... using mental arithmetic tricks. I'm not going to teach you how to do long addition on paper, although I will assume you are comfortable with doing it. This Instructable is for those times when you don't have paper or a calculator handy.
Step 1: Borrow Ones
"From where?", I hear you cry. Essentially from nowhere- sometimes you can make an addition (or sometimes a subtraction or multiplication) easier by borrowing a one, doing the sum and putting the one back. Example:
13+29 is a bit of a pain to do in your head. If you like the previous technique you could call it thirty-twelve, but to do that precise addition is a little awkward. 13+30 is much easier, however. Borrowing ones essentially means you say "13+29 is hard- I'm going to borrow a one and add it to the 29, 13+30 is 43, then I put back the one I borrowed so the answer is 42". It sounds complicated, but with a little practise I now do all of those steps in my head much faster than just trying to add 13+29.
Borrowing ones can also work in reverse by "burying" a one you would rather ignore for the time being- for example, 13+31 can be turned into 13 + 30 = 43 and then you add the one you ignored earlier, but additions that become easier by "burying" a one are less common that those made easier by borrowing.
Pros: simple, conceptually easy
Cons: limited application, doesn't allow very large additions.
Step 2: Match Pairs
What is special about the pairs (1 and 9), (3 and 7), (4 and 6)?
They sum to 10. There are only five of these pairs so anyone who has spent a while doing maths should instantly recognise them, and they can make your life a whole lot easier when doing additions.
Take 13+27. Recognising the matched pair 3 and 7 in the ones place tells you the answer will be a multiple of ten, so you have to add 10, 20 and the 10 resulting from (3+7) to get 40- you have effectively simplified the addition from 13+27 to (1+2+1) * 10. Good for you.
If you become proficient at this technique, you may be able to combine it with the previous technique- if you saw 14+25 you might recognise that you can borrow a one to turn it into 14+26 then use this technique.
Pros: very quick, conceptually simple, quite general
Cons: requires practise for recognition, slightly limited application.
Step 3: See Subtractions
This is a similar technique to borrowing ones, but can be extended to numbers other than one. If you tried to do 98+97 using traditional means there would be a lot of carries. The easier way to do this addition is to recognise that 98 = (100-2) and 97 = (100-3) so the result is the sum of these, namely (200-5) or 195. Many people can do this intuitively given a case like 99+99=198, but with a little practise you can make it much more general by recognising larger subtractions. The result may become very confusing if the subtractions sum to more than 10 (for instance, 93 + 94 = 200 - 13), so beware of replacing an awkward addition with an equally awkward subtraction.
Pros: general, work with large numbers
Cons: needs a conceptual shift, requires some practise
Step 4: Make Groups
The previous few techniques have been delaing with adding two large-ish numbers. This one is more suited to "restaurant bill" type additions where you are adding a lot of very small numbers (or not very small, depending where you choose to eat).
This technique uses the fact that most people who have done maths for any length of time recognise most single-digit additions without even having to think about them- 2+2 is the obvious example but any maths student should be able to do 5+7 or 8+8 without too much thought.
When confronted by a string of numbers to add, you can simplify it by spotting groups that you can combine using the instinctive additions above. As a simple example, 2+2+3+4 could become 4+3+4 by pairing the twos, then 8+3 by pairing the 4s. If the numbers are out of order the pairs may be scattered, but in a longer example like the above you could even group all of the similar digits together and turn them into a multiplication, so you could start 2+5+7+2+7+3+8+2+4+6+2 by removing the four twos and adding an eight.
Pros: conceptually simple, general
Cons: requires holding a lot of working in your head, at risk of "losing count" and making mistakes.
Step 5: Running Total
In a similar manner to the previous technique, a string of small additions may be best handled by simply keeping a running total as the numbers are added in order. 2+2+3+4+5 = 4+3+4+54 = 7+4+5 = 11+5 = 16. A more experienced mental mathematician could probably reduce the number of steps with larger recognitions- if you recognise from sight that 2+2+3=7, compress that into one step and skip to 7+4+5.
Pros: conceptually simple, very general
Cons: not a great speed improvement
Step 6: Multiply
"But wait," I hear you cry, "this Instructable is meant to be about adding!"
But we've already invented a handful of new digits and borrowed numbers from out of thin air, so a simple multiplication shouldn't be too hard.
What is 55+66? The conventional method involves lots of carries and is (like many of the examples in this Instructable) a bit awkward. However, I'm sure many people will notice that 55 is 11*5 and 66 is 11*6, so the sum is naturally 11*11. Depending on how far your multiplication tables went you may know 11*11=121 off the top of your head, but other multiplications (the 5 and 9 times table, for instance) feature a lot of rules for working them out so may be of use in this technique.
63+18 might be a pain to work out in your head explicitly but a knowledge of the 9 times table will tell you that it is 7*9 + 1*9 = 9*9 = 81. For small numbers this method may not be much quicker than simple addition but certain problems may be greatly simplified.
Step 7: Runs
This is the most obscure of the techniques I am going to present. It's not as simple a method as the others and has limited application, but also has some incredibly powerful uses. The story goes that when he was a child, the mathematician Carl Friedrich Gauss was asked to mentally add all the integers between 1 and 100 to keep him busy for a period. It didn't work, as he summed all the number in seconds by realising that they form a simple pattern.
Gauss saw that he could break the numbers down into matched pairs, remove an amount from the high number and add it to the low number to make all the pairs the same. Taking the numbers 1 to 5 as a simpler example, you would take 2 from the 5 and add it to the 1, take 1 from the 4 and add it to the 2. This turns 1+2+3+4+5 into the rather more manageable 3+3+3+3+3 or 15.
To do this for the numbers 1 to 100 is a little trickier because there are an even number of them so the pairs end up summing to 50.5 (the mean average of all the numbers). The general rule is that to add a run of increasing integers, add the first and last together, halve the value and multiply by the number of numbers you are adding together. To go back to our smaller example, 1+5=6, 6/2=3 and 3*5=15. The diagram below shows in a more visual way why this works. The intricacies of this technique are outside the scope of this Instructable, but for now you can remember the formula (first+last)/2 * number of numbers
Again, this method isn't particularly general because it requires an addition of the form 1+4+7 (properly, an arithmetic series), but when it is applicable it is a very powerful method (as Gauss demonstrated). Runs of evenly-spaced additions are unlikely to come up much in "real life" (unless you go to restaurants with mathematicians) but pure maths features them more often.