Mechanical Calculation

Public summary: 

Electronic calculators are everywhere but how can we get these machines to understand simple tasks like addition? Explore some of the first decimal calculators and then explore modern binary arithmetic.

Adding numbers using mechanical calculators.
Useful information
Kit List: 

The binary adder (needing repairs/replacement)
A bag of marbles
Decimal Gear Calculator (To Be Made)
The Mystery Calculator Game
Simple Pocket Calculator (need to find)

Packing Away: 

Old Adder goes into cardboard box.
New pieces live in the Maths box.


NB - This needs someone to print all the plans off thingiverse before it can be used.

A mechanical binary adder, which adds two numbers represented by marbles.
Get people to figure out what it does by repeatedly adding one, see if they can spot how to read a number off or add a number other than one. Talk about how this is how computers represent numbers. Why do they use something different? Easy to store in this format instead of 10 possible digits we only have one, we can then store this as +ve or -ve, on/off or something like that.
There's also options to talk about logic, very easy switches for operations AND and XOR (exclusive or). How do we add numbers using these? AND to see if we need to carry one, XOR tells us if what we need in that bit.
You can also demonstrate overflow by adding big numbers together.
You could talk about how computers store negative numbers, sign and magnitude is probably most obvious way of doing it. There's also 2's complement. Or you could talk about fractions and storing the number of halves, quarters... or as numerator denominator.
If you wanted you can even get people to play about with finger binary. We can count up to 31 on one hand by using binary and whether a finger is bent or not.

The decimal gear calculator is another thing you can use. This hopefully shows it's more difficult than binary as we can't use on/off. Most old mechanical computers used gears so this was how they worked but with electricity we can make smaller computers. To add number a to b we need to set the machine with a displayed at the bottom, we then turn the units, tens and hundreds gears separately each a number of clicks corresponding to the number in the units,tens and hundreds place of b. You can see the additional gears at the back are turned and offset slightly back, when we 'overflow' in one column they knock the next one. If we were going to do a very large calculation and weren't sure if it would be too large for the calculator when the final gear overflows we could use the same idea to indicate that, original mechanical computers would often ring a bell.

The "Mystery Calculator" game is formed of a couple of cardboard sheets with numbers on. The idea is to get a child to pick a number from 0 to 63, you then lay out the cards and ask them to pick out those with there number on, just by adding up the top left corners you can guess there number. How is it working, well look at the numbers on the cards, the first card only has odd numbers on it, the final card has everything from 32 to 63. Maybe we should think about binary? What happens if we write these numbers out, we'll find being on card n means you have a 1 in the nth place of your binary expansion.
This leads on to questions about how much information we have and how this relates to binary, with 6 cards and you picking out those with your number on answers 6 yes or no questions about your number. So we'd expect to be able to determine the number from 2^6 possibilities but no more. These questions are really nice because we always need these 6 questions to determine the number, every possible way of answering the questions matches a number and it doesn't matter about order or anything like that.
Obviously we could ask different questions, one way might be asking "Does 2 divide your number?", "Does 3 divide your number?", all the way up to "Does 61 divide your number?". But that's 18 questions! Also say we ask "Does 47 divide your number?" and the answer is "Yes" we know the number is 47, we only needed that question. Do these questions actually identify the number uniquely? What about 6, 12, 18, etc. We need more questions about if 4, 9, etc. divide the numbers. So what's the problem with this choice of questions that means they need a lot more to get the same information. Well there's lots of overlapping information.
If I have "Does 43 divide your number? Yes" then I know the answer to every other question as I know your number is 43.
If I have "Does 43 divide your number? No" all I know is your number isn't 43, so I've gained very little.
If you picked numbers at random I only had a 1/64 chance of getting a yes and most of the time I get little information.
The original questions have no overlap and this set has several ways options of what to ask. The originals also exclude exactly half the options at each question whereas these are less even.
We can also find a lot more structure in the originals. "Is your number 32 or larger?" If Yes take away 32 from your number for the next question. "Is your number 16 or larger?" If yes takeaway 16 etc. This is one way of expressing these questions.

If you wanted to talk about logic gates you can build some out of dominoes, this is a little painful however there are printed out templates which you can carefully place dominoes on. See if you can figure out how they work. You'll need to knock both inputs simultaneously as timing is important.

Risk Assessment
Date risk assesment last checked: 
Tue, 01/01/2019
Risk assesment checked by: 
Date risk assesment double checked: 
Fri, 12/01/2018
Risk assesment double-checked by: 
Josh Garfinkel
Risk Assessment: 

A mechanical binary calculator and other calculation devices.

Hazard Risk Likelihood Severity Overall Mitigation Likelihood Severity Overall
Model Possibility of sharp edges or splinters on the model, if damaged. 2 2 4 Check for damage before demonstrating, and avoid knocking or dropping during use.
Call first aider in event of injury.
1 1 1
Marbles Marbles could present a slip hazard if dropped. 2 2 4 Keep unused marbles in a container, and keep track of how many there are.
Call first aider in event of injury.
1 1 1
Marbles Children may eat marbles. 2 2 4 Watch children with the marbles, only give them one at a time to immediately put in the machine. Call first aider in event of injury. 1 1 1
Gears Gears may trap fingers. 2 2 4 Make sure only one person using calculator and turning gears properly.
Call first aider in event of injury.
1 1 1