Here's a technique that I sometimes use for multiplying two-digit numbers in my head. It has the advantage that you don't have to use most of the multiplication table at all - just addition and subtraction, and division by 2. It's based on the well-known
difference of two squares formula from algebra, but you don't need to know any algebra to apply the technique. It has similarities with the old
quarter squares method but is explicitly designed for mental calculation. I'm not aware of anyone else who uses this exact method.
You will need to know the perfect squares up to 25 x 25. This isn't as daunting as it may sound, as the traditional multiplication table takes you up to 12 x 12, and there are a number of mnemonics that can help you remember most of the rest.
The "pivot" rule
As a simple example, suppose you want to multiply 7 by 13. (For the moment, we'll stick to examples where both numbers are odd, or both are even.)
Imagine
a seesaw. At the two ends of the seesaw are the two numbers you want
to multiply. In between the two numbers there's a regular scale, as on a ruler. The "pivot" number is the halfway point between the ends of the seesaw. So if you have 7 at one end and 13 at the other, the pivot will be 10. You might be able to spot this straight off, but if not you can calculate it by adding the end numbers together and dividing by 2; 7 + 13 = 20, and 20/2 = 10.
Now calculate the distance from the pivot to either end of the seesaw: 10 - 7 = 3, or 13 - 10 = 3. It doesn't matter which end you choose.
Now calculate the square of the pivot: 10 x 10 = 100
and the square of the distance from the pivot: 3 x 3 = 9
Finally subtract the second from the first: 100 - 9 = 91.
And that's your answer!
In this case it was relatively easy because the squares of 10 and 3 are well known. The key to this technique is knowing how to compute the square of
any two-digit number. Fortunately, you only need to know the squares of the first 25.
Memorizing the first 25 perfect squares
The first twelve perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.
13 x 13 = 169
- anagram of the next one
14 x 14 = 196 - anagram of the previous one
15 x 15 = 225 - all squares of numbers ending in 5 end in "25" (cf. 25 squared)
16 x 16 = 256 - 2 to the power of 8 (crops up a lot in computing)
17 x 17 = 289
18 x 18 = 324 - "graph" of digits dips down 1 and up 2 (cf. 24 squared)
19 x 19 = 361
20 x 20 = 400 - easily derived from 2 x 2 = 4 and 10 x 10 = 100
21 x 21 = 441 - 12 squared backwards (21 is 12 backwards)
22 x 22 = 484 - think 11 squared = 121, times 4 (also note 22 and 484 are both palindromic)
23 x 23 = 529
24 x 24 = 576 - "graph" of digits dips up 2 and down 1 (cf. 18 squared)
25 x 25 = 625 - all squares of numbers ending in 5 end in "25" (or think of old 625-line TV)
I don't know any easy way of memorizing 289, 361 and 529. My old school was near a road called the A361 but that's unlikely to be much help to anyone else! Hopefully you can find your own personal connections for the numbers.
When you've mastered these you should be able to multiply any two odd or two even numbers whose sum is 50 or less.
Multiplying two odd or two even numbers whose sum is 50 or less
As an example, try 14 x 22.
The pivot is 18. (14 + 22 = 36; 36/2 = 18.)
The distance to the pivot
is 4 (18-14 or 22-18).
18 squared is 324, and 4 squared is 16.
So the answer is 324 - 16 = 308.
Another example: 17 x 27.
The pivot is 22. (17 + 27 = 44; 44/2 = 22.)
The distance to the pivot
is 5 (22-17 or 27-22).
22 squared is 484, and 5 squared is 25.
So the answer is 484 - 25 = 459.
Try these for yourself (solutions in white, with method):
13 x 27
Pivot = 20; distance = 7; answer = 400 - 49 = 351
14 x 34
Pivot = 24; distance = 10; answer = 576 - 100 = 476
17 x 21
Pivot = 19; distance = 2; answer = 361 - 4 = 357
18 x 28
Pivot = 23; distance = 5; answer = 529 - 25 = 504
Deriving the next 25 perfect squares
Once you feel comfortable with the first 25 perfect squares, there's a simple rule that allows you to derive the next 25, from 26 to 50. To explain it we can use the seesaw analogy again. Call the number you want to square the "base" number.
