only one value ofnfor which

12 +2^2 +3^2 +···+n^2

is a square. What is this value?

5.Ramanujan’s house number.

6.An unidentified country has 7-digit population – and everyone has

been given a National ID Number, sequentially from one, allocated

by no identifiable logic.

The Censure Minister has chosen three names at random, and is

finding their ID number on the computer. When the first number

appears on the screen, the Government’s mathematical whiz-kid

informs the Minister that there is precisely a 50-50 chance that the

other two numbers will both be less than the one just displayed.

What is the population, and what is the first number?^1

(^1) Problem 2585,JRM, 31 (2002–2003) 71.