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### Number and algebra

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# 3D Treasure Hunt

*This problem follows on from Treasure Hunt, which you may wish to explore first.*

Imagine you are looking for treasure in a strange three-dimensional treasure island, where treasure can be hidden at any point on a grid defined by three coordinates.

When you enter a guess in the interactivity below, it tells you the shortest distance you would have to travel along grid lines to reach the treasure.

For example, if the treasure is at (1, 2, 3) and you enter (3, 3, 3), the interactivity will tell you that the treasure is a distance of 3 steps away (2 steps in the x direction, 1 in the y direction and 0 in the z direction).

**Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of moves?**

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Challenge Level

Imagine you are looking for treasure in a strange three-dimensional treasure island, where treasure can be hidden at any point on a grid defined by three coordinates.

When you enter a guess in the interactivity below, it tells you the shortest distance you would have to travel along grid lines to reach the treasure.

For example, if the treasure is at (1, 2, 3) and you enter (3, 3, 3), the interactivity will tell you that the treasure is a distance of 3 steps away (2 steps in the x direction, 1 in the y direction and 0 in the z direction).

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?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?