Imagine you have six different colours of paint. You paint a cube
using a different colour for each of the six faces. How many
different cubes can be painted using the same set of six colours?
Six points are arranged in space so that no three are collinear.
How many line segments can be formed by joining the points in
Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in
every possible order to give 5 digit numbers. Find the sum of the
In how many ways can a pound (value 100 pence) be changed into
some combination of 1, 2, 5, 10, 20 and 50 pence coins? There are
more than 4000 possibilities. Lots of people have sent lists of
some of the possibilities but nobody has cracked this one yet.
Perhaps people were put off by the length of the question but it
described how to solve the problem.
Solution to tough nut Small Change by John Lesieutre and
Tony Cardell, both 14, both State College Area High
The method we used to solve this problem was rather simple. The
number of ways to make n pence out of 1 and 2 pence coins is
(n/2)+1 is n is even and ((n+1)/2) if n is odd. This is given in
the hint, and is also obvious. A calculator program was then
written which checked all combinations of up to 2 fifty pence
coins, up to 5 twenty pence coins, up to 10 ten pence coins, and up
to 20 five pence coins, and then filled up the rest of the amount
needed to reach 100 pence with every combination of 1 and 2 pence
coins. There are a total of 3*6*11*21 (There can be 0 of a
coin!)=4158 different combinations to try. For each combination,
the program calculated the number of ways to finish the 100 pence
using 1 and 2 pence coins, and added this number to a variable E.
When all 4,158 combinations had been checked after 5 minutes and 15
seconds, E=4,562 and therefore this is the number of ways to make 1
pound out of 1, 2, 5, 10, 20, and 50 pence coins.