¿ìè¶ÌÊÓƵ

#25 Triangles in circles

Set by Sam Hartburn

If you have six equally spaced points on a circle and join three of them, you can make three different triangles. (Two triangles are counted as the same if one is a mirror image or rotation of the other.)

How many different triangles can you make from seven equally spaced points?

And how many can you create from 12 equally spaced points?

Solution next week

 

#24 Coin flips

Solution

A triangle with six coins can be made to point downwards by moving two of the corner coins to the opposite side, for example:

For a triangle with four rows, we need to move the three corner coins to invert the triangle. If we consider triangles of all sizes (starting from a triangle with two rows), the number of coins in each triangle is 3, 6, 10, 15, 21, 28… and the number of coin moves needed to flip each triangle upside down is 1, 2, 3, 5, 7, 9 and so on. The general rule is to divide the number of coins by 3, ignoring any remainder. For larger triangles, instead of moving single coins, you need to move triangles of coins from each corner.