Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

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?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

Spotting the Loophole

Age 14 to 16 Challenge Level:

In order to solve this problem you will need to think about what makes a loop closed instead of open.

In two dimensions the overall changes in the x direction around the loop must cancel out and the overall changes in the y directions around the loop must also cancel out. You can think about these two directions separately if you cannot quickly spot how to form a closed loop.

Visually you can use lots of intuitive shortcuts, such as that 'long' arrows may need multiple 'short' arrows to close up into a loop.

Of course, when you think that you have spotted a loop you will need to check carefully to see if you are correct by adding up the vectors exactly.