Indices

problem

Really Mr. Bond

Age

14 to 16

Challenge level

Can you explain what's happening with these numbers?
problem

Big, Bigger, Biggest

Age

16 to 18

Challenge level

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
problem

Powerful Factors

Age

16 to 18

Challenge level

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.
problem

Cube Roots

Age

16 to 18

Challenge level

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.
problem

Pythagoras Mod 5

Age

16 to 18

Challenge level

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.
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.
article
More Sums of Squares

Tom writes about expressing numbers as the sums of three squares.
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
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
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.