I start with a red, a blue, a green and a yellow marble. I can
trade any of my marbles for three others, one of each colour. Can I
end up with exactly two marbles of each colour?
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
Well explained solutions to this
problem came from Anna of West Flegg Middle School, Avishek, Mark,
Thomas, James and Ricardo of Simon Langton Boys' Grammar School,
and Nodoey from Singapore.
I start with three marbles, 1 red, 1 green and 1 blue. I can
trade any one marble for two others, one each of the other two
colours. However many times I do this it is impossible to have a
difference of 5 between the number of red and blue marbles because
the difference between these two numbers is always even.
Suppose I trade one marble of any colour (say blue) for two
others, then I will have two reds and two greens and no blues. If I
then trade one of my marbles I will have one blue, one of another
colour and three of the third colour. Each time I trade I receive
one more marble in total but, more importantly I have alternately
an even number of each colour then an odd number of each colour,
then an even number, and so on. For example:
R B G
1 1 1
2 0 2
1 1 3
0 2 4
1 3 3
So the number of blue and the number of red marbles are always
both even or both odd (this applies to every colour). So the
difference between the number of reds and blues is always even and
can never be five.
A further challenge:
You might like to improve on this
solution by using vectors in the proof. You can learn about adding
vectors by reading the article A Knight's
Journey and apply the ideas to the