Inspirations from Easter Conferences - September 2006, Stage 4&5

Problems

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Take Ten Sticks

Stage: 4 Challenge Level: Challenge Level:1

Take ten sticks in heaps any way you like. Make a new heap using one from each of the heaps. By repeating that process could the arrangement 7 - 1 - 1 - 1 ever turn up, except by starting with it?

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Odd Stones

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

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There's Always One Isn't There

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Take any pair of numbers, say 9 and 14. Take the larger number, fourteen, and count up in 14s. Then divide each of those values by the 9, and look at the remainders.

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Number Chains

Stage: 5 Challenge Level: Challenge Level:1

Find all the periodic cycles and fixed points in this number sequence using any whole number as a starting point.

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Spirostars

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

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Spiroflowers

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Analyse these repeating patterns. Decide on the conditions for a periodic pattern to occur and when the pattern extends to infinity.