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.

Hiking the Hill

Age 14 to 16 Short Challenge Level:

Answer: $2\frac23$ miles per hour

Total distance and total time
Uphill: $6$ miles, $2$ miles/hour takes $3$ hours

Downhill: $6$ miles, $4$ miles/hour takes $1\frac12$ hours

Average speed: $\dfrac{6+6\text{ miles}}{3 + 1\frac12\text{ hours}} = \dfrac{12}{4\frac12}$ miles per hour
$=\frac{24}{9}=\frac83=2\frac23$ miles per hour

Weighted average
Uphill: 2 miles per hour
Downhill: 4 miles per hour
Takes twice as long to go up as down
$\Rightarrow$ spends twice as long going up as down
$\Rightarrow$ spends twice as long travelling at 2 miles per hour as at 4 miles per hour

$\therefore$ average speed $=\frac{2+2+4}{3}=\frac83 = 2\frac23$ miles per hour

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.