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
Thank you Kang Hong Joo for this solution:
Now, let the flag split into three parts. In the first part, 5
colours can be put into it. In the second part, only 4 colours can
be put into it, as it cannot be the same colour as the first part.
In the third part, only 4 colours can be put into it, as it cannot
be the same colour as the second one yet it can be the same colour
as the first one. Thus, the number of tricolour flags are 5 x 4 x 4
= 80 with 5 available colours.
If there are 256 colours, by the same reasoning, the number of
tricolour flags possible are 256 x 255 x 255 = 16 646 400.
Helen Battersby Year 9 The Mount School York
explained the reasoning as follows:
For 2 colours there are only 2 possible patterns.
And for 3 colours there are 12, as given in the question.
Rearranging the 12 solutions given into ones beginning with the
same colour, I could see that each beginning had 4 different
combinations. So with 3 colours the calculation is 3 x 4 = 12.
Looking at 4 colours, I could see that from the logic in this, each
beginning (like the dots as shown) would have 9 different
combinations. There can be 4 beginnings as there are 4 colours, so
altogether there are 4 x 9 = 36 flags.
This suggests a rule$c \times (c - 1)^2$ where $c$
stands for the number of colours.
Well done all the following for your excellent work on Flagging:
Patrick Coleman, Scott Reynolds , St Peter's
College, Adelaide, Australia, Michael Swarbrick
Jones , Y7 Comberton Village College, Cambridgeshire,
Catherine Harrison, Joanne Barker, Rachael Evans, Sheila
Luk and Cheryl Wong , Year 9 The Mount
School York, P. Cresswell , City of Norwich School
and James Page , Hethersett High School,