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# Overarch 1

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### Overarch 2

### Stonehenge

### Maximum Flow

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Age 16 to 18

ShortChallenge Level

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.

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Explain why, when moving heavy objects on rollers, the object moves twice as fast as the rollers. Try a similar experiment yourself.

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.