problem Favourite Tens Age 16 to 18 Challenge level When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?
problem Favourite Negative power Age 14 to 16 Challenge level What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
problem Favourite Telescoping series Age 16 to 18 Challenge level Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.
problem Favourite Power quady Age 16 to 18 Challenge level Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
problem Favourite Climbing powers Age 16 to 18 Challenge level $2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?
problem Favourite Sums of squares Age 16 to 18 Challenge level Can you prove that twice the sum of two squares always gives the sum of two squares?
problem Favourite How many solutions? Age 16 to 18 Challenge level Find all the solutions to the this equation.
problem Favourite Perfectly square Age 14 to 16 Challenge level The sums of the squares of three related numbers is also a perfect square - can you explain why?
problem Favourite Giants Age 16 to 18 Challenge level Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?
problem Rachel's problem Age 14 to 16 Challenge level Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!