This technique is based on the methods of mathemagician Arthur Benjamin ( see his book: The mathemagician's guide to lighting calculations and amazing math tricks). I included some special cases which allow for even quicker calculations :o)
The first trick I will be sharing is exceptionally easy and intuitive. It concerns the case where you have two double digits which both start with a 1:
Rule for numbers starting with 1: Select any of the two numbers add the second digit of the second number, then multiply by 10 and add what you get when multipliying the last two digits of both numbers
Pretty lengthy rule but once you get some examples done you will see it is pretty easy and quick to perfrom:
Example 1: Let's consider 13x14. Let's pick 13 then we need to add 4, which gives us 17, multiply by 10, which gives 170, and then add 3x4, and the total becomes 182.
Example 2: Let's consider 16x17. We pick 16, then we need to add 7, which gives 23, then multiply by 10, which gives 230 and finally add 42, putting the total at 272. Easy. :o)
What is the general version of this rule when we consider numbers starting with the same digit? With a bit of math you can show that it is this:
Rule for numbers starting with same digit: Square the first digit multiply it by 100 then add the sum of two last digits times the first digit times 10 and add the multiplication of the last two digits.
Again let's do some examples to get the hang of it.
Example 3: We will compute 24x27. So 2^2x100 + 2x(4+7)x10 + 4x7 = 400 + 220 + 28 = 648. It might look a bit complicated but the idea here is that the way that computation has been split makes it easy to solve.
Example 4: Now we do 43x42. So 4^4x100 + 4x(3+2)x10 + 3x2 = 1600 + 200 + 6 = 1806.
Example 5: Just one more: 57x56 = 5^5x100 + 5x(7+6)x10+7x6 = 2500 + 650 + 42 = 3192.
The computation get more complicated the larger the digits are. But once you get the hang of this for smaller digits you also get quite quickly good at larger digits.
General rule for multiplying two double digit numbers: Multiply the first two digits and then multiply them by 100, then add to that the multiplication of the two cross-opposed numbers times ten and finally add to that the multiplication of both last digits.
Again this rule becomes way more simpler after an easy example.
Example 6: Take 21x34. Using the general rule we have 2x3x100 + (2x4+1x3)x10 + 4 = 600 + 110 + 4 = 714 .
It gets a bit more complicated if you consider larger digits. So let's try one of those:
Example 7: Now we solve 56x78. So then we have 35x100 + (5x8+6x7)x10+48= 3500+820+48=4368. That's not so bad :o)
There actually lots of special sub-cases for which you can find special rules such that you get the solution in fewer steps. One of these is multiplication by 11 which we discussed in this post. Another one is squaring numbers which I will write about in my next post.
Reference
[1] A. Benjamin. Secrets of mental math: The mathemagician's guide to lighting calculations and amazing math tricks.
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