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Just Opposite

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

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Corridors

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

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Hypotenuse Lattice Points

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

Ladder and Cube

Age 14 to 16 Challenge Level:
A cube (1 metre by 1 metre by 1 metre) has one face on the ground and one face pushed up against a wall.

A 4 metre long ladder is leaning against the wall, just touching the cube.




How high is the top of the ladder above the ground?