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Building on the solution to Problem 1, the following diagram combines
that answer with the chequered pattern:
You can see that there are 2 white and 2 black triangles covered in each
case. This pattern was first noted in a solution to an Year 8-9 question
where the Problem 2 was posed. We will let the Hope and Dream
Team explain.
Do: First of all, we coloured half of the
triangles like this:
After working out a few answers, we suddenly
found out that there could only be three counters in the white triangles
and three counters in the black triangles.
The team went on to use this idea to find a number of different ways
of placing the counters. Then, we had an email from them in which one of
the team said:
Actually, I was a little bit confused when I
got this magical rule, I really didn't know why it was like that. 
Have you got the answer? I think it should have a connection with
symmetry. It's a symmetry so the numbers of them must be the same. Then
we have to put 3-black and 3-white-counters in the hexagon. I don't know
whether it's right or wrong.
I hope that you can put it on the webpage, and share the idea with other
teams if possible! I'd like to leave my E-mail to them. My E-mail
Address is mystery968@hotmail.com
- hopefully they can exchange ideas with me! 
So ... if you have a moment, please think about this very special
idea, and if you can tell the Hope and Dream Team why their idea is
correct, they will be really pleased to hear from you. But, when you do
email them, please send a copy to hexagonia@naturalmaths.com.au
... because we would really like to know the answer too!
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