### 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.