As well as the balloons, cake and crisps, we had some party games, well, activities for our guests. They could have a go at the Turing test, at breaking enigma codes, and my contribution was for them to be a small part in a computer, a 2-bit adder. The aim was to show how the innards of a computer processor are little more than a whole load of switches and it doesn’t matter much (at least to a mathematician like Turing) what these switches are. I hoped this would help show that computers are more than black boxes, and help put add some context to what electronic computers were about when Turing was working. (And, yes, I do know that it was Shannon not Turing who developed the theory.)
So, it starts with a switch that can turn another switch on and off. Here’s a simulation of one which uses a transistor to do that. If you click on that link a java window should open that shows a simple circuit. The input on the left is at a Low voltage, the output is at a low voltage. Click on the input to set it to a High, and it will turn on the transistor, connecting the output to the high voltage source, so the output goes High. So by setting the input to high voltage (presumably by pressing a switch) you can set the output to high voltage. You’re allowed to be under-impressed at this stage. (Make sure you close any windows or browser tabs opened by that link, leaving them open might cause later examples not to work)
Turing didn’t have had access to transistors. At the time he worked these switches were electromechanical relays, a physical spring-loaded switch that was closed by the magnetic attraction between a coil and permanent magnet when a current ran through the coil. Later, vaccuum tube valves were available to replace these, but much to Tommy Flowers chagrin, Turing wasn’t at all interested in that. For mathematicians the details of the switching mechanism are a distraction. By not caring, maybe not even knowing, about the physics of the switch Turing was saved from worrying about a whole load of details that would have been out of date by the 1960s; as it is his work is still relevant today. This illustrates my favourite feature of Mathematics, which is that maths is the only subject where it is best not to know what you are talking about.
Back to this thing of turning a voltage signal high or low by turning a voltage high or low.That may be underwhelming, but put two of these next to each other and something interesting happens: the output will only be High if both the inputs are. In other words the output is High if both input 1 AND input 2 are high. That’s mathematics: a simple logic calculation. You can try it out in the simulation. You can also try other arrangements that show an OR logic calculation and an XOR calculation, that is an exclusive OR, the output is high if on of input 1 or input 2 is high but not both. We call these circuits logic gates. Remember to close all windows and browser tabs when going from one simulation to another.
This is where we leave electronics and start using the audience. My colleague and I each had a flag and we gave everyone in the audience a flag. We were the inputs, they had to be logic gates; they had to raise their flag if she AND I both raised ours, or if she OR I had a flag up, or if she or I, but not both of us raised a flag (the XOR calculation).
The next trick was to show how these logic calculations relate to adding numbers together: A+B = S. First, of course, the numbers must be represented as binary with a low voltage/flag down equivalent to the digit 0 and high voltage/flag up equivalent to the digit 1. And we have to do the addition one digit at a time, starting from the units. Adding the first digit, the units, is easy enough. 0+0 = 0, 0+1=1, 1+0=1, 1+1=0 with 1 to carry. Think of that input 1 + input 2 = output, where the output can either be the digit for the sum or the digit to carry. For the sum, the output is 1 if either input 1 or input 2 is high, but not both, so S = input 1 XOR input 2; and we carry 1 if = input 1 AND input 2 are 1. The second and subsequent digits are harder since we need to add the digit from each number and the carry, but it’s not too difficult.
We can use logic gates to do the calculation for each bit of the addition. The circuit looks like this:
You can hopefully see how bit one of the sum if the XOR of the inputs for bits one of the numbers A and B, and the carry to the calculation of the second bit is the AND of these inputs. Again there is a simulation you can try, you might need to stretch the JAVA window to see all the circuit. Try 1 plus 1 (01+01 = 10 so set inputs A1 and B1 High, A2 and B2 Low to gives Output S1 Low and Output S2 High). And 2 + 2 (10 + 10).
We implemented this circuit using our audience of flag-wavers. We put pupils on the front row to be the inputs, pupils on the next row to be gates 1-4, and so on, making sure that each one knew at whom they should be looking and what condition should be met for them to raise their flag. We ran this three times, and each time it worked brilliantly. OK, so we could only add numbers less that 3, which isn’t much computing power, but given another 35 people we could have done eight-bit addition. And I’m pretty sure that we could have managed flip-flops and registers, but we would need something like 10,000 pupils to build a processor equivalent to an 8086, so the logistics might be difficult.