You may also like

problem icon

In Particular

Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11. Then find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers.

problem icon

For What?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

problem icon

Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Triangles Within Squares

Age 14 to 16 Challenge Level:
There is a very strong connection between this problem and the "Sequences and Series" problem.

Can you see the rectangles made from two triangular numbers in the square?

Can you explain why there is always one left over?