What follows is a motley and pretty random collection of techniques or shortcuts, collected over many years, to help speed up calculations in your head. They are not in any particular order, and I will not pretend that all of these are the full toolbox by any means, but they are probably the most accessible ones. You will be spared all of the algebraic stuff supporting what follows, but I will be happy to supply the same to anyone that wants it.

Mentally multiplying different two digit numbers together

Imagine two separate sets of numbers represented by A,B and C, D and imagine them set out in the following format
A B
C D
Now consider these actual numbers in those positions
68
47

To mentally multiply these together:

• first calculate AxC [6×4]
• this results in 24 to which 00 is appended, giving 2400
• we then calculate AxD and BxC [6×7, 8×4], add them together and append 0 to give 740
• this is added to 2400 to give 3140
• to complete the calculation, calculate BxD [8×7] and add the result of 56 to 3140
• so the answer is 3196

 Multiplying different three digit numbers together

When young, I became easily bored on car journeys. In those days most cars I saw on the road had three digit number plates, and I became grabbed by the idea of trying to multiply them together in my head.
What follows is the system that eventually evolved: my guess is that it will sound slightly odd, but it’s just an extension of the above really. The visual elements of it made it simpler and quicker to use and I’ve stuck with this to the point that over the years it has become automatic and second nature.
Imagine two separate sets of numbers represented by A,B, C and D,E, F and imagine them set out in the following format
A B C
D E F
Now consider these actual numbers in those positions
389
764

To mentally multiply these together:

• first calculate AxF, BxE, CxD and add them together. This results in 123 to which 00 is appended..
  Now, at this point, the bored child imagined two fields next to each other with a fence separating them: the right hand field as you look at it is for chicken eggs, and the left hand field is for chicks: the last three digits of the first calculation [300] always occupies right hand field like so [012 ][300]
• We then calculate BxF and CxE, add them together and append 0 to give 860. This is added to the right hand field and as you can see, it will also increase  the number of chicks by 1 to 13 in the left hand field i.e. [013][160] ……….an egg hatched!
• To finalise and complete the number in the eggs field, calculate CxF and add the result [36] in as so [013][196] – and, at this point, the number in the right hand field is fixed and cannot change.
• And here is the very place to make an extremely important point. When people start experimenting with these sort of mental gymnastics, the question will usually arise as to how one can guarantee retaining the fixed part of the answer [196 in this one] whilst working on another part. The answer lies in storage and retrieval to and from memory using the Major System (see Pi Memorisation page). So, in this case, you could store the 196 egg number in a place in your memory as the image  Toy BuSH for example. It doesn’t need to make sense! – it just needs to be visual and bizarre enough to stay in your memory long enough before retrieving it. The more you practice storage and retrieval, the quicker and  easier it gets. 
• So then ,just carry on calculating the chicks…………. then when you have finished with the field of chicks, just reach back into your mind for the image you left in the field of eggs – because, trust me,  it will be there – and convert it back to numbers. With practice,  you will need to do this less  and less often for smaller numbers — though it is still a technique to retain for more complex calculations.
• Anyway – back to this calculation: we then calculate BxF and CxE, add them together to give 74. This is now added to the chicks in the left hand field like so: [087][196]. 
• Then finish everything by calculating AxD and append 0 to give 210 which is then also added to the chicks in the left hand field to give the final answer 297196.

Again, I know this all looks really horrible written down, but this systematic approach really makes it easier to move through the gears mentally, in the same way every time. OK, but what is all this crap about chicks and eggs anyway? Well, all I can say is that it does stick in the mind much more to think of 297 chicks and 196 eggs in seperate fields than just the formless numbers themselves….. the brain holds numbers better as objects in spaces it seems.

Last words

• For the purpose of the website, I’m going to stop at three numbers here but, if you’re feeling adventurous, you can develop and apply the above principles to 4 or even 5 digit numbers. 
• Obviously, with large numbers, the technique changes, and mine is to extend the two digit approach outlined above, but with the alpha letters shown there becoming groups of two or three numbers and so treated accordingly 
• Look, over more than three digits I won’t pretend it’s going to be easy to start with  – but it is very doable if you stick with it and you can become quicker with practice…….. though 5 digits certainly can give you a headache! Let me know if the masochists among you want to know my detailed approach with this.