### 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.

# Black and White Socks

##### Age 14 to 16 ShortChallenge Level

There are $20+n$ socks in the drawer altogether, so $\dfrac{n}{20+n}=\dfrac{1}{n}$.
$$\begin{split}&\frac{n}{20+n}=\frac{1}{n}\\ &\Rightarrow n=\frac{1(20+n)}{n}\\ &\Rightarrow n^2=20+n\\ &\Rightarrow n^2-n-20=0\\ &\Rightarrow (n+4)(n-5)=0\\ &\Rightarrow n+4=0\hspace{3mm}\text{or}\hspace{3mm} n-5=0\\&\Rightarrow n=-4\hspace{7mm}\text{or}\hspace{3mm}n=5\end{split}$$ $n$ must be positive as there cannot be a negative number of white socks, therefore there are $5$ white socks in the drawer.