# Maths item of the month

### April 2015

Consecutive Fibonacci Squares

The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34 ... (each number is the sum of the previous two numbers in the sequence and the first two numbers are both 1).

The sums of the squares of some consecutive Fibonacci numbers are give below:

1^{2} + 1^{2} = 2

3^{2} + 5^{2} = 34

13^{2} + 21^{2} = 610

Is the sum of the squares of consecutive Fibonacci numbers always a Fibonacci number?

### March 2015

4 points, 2 lengths

In how many ways can you arrange 4 distinct points in the plane so there are exactly two different distances amongst the 6 pairs?

e.g. if the four points are at the corners of a square then the four sides are the same length and the two diagonals are the same length.

### February 2015

1, 2, 3, 4

Find two quadratic functions f(*x*), g(*x*) so the equation f(g(*x*)) = 0 has the four roots *x* = 1, 2, 3, 4.

Is it possible to find three quadratic functions f(*x*), g(*x*), h(*x*) so the equation f(g(h(*x*))) = 0 has the eight roots *x* = 1, 2, 3, 4, 5, 6, 7, 8?

### January 2015

Happy 2015: A Triple of Triples

2015 is the product of 3 distinct primes: 5×13×31

2014 and 2013 are also the product 3 distinct primes.

Can you find a smaller triple (*n*, *n*+1, *n*+2) where *n*, *n*+1 and *n*+2 are all the product of 3 distinct primes?

Are there any quadruples (*n*, *n*+1, *n*+2, *n*+3) where *n*, *n*+1, *n*+2 and *n*+3 are all the product of 3 distinct primes?