### 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?

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?

### 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?

# Lost on Alpha Prime

##### Age 14 to 16 Challenge Level:

On the planet Alpha-Prime the city of Omega is built on a grid system in three dimensions. This means that every corner on the grid is defined by three coordinates such as (3,5,6). At the origin (0,0,0) is the city zoo, from which a metagiff (looks a bit like a giraffe but has a deadly bite) has escaped. It is your job to find him.

The city sheriff has installed a sophisticated tracking system which will tell you how far the metagiff is away from any coordinates you enter. It does not give the shortest (diagonal) distance but the distance in blocks, and you do not have to be at the point whose coordinates you enter.

For example, if the metagiff is at (1,1,1) and you key in (3,3,3) the tracking system will tell you that the point is 6 blocks from the metagiff (2 blocks along, 2 blocks up and 2 blocks across).

You need to get to the metagiff as quickly as possible so you want to enter as few coordinates as possible in order to locate its position. The system works on the principle of using the corner nearest the metagiff for its point of reference.

Can you give a reliable strategy for choosing coordinates that will locate the metagiff in the minimum number of moves?