Imagine a seesaw with the pivot at 25 and the base number at one end. The number at the other end will be 50 minus the base number. For instance, if you want to square 33, then 17 will be at the other end.
Now square this number: 17 x 17 = 289 (from the list above)
Now take the distance from the pivot to one end of the seesaw and multiply by 100: 8 x 100 = 800
Add these two together: 289 + 100 = 1089
Another example: find 42 x 42.
Pivot is 25, 8 is at opposite end. Distance is 17.
So 42 squared is 8^2 + (17 x 100) = 64 + 1700 = 1764
A few for practice:
49 x 49
1 at opposite end. Distance is 24. 1^2 + (24 x 100) = 1 + 2400 = 2401
29 x 29
21 at opposite end. Distance is 4. 21^2 + (4 x 100) = 441 + 400 = 841
39 x 39
11 at opposite end. Distance is 14. 11^2 + (14 x 100) = 121 + 1400 = 1521
36 x 36
14 at opposite end. Distance is 11. 14^2 + (11 x 100) = 196 + 1100 = 1296
If you practise using the squares from 26 to 50 often enough you'll find
that the values start to come automatically after a while. Here's the full list for reference:
26 x 26 = 676 ( = 576 + 100) 38 x 38 = 1444 ( = 144 + 1300)
27 x 27 = 729 ( = 529 + 200) 39 x 39 = 1521 ( = 121 + 1400)
28 x 28 = 784 ( = 484 + 300) 40 x 40 = 1600 ( = 100 + 1500)
29 x 29 = 841 ( = 441 + 400) 41 x 41 = 1681 ( = 81 + 1600)
30 x 30 = 900 ( = 400 + 500) 42 x 42 = 1764 ( = 64 + 1700)
31 x 31 = 961 ( = 361 + 600) 43 x 43 = 1849 ( = 49 + 1800)
32 x 32 = 1024 ( = 324 + 700) 44 x 44 = 1936 ( = 36 + 1900)
33 x 33 = 1089 ( = 289 + 800) 45 x 45 = 2025 ( = 25 + 2000)
34 x 34 = 1156 ( = 256 + 900) 46 x 46 = 2116 ( = 16 + 2100)
35 x 35 = 1225 ( = 225 + 1000) 47 x 47 = 2209 ( = 9 + 2200)
36 x 36 = 1296 ( = 196 + 1100) 48 x 48 = 2304 ( = 4 + 2300)
37 x 37 = 1369 ( = 169 + 1200) 49 x 49 = 2401 ( = 1 + 2400)
50 x 50 = 2500
Distance is 11. 14^2 + (11 x 100) = 196 + 1100 = 1296
Multiplying two odd or two even numbers whose sum is 100 or less
So now you have enough knowledge to multiply any two odd or even numbers whose sum is 100 or less! Here's an example: 44 x 24. You might find it difficult to hold all the necessary information in
your head at first, so write down the pivot if
necessary.
The pivot is 34, and the distance is 10.
34 squared is 1156 (either from the list above or by doing 16^2 + 900).
10 squared is 100, so the answer is 1156 - 100 = 1056.
Another example: 47 x 29.
Pivot = 38, distance = 9.
38 squared is 1444 ( = 12^2 + 1300), 9 squared is 81.
Answer is 1444 - 81 = 1363.
Practice examples:
38 x 22
Pivot = 30, distance = 8. 30^2 = 900, 8^2 = 64, answer = 836
37 x 27
Pivot = 32, distance = 5. 32^2 = 1024, 5^2 = 25, answer = 999
46 x 32
Pivot = 39, distance = 7. 39^2 = 1521, 7^2 = 49, answer = 1472
63 x 31
Pivot = 47, distance = 16. 47^2 = 2209, 16^2 = 256, answer = 1953
You probably found with the last example that the numbers were quite a long way from the pivot and it was difficult to do the final subtraction in your head, which is a limitation to this as a purely mental technique. Nevertheless you can always write the final step down - the number of operations you need to do is still fewer than in conventional long multiplication.
