Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

Ladder and Cube

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?

Bendy Quad

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.

Adding to 400

Age 14 to 16 Short
Challenge Level

Answer: $192$, $96$, $64$ and $48$

first integer: $x$

second integer: $\frac12{x}$

third integer: $\frac13x\\$
fourth integer: $\frac14x$

$$\begin{align}x + \tfrac12x + \tfrac 13x + \tfrac 14x &= 400\\ \Rightarrow \tfrac{25}{12}x &= 400\\ \Rightarrow x &= 192\end{align}$$

Therefore the four numbers are, in order: $192$, $96$, $64$ and $48$.

This problem is taken from the UKMT Mathematical Challenges.

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

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Building Tetrahedra

Ladder and Cube

Bendy Quad

Adding to 400

Age 14 to 16 ShortChallenge Level

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Challenge Level