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From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

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Lap Times

Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the track.

Doesn't Add Up

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

Here is an 8 by 8 square that has been divided into four pieces:

Here is a rectangle made from the same four pieces:

You might like to try cutting out the shapes from a piece of paper.

It seems that the same pieces can make two shapes with different areas!
Can you explain where the extra area comes from?

Here are some questions you might like to consider:

Can other squares be split up and rearranged to make rectangles with a different area?

Are there other square/rectangle pairs where the areas differ by 1 square unit?

Is there a pattern in the sizes of squares that can be arranged in this way?