Multiply two two-digits numbers in your head

Ever had to multiply two two-digit numbers without a pocket calculator? Yes? Well how about doing the same without even pen and paper? This happend to me recently when I was in the garden and too lazy to get inside fetch my mobile. I would have been 10 times faster if I had used my feet, but I gave my head a try.

But back to topic, 40 times 40 is 1600, fairly easy, we only have to multiply two digits (4 * 4) and add two zeros, but what about 45 times 37? What I now do in such a case is to visualize a grid in my mind's eye similar to squared paper. I put both numbers with a couple of squares distance. Those figures are colored, the left one I paint blue the right one red. In addition, whenever I focus on one of those figures (I tend to also move my eyes in the according direction) I imagine a burbling sound for the blue figure (water) and a crackling sound for the red one (fire), this helps me to not confuse the numbers.

		4	5			3	7

When I feel confident that I have fixed those numbers in my mind I start to multiply the first blue, burbling digit with the first red, crackling digit, in our case: 4 * 3 = 12. I then add a zero and place it right aligned three squares above the left blue starting number, just like so:

	1	2	0
		4	5			3	7

Next I take the first red, crackling digit again and multiply it with the second blue, burbling digit: 5 * 3 = 15. Without adding an additional zero I place it right under the first product.

	1	2	0
		1	5
		4	5			3	7

Now I try to be fast in summing up both numbers (120 + 15 = 135) and instantly replace them with the result. By being quick it helps to wipe out the previous two numbers from my mind's grid which is importat since we don't need them anymore. Now I am looking at the following grid:

	1	3	5
		4	5			3	7

After focussing a short moment on the new blue, burbling 135 I proceed in taking the second left red crackling digit, multiply it with the first blue, burbling digit: 4 * 7 = 28. Again a zero is added to the result but this time I put it above the left red burbling starting number and get the following grid:

	1	3	5		2	8	0
		4	5			3	7

Next I take the second red, crackling digit, multiply it with the second blue digit: 5 * 7 = 35. I then put the result under the previous product like so:

	1	3	5		2	8	0
						3	5
		4	5			3	7

Again, as quick as possible, I sum up those two red crackling numbers: 280 + 35 = 315 and replace them with this result, wiping out the numbers 280 and 35 from the grid in my mind's eye. Take a second to check if you still have that blue, burbling 135 from a second ago and you are confident about the correctness of the results. If so, its time to also clear the starting numbers 45 and 37 we don't need them anymore. Only two numbers are sitting in your grid now, that is a blue, burbling 135 on the left and a red, crackling 315 on the right.

	1	3	5		3	1	5
		4	5			3	7

Before we sum up those two we have to know that in my mind water is more powerful than fire since the first is able to extinguish the latter. To honor that fact I again a zero to the blue, burbling number. At the same time I move the left, crackling number under the blue one like this:

	1	3	5	0
		3	1	5

Take a second until everything is neatly aligned. With a final "flosh" the water number extinguishes the fire number by adding it to itself. 1350 + 315 = 1665. I wipe out the blue and red number, place the result in the middle of the grid and color it purple. Why do we need this post-treatment? Because I tend to loose that figure as fast as I created it if I don't. By this extra work I still remeber this number minutes after, here it is:

		1	6	6	5

The reason this algorithm worked for me is that I can divide and conquer. I only have to mulitply one-digit numbers which normally is done fast and reliably. All that hocus-pocus with color, sound and spacial distribution I only use to keep the numbers in my mind's grid, when you try to remember abstract things like numbers you will see how hard it is to tame them so that tjey stay where and what they are, well at least in my case.

Admittedly it takes some time to get familiar with that algorithm and you have to practice it on a regular basis. But nothing is more exciting than someone asking you to multiply 56 and 78 and you spit out 4368. Granted that won't happen often :) But then again, if you are stuck in a waiting line it is more fun to do some mental arithmetic then to count fluffs on the coat of the person in front. Give it a try.