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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?

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Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

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Isosceles Triangles

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Route to Infinity

Stage: 3 and 4 Challenge Level: Challenge Level:2 Challenge Level:2

Take some time to look at the route followed by the arrows in this diagram.
Then look away and describe the path to a friend.

If the pattern of arrows continues for ever, which point will the route visit immediately after (18,17)? Explain how you know.

How many points will be visited before the route reaches the point (9,4)?
Explain how you worked it out.

Which point will be the $1000^{th}$ to be visited?

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