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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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Find the vertices of a pentagon given the midpoints of its sides.

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Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.


Stage: 4 Challenge Level: Challenge Level:1

No one sent us a complete solution to this problem, but here is the start of a solution which shows some good mathematical thinking and a systematic way of working.

We decided to investigate what happens if you keep the $x$ coordinate the same and change the $y$ coordinate. Here is our table to show the number of squares and gridlines crossed when the $x$ coordinate is $5$.

We noticed a pattern for the number of squares crossed, which works for every $y$ coordinate we tried except $5$. If you add together the $x$ and $y$ coordinates and take away 1, you get the number of squares crossed. This is because for example to get from (0,0) to (5,3) you go along 5 squares and up 3 squares, meaning that you travel through 8 squares altogether, but that counts the corner square twice so you need to take away 1.
It doesn't work for (5,5) because you go diagonally through the corners of the squares instead of cutting through the edges.

Can anybody build on this thinking and explain the patterns found?