What is the fastest way to multiply two digit numbers?
The way I learned it:
$$\begin{array}{rrl} & 18 & \\ \times & 17 & \\ \hline & 156 & (10\times 10 + 8\times 7) \\ + & 70 & (10\times 7) \\ + & 80 & (10\times 8) \\ \hline = & 306 &\\ \hline \end{array}$$
In general:
$$\begin{array}{rrl} & {\rm AB} & \\ \times & {\rm CD} & \\ \hline & {\rm ACBD} & \text{(pad 0 if neccesary)} \\ + & {\rm BC}0 & \\ + & {\rm AD}0 & \\ \hline \end{array}$$
This is just dividing it into polynomial: $(10+7)\times (10+8)$. And it allows me to visualize easier than the traditional method.
The absolute fastest way to multiply two-digit numbers is to already know the result, by having learned the two-digit multiplication table just like you learned the one-digit multiplication table. Of course it means a lot of work up front.
My method for multiplying some two digit numbers is not going to work well in all cases, and has the draw back that you must memorize all the squares. But, I present it just as another trick you can learn. I +1'd FiniteA's answer as that seems like a nice way that works well in just about any case, and takes very little memorization or skill, i.e., it boils it down the simplest parts.
I would like to point out that my method is probably faster when the two numbers are relatively close together and when their difference is an even integer.
Using the fact that $(x - a)(x + a) = x^2 - a^2$, if you memorize perfect squares, you can do some multiplications pretty quickly.
Example 1: $24 \times 26 = (25 - 1)(25 + 1) = 25^2 - 1^2 = 625 - 1 = 624$
Example 2: $24 \times 27 = 24 + 24 \times 26 = 24 + 624 = 648$
Example 3: $24 \times 32 = 28^2 - 4^2 = 784 - 16 = 768$
1) Practice.
2) Practice some more.
3) ????
4) Profit!
Here are the things to practice:
(1) To compute $x^2$: use the identity $x^2=(x+a)(x-a)+a^2$, with $a$ chosen to make $x+a$ as round as possible. This is especially fast for numbers near to $50$ or $500$ or $5000$ and so on. I can do squares of numbers near $500$ in about 2 seconds this way. Example: $46^2=50*42+4^2$, further simplified if one simply remembers $50^2=2500$ and therefore that $50*42=2500-400$.
(2) In general, round up or down to the nearest multiple of $10$ and then correct: for example, compute $93*42=100*42-7*42$. This is especially useful if you don't need the exact value---then you get a good approximation very quickly. The one choice you have to make here is which of the two numbers to round, and you should do this to maximize the resulting simplification. In the example I chose, it's better to round $93$ up to $100$, since multiplying by $100$ is slightly easier than multiplying by $40$.
(3) As you begin doing mental arithmetic with larger numbers, you will realize that the primary obstacle is not speed but space: you will run into the problem that you cannot reliably store more than a few digits in your head at a time. To overcome this, you will need a mnemonic. One relatively painless way is described in Art Benjamin's book "Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks": turn numbers into phrases, poems, stories or songs!
The mental calculators' trick is to start from the left :
$$
\begin{array}{r}
17\\
\times 18
\end{array}
$$
$$
\begin{array}{c l}
\text{'cross' computation} &\text{ partial result}\\
1\cdot 1=1 & 1\\
1\cdot 8+7\cdot 1=15 & 25\\
7\cdot 8=56 &306
\end{array}
$$
Another example
$$
\begin{array}{r}
237\\
\times 543
\end{array}
$$
$$
\begin{array}{c l}
\text{'cross' computation} &\text{ partial result}\\
2\cdot 5=10 & 10\\
2\cdot 4+3\cdot 5=23 & 123\\
2\cdot 3+3\cdot 4 +7\cdot5=53 &1283\\
3\cdot 3+7\cdot 4=37 & 12867\\
7\cdot 3=21 & 128691\\
\end{array}
$$
The same trick (working from left to right) is used for other operations. Neat and fast!