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
Well done Michael Brooker, age 9, home educated and Chris
Waterhouse, Hethersett High School, Norfolk, for spotting the
Fibonacci sequence here and finding that there are 89 different
ways to lay the 10 paving slabs. Each slab is 2 foot by 1 foot and
a path has to be made 2 foot wide and 10 foot long from my back
door into my garden, without cutting any of the paving slabs. Chris
drew all the different patterns for 1,2,3,4 and 5 slabs and you can
try this for yourself. Here is Michael's solution:
I make the answer 89.
Having spotted the Fibonacci sequence it is necessary to PROVE
that it works to give the right answer for 10 slabs, or better
still that it woks for any number of slabs.
Supposing you have to lay n slabs. When you have laid (n - 2)
slabs, in any one of the many possible ways, you can lay 2 more,
making n altogether, laying them 'lengthwise' along the path (like
slabs 2 and 3 in the diagram). All these arrangements will give you
different patterns from the ones you get by laying (n - 1) slabs,
then laying just one more across the path (like slab 4 in the
diagram). This proves that the number of patterns for n slabs is
the sum of the number of patterns for (n - 2) and for (n -1) which
is the rule for generating the Fibonacci sequence.
Another way to do this problem is to work out the number of
arrangements with 5 pairs lengthwise and no slabs across the path
(just 1); the number with 4 pairs lengthwise and 2 across (15
ways); the number with 3 pairs lengthwise and 4 across (35 ways);
the number with 2 pairs lengthwise and 6 across (28 ways); the
number with one pair lengthwise and 8 across (9 ways); and the
number with no pairs lengthwise and all 10 across (1 way). Although
this is not so efficient as using the Fibonacci sequence it does
provide a challenge in itself and, for those who like to do it, we
have given the answers here so they can check their work.