A couple of problems

Here’s a couple of problems that were shared with me recently, that I will be taking along to MathsJam this month.

1) Imagine 6 people standing in a row facing you, with numbers on their shirts: 1 2 3 4 5 6
They shuffle themselves in the following way: the person on the far left stands in the middle, and then the next person stands on his right, the next person stands on his left, the next person stands on the right end of the line, and so on, so you can now see the people in the order 6 4 2 1 3 5.

Then everyone repeats the shuffle again… and again… how long until they are back in the original order? What if we started with a different number of people?

2) Take a circle, and draw a chord AB. Then choose another point C on the circle to create the chord BC. Then choose another point D to create the chord CD. Then construct the line from D parallel to AB; this intersects the circle again at E. Then construct the line from E parallel to BC; this intersects the circle again at F. Finally, construct the line from F parallel to CD; this intersects the circle again at G. What can you say about point G?

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