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?

Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Collatz-ish

Stage: 4 Short Challenge Level:
See all short problems arranged by curriculum topic in the short problems collection

The first term of a sequence of positive integers is $6$. The other terms in the sequence follow these rules:

if a term is even then divide it by $2$ to obtain the next term;
if a term is odd then multiply it by $5$ and subtract $1$ to obtain the next term.

For which values of $n$ is the $n$th term equal to $n$?

If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.

This problem is taken from the UKMT Mathematical Challenges.