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?

Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Indigo Interior

Stage: 4 Short Challenge Level:
Let the centre of the circle be $O$ and let $A$ and $B$ be corners of one of the shaded squares, as shown.

As the circle has area $\pi$ square units, its radius is $1$ unit. So $OB$ is $1$ unit long.

Let the length of the side of each of the shaded squares be $x$ units.
By Pythagoras's Theorem, $OB^2 = OA^2 + AB^2$, that is $1^2 = (2x)^2 + x^2$.
So $5x^2=1$. Now the total shaded area is $8x^2 = 8 \times \frac{1}{5} = 1 \frac{3}{5}$ square units.

This problem is taken from the UKMT Mathematical Challenges.