Multiplying an odd number by an even number
If you've got this far you're almost certainly shouting "Well that's all very well if they're both odd or even, but what if there's one of each?"
There are various ways of dealing with this, but the simplest is to subtract 1 from the higher number, perform the calculation as normal, and then add the lower number. So for 26 x 37 you'd do 26 x 36, which is 31^2 - 5^2 = 961 - 25 = 936, and finally add 26 to give 962.
Another example: for 33 x 42, calculate 33 x 41 = 37^2 - 4^2 = 1369 - 16 = 1353, and add 33 to give 1386.
Try the following:
13 x 18
13 x 17 = 15^2 - 2^2 = 225 - 4 = 221 and 221 + 13 = 234
16 x 29
16 x 28 = 22^2 - 6^2 = 484 - 36 = 448 and 448 + 16 = 464
23 x 32
23 x 31 = 27^2 - 4^2 = 729 - 16 = 713 and 713 + 23 = 736
17 x 44
17 x 43 = 30^2 - 13^2 = 900 - 169 = 731 and 731 + 17 = 748
Multiplying any two two-digit numbers
All that remains now is to learn the technique for squaring two-digit numbers between 50 and 100, and in theory you should be able to multiply any two two-digit numbers. The sum of any two two-digit numbers is always less than 200, so the pivot will always be less than 100. You might find that the final subtraction becomes rather unwieldy with the higher numbers and the method is no longer useful. Nevertheless I'll include this section for completeness. There are different methods for numbers above and below 75.
Squaring a number from 51 to 75: subtract 50 from the number, take the square and then add on the original number less 25, multiplied by 100. Note that the number of hundreds you add on is the "pivot" between the number and the number less 50.
E.g. for 63 squared: 63 -50 = 13, 13 squared = 169 and add (63 - 25) x 100 = 3800 to give 3969.
You might find it useful to imagine a seesaw with 63 at one end and 13 at the other; the pivot will be at 38, so you add on 38 x 100 to 13 squared.
Another example: for 72 squared, 72 - 50 = 22, 22 squared = 484 and add (72 - 25) x 100 = 4700 to give 5184. (47 is the "pivot" between 22 and 72.)
Squaring a number from 76 to 100: subtract the number from 100, take the square and add on 100 less twice the original number, multiplied by 100. In this case you might imagine a seesaw with the pivot at 50, and the number of hundreds you add on is the entire length of the seesaw.
E.g. for 93 squared, think 100 - 93 = 7. The length of the seesaw is 93 - 7 = 86. So the answer is 7^2 + (86 x 100) = 49 + 8600 = 8649.
Another example: for 78 squared, 100 - 78 = 22, length = 78 - 22 = 56, answer is 22^2 + 5600 = 484 + 5600 = 6084.
Here's a sample multiplication sum using these squares: 57 x 79.
The pivot is 68. 68 squared is 18^2 + 4300 = 4624.
The distance is 11, and 11^2 = 121. So the answer is 4624 - 121 = 4503.
Another example: 62 x 92.
The pivot is 77. 77 squared = 23^2 + 5400 = 5929.
The distance is 15, and 15^2 = 225. So the answer is 5929 - 225 = 5704.
If you dare, here are some examples to try:
56 x 74
Pivot = 65, distance = 9. 65^2 = 15^2 + 4000 = 4225, 9^2 = 81, answer 4144
67 x 81
Pivot = 74, distance = 7. 74^2 = 24^2 + 4900 = 5476, 7^2 = 49, answer 5427
81 x 97
Pivot = 89, distance = 8. 89^2 = 11^2 + 7800 = 7921, 8^2 = 64, answer 7857
66 x 86
Pivot = 76, distance = 10. 76^2 = 24^2 + 5200 = 5776, 10^2 = 100, answer 5676
Don't worry if you've given up by this point! I don't find numbers of this size easy by this method either. In practice I might use a variety of different techniques. E.g. for 56 x 74 I might think
56 = 7 x 8
8 x 75 = 600 (6 is three-quarters of 8), so 56 x 75 = 7 x 600 = 4200
So 56 x 74 = 4200 - 56 = 4144
I hope some of this has been useful (or at least interesting) anyway!
