Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square. Three of the numbers that he found are a = 18530, b=65570, c=45986. Find the fourth number, x. You could do this by trial and error, and a spreadsheet would be a good tool for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then focus on Q^2-R^2=b-c which is known. Moreover you know that Q > sqrtb and R > sqrtc . Use this to show that Q-R is less than or equal to 41 . Use a spreadsheet to calculate values of Q+R , Q and x for values of Q-R from 1 to 41 , and hence to find the value of x for which a+x is a perfect square.

Ordered Sums

Stage: 4 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. For example, a(4) = 5 because:

 4 = 2 + 2 2 + 1 + 1 1 + 2 + 1 1 + 1 + 2 1 + 1 + 1 + 1.

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.