Indices

problem
How many solutions?

Age

16 to 18

Challenge level

Find all the solutions to the this equation.
problem
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
Growing

Age

16 to 18

Challenge level

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)
problem
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
More mods

Age

14 to 16

Challenge level

What is the units digit for the number 123^(456) ?
article
Card shuffle

This article for students and teachers tries to think about how long would it take someone to create every possible shuffle of a pack of cards, with surprising results.
article
Learn about number bases

We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.
article
Modulus arithmetic and a solution to dirisibly yours

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
article
More sums of squares

Tom writes about expressing numbers as the sums of three squares.
article
Sums of squares and sums of cubes

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.