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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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Be Reasonable

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

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The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

Road Maker 2

Age 16 to 18 Short Challenge Level:

Why do this problem?

Provide training in conjecture, mathematical analysis and proof. This difficult problem requires students to realise that it is possible to use counting (a discrete process) to somehow categorise different paths. This gives the power to make general statements.

Possible Approach?

First students need to calculate the end points of a few simple roads. Although there are rational and irrational endpoints, group discussion should lead to the conclusion that root 3 should be involved in the irrational endpoints in some way.

Key Questions?

If you have a valid road, how can its endpoint change with the addition of a single tile? Two tiles?
Can you make any 'families' of roads which are similar, yet of different lengths? Can you create expressions for the lengths of these families of roads?