Orders Please

Programme Outline

The Wizard's calculator appears to be malfunctioning. Lisa works out that, while it doesn't work as one might expect, it is at least being consistent: when fed a calculation, it will always perform bracketed expressions first, then multiplication and division, and finally addition and subtraction.

Start to 01:25 - Setting the scene Lisa and the Guard run to help the Wizard, who's just been blown up by his latest mechanical contrivance.
01:25 to 03:48 - The 'Poke-It Calculator' The Wizard describes his 'Poke-It Calculator', similar to the electronic pocket calculator. He's trying to run it in by doing calculations to produce the numbers from 1 to 20. Unfortunately, the calculator objects to being fed any number other than 4. Having demonstrated the workings of the machine, he leaves Lisa to it.
03:48 to 04:22 - The Mathematical Encyclopaedia The Encyclopaedia opens to page 'more than one', and proceeds to describe how large numbers have always presented problems. There are still situations where we don't bother counting lots of things – the number of trees in a forest, for example.
04:22 to 06:05 - Progress Lisa has managed to fill in a fair chunk of the table, including 17: 4x4+4÷4=17. Unfortunately, she's having trouble with 2. 4+4÷4=5, according to the machine. Lisa thinks she knows why: it's doing the division first, rather than working from left to right.
06:05 to 08:45 - Brackets There's still a pair of tiles left: brackets. These force the machine to do one part of a calculation before another one, which finally allows them to do (4+4)÷4=2. A further example demonstrates the order of operations: brackets first, then multiplication and division, and finally addition and subtraction.
08:45 to 09:33 - Another example The Wizard explains that he really wanted to solve a completely different set of puzzles, such as the calculation (3+2)x4+1=21. He heads off to have another try with the machine.
09:33 to End - Substitution Lisa and the Guard have completed the table - they've made all the numbers up to 20 using just the operators and the number 4. Now the Wizard wants to feed his calculation in, but since it uses numbers other than 4, it might blow up the machine. Lisa suggests using the existing calculations, surrounded by brackets, in place of the Wizard's numbers. Instead of the number '3', they use '4-4÷4', and so on. The result is a monster equation, which the calculator finds too demanding. It asks for a cup of tea to help it along.