### Upsetting Pitagoras

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

### Lunar Leaper

Gravity on the Moon is about 1/6th that on the Earth. A pole-vaulter 2 metres tall can clear a 5 metres pole on the Earth. How high a pole could he clear on the Moon?

### Seriesly

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

# Universal Time, Mass, Length

### Why do this problem?

This problem places units in an interesting context which can lead on to other problems. It requires relatively little technical skill, allowing students to focus solely on the process of changing units.

### Possible approach

Discuss the problem as a class. Is everyone familiar with each of the units mentioned? Throughout, encourage numerical order of magnitude checks of the answers.

### Key questions

• Before changing units, do you expect the answer to be numerically larger or smaller after the change?
• Relative to the natural scales of the universe are we small, heavy, slow?

### Possible extension

Consider the meaning of the statement 'the natural scales for the universe'. Relate these units to everyday objects. Work out the ratios for a person / a proton/ the sun / the galaxy / a neutron star in the natural units. How do they compare G, c, and h?

### Possible support

You could treat this simply as a change of units question.