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Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

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Lunar Leaper

Gravity on the Moon is about 1/6th that on the Earth. A pole-vaulter 2 metres tall can clear a 5 metres pole on the Earth. How high a pole could he clear on the Moon?

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Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

Overarch 1

Stage: 5 Short Challenge Level: Challenge Level:1


Note: you may first like to play the Tower Rescue game which will give you a feeling for the mechanics of building the structures described in this problem.

A uniform square tile of side 20cm is placed on the ground and another identical tile placed on top so that it overhangs as far as possible without toppling over, as in the following diagram:

Clearly the top tile can be placed as far as half of the way along the base tile without toppling over, so that the overhang will be of length 10cm.

These two tiles are then joined in this configuration and placed on top of a third tile so that the whole construction just balances. What will the size of the overhang be in this case?

In a similar way, find the maximum overhang when three tiles are balanced in this way on top of a fourth tile.

For a more challenging extension of this please see Overarch 2.

Investigation: Try making a tower of CDs or other similar objects in this fashion. How will the fact you are using real world objects affect your answers? Think carefully about the physical effects at play and how you could model these mathematically.

Click here for a poster of this problem.