Variable Behaviour

Programme Outline

The Wizard has promised to make onion soup for the village. Unfortunately, he has only three onions, and his calculations suggest he'll need exactly 29. The Guard, ever resourceful, is attempting to raise onion money through street theatre. Lisa, on the other hand, investigates two machines the Wizard has left lying around. The first appears to multiply the number of onions, the multiplication factor being based on an input on the front of the machine. By compiling a table of results and drawing a graph, Lisa works out exactly what the machine is doing. The second machine adds rather than multiplies; by finding the right combination of inputs, Lisa and the Guard manage to turn the Wizard's three onions into 29.

Start to 01:15 - Setting the scene The Guard greets Lisa with a bit of mime, which he's taken up in an effort to raise funds for the Wizard. Unfortunately, he's not much good. The Wizard needs money to buy onions, since he has only three and needs 29 to make some soup. He's off experimenting, but has left behind two large machines.
01:15 to 02:10 - The number machine The first machine takes a number tile as an input on the front. The operator then drops an object in the top, and a number of similar objects are ejected. When the Guard enters '2' as the input and drops a pepper in, he receives five peppers for his trouble. He leaves Lisa to work out what's going on, while he heads off to devise a back-up plan.
02:10 to 02:45 - The Mathematical Encyclopaedia The Encyclopaedia opens to page 'n'. Book pages are traditionally numbered in sequence, and this page is no different. The trouble is, it hasn't yet decided which number it wants to be, so it could go anywhere.
02:45 to 07:50 - The first machine Lisa has completed her investigations of the first machine. She dropped in just one pepper at first, and built a table of results showing how the number of peppers the machine spat out changed with the input number on the front. By drawing a graph, she can predict the input value needed for a desired output. By her reckoning, the machine is mapping the input number of peppers (P) and the input value (i) like so: P --> (i + 3)P Thus, if she puts the zero on the front and feeds in two peppers, she receives (0+3)x2 = 6 peppers as output. Similarly, an input of -3 turns the machine into a pepper-eating box: no peppers are output. Unfortunately, Lisa and the Guard can't find a combination of inputs which yields the required 29 peppers. It's time to investigate the second box.
07:50 to 8:30 - Variables The Wizard introduces the term 'variable', and reinforces what Lisa is discovering: that the number on the front of the box affects the behaviour of the box.
08:30 to 11:40 - The second box Lisa has found that the number of peppers the second machine chucks out is the number she puts in, plus the number on the front, plus two. That is, the box is doing: C --> C + j + 2 where C is the number of peppers input, and j is the input number. So they need to work out which numbers to use as inputs for either box, and how many peppers to input. After much deliberation, the Guard spots that if i = 5, j = 3, and they start with three peppers, they should end up with 29.
11:40 to end - Onions Just when Lisa and the Guard have worked out what to do, and are wondering if what works for peppers will also work for onions, the Wizard enters. They manage to persuade him to entrust his three good onions to them, and they start to feed the machines. Eventually, they have 29 onions, and the Wizard can make his soup. Lisa and the Guard wouldn't mind a spot of soup after their exertions, but of course, for two more people he'll need more onions...