2 out of 3
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
2 out of 3
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
2 out of 3
If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?
2 out of 3
Explore a number pattern which has the same symmetries in different bases.
2 out of 3
A point moves around inside a rectangle. What are the least and the greatest values of the sum of the squares of the distances from the vertices?
2 out of 3
Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
2 out of 3
A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?
2 out of 3
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
2 out of 3
How many tours visit each vertex of a cube once and only once? How many return to the starting point?
2 out of 3
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
