Coordinates

problem
A Cartesian puzzle

Age

7 to 11

Challenge level

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.
problem
A tilted square

Age

14 to 16

Challenge level

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
problem
Square pair circles

Age

16 to 18

Challenge level

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.
problem
Parabella

Age

16 to 18

Challenge level

This is a beautiful result involving a parabola and parallels.
problem
Hypotenuse lattice points

Age

14 to 16

Challenge level

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?
problem
Four points on a cube

Age

16 to 18

Challenge level

What is the surface area of the tetrahedron with one vertex at O the vertex of a unit cube and the other vertices at the centres of the faces of the cube not containing O?
problem
Plane to see

Age

16 to 18

Challenge level

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.
problem
Napoleon's hat

Age

16 to 18

Challenge level

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?
problem
Cushion ball

Age

16 to 18

Challenge level

The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?
problem
Ladder and cube

Age

14 to 16

Challenge level

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?