### 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.

# Nested Square Roots

##### Age 14 to 16 ShortChallenge Level

Squaring both sides
Let $x=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}}$

\begin{align}x^2&=\left(\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}}\right)^2\\ &=3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}\\ &=3+2x\end{align}

So $x^2-2x-3=0\Rightarrow (x-3)(x+1)=0$, so $x=3$ or $x=-1$.

Since $x$ is positive, $x=3$.

Check: $\sqrt{3+2\times(3)}=3$

'Working it out' using a sequence
Let $x=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{...}}}}}$

We can approximate $x$ by imagining that we are going to find that large square root as a calculation.

We can start by finding $\sqrt{3}$, and then double it, add $3$, and find the square root of that, and so on.

So if we let $a=\sqrt{3}$, and then $b=\sqrt{3+2\times a}=\sqrt{3+2\sqrt{3}}$, then $c=\sqrt{3+2\times b}=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3}}}$, and $d=\sqrt{3+2\times c}=\sqrt{3+2\times\sqrt{3+2\times\sqrt{3+2\times\sqrt{3}}}}$, then the sequence $a, b, c, d, ...$ will get closer and closer to $x$.

Using a calculator, or a spreadsheet, $a=1.73205$, $b=2.54246$, $c=2.84340$, $d=2.94734$, $e=2.98239$, $f=2.99413$, ...

It looks like they are getting closer and closer to $3$, so it looks like $x=3$. If we got to $3$ in our sequence, then the next number in the sequence would be $\sqrt{3+2\times 3}$, which is $\sqrt{3+6}=\sqrt{9}=3$. So if the sequence ever gets to $3$, it will stay at $3$ forever, which strongly suggests that $x=3$.

You can find more short problems, arranged by curriculum topic, in our short problems collection.