Multiplication tables revisited

Mental Calculation, Mathematics, and Puzzles
1

Some people on the forum want to memorize multiplication tables.

This can be done using your preferred number coding system like the Major System:
a TiT and a TooTh are building a TeNT,
etc.

The multiplication tables 1-1000 contain a million numbers.
Since 318 X 524 is the same as 524 X 318, you can half the amount of numbers.
However, it is still a staggering amount of data to memorize.

In my opinion it generates a lot more mental plasticity if you can actually quickly calculate them.
Let’s see if we as a forum can come up with ways to quickly do this.

Let’s start simple. Most people know the table 1-10, maybe even 1-12:

The middle line is a line with squares and the numbers below it are replicated above it.
This means half the work.

Now we are going to cut the work in another half as well.
Starting from the squares (1, 4, 6, 16, etc.), all cells 1 step diagonally removed are 1 lower.
Starting from the squares (1, 4, 6, 16, etc.), all cells 2 step diagonally removed are 4 lower.
Starting from the squares (1, 4, 6, 16, etc.), all cells 3 step diagonally removed are 9 lower.

See the pattern?
Here it is in visual form:
mult.jpg

So if you learn the squares, it is easy to move from place to place.

Here is an example:
8 X 12 = 10 X 10 - 2 X 2 = 100 - 4 = 96

Put in another way: the average of 8 and 12 is 10. 10 squared is 100, 8 and 12 are 2 removed from 10, so we need to subtract 2 squared or 4 from 100, 100 - 4 = 96.

Now look at the numbers and see if you can do this visually. Start with 100, then move 2 cells to 96.

4 Likes
1x99 Multiplication Table
2

I see Arthur Benjamin doing some 2 digit multiplication really fast, that i believe it was too common to him, like the multiplication table. So i think that by memorizing 2 digit multiplication, i can improve speed while make bigger calculation. You are right that we can use our memory system to handle this however 2 digit system will create duplicate image and easy for mistake. Before i had memorized cube of 2 digit number by AO (i not use P as it’s really hard for me to distinguish the P each other) + Major, but when using that system for other thing i often facing the ghost image. It’s not comfortable at all, so i stopped to revisit the cube for a time. Bad result is now my recall gone. Now i learn that i should reserve a unique journey/peg to memorize any specific task for long term.
For 2 digit multiplication, i may try to build a separate pegs for it.

3

Arthur Benjamin explains somewhere that this is the way he started remembering the 1-100 multiplication tables.
At some point when he was young he realised this relation with the squares I just described.

Think about it, If you remember the squares (only 100 in a sea of 10,000 possible muliplications), you can do half of the 10,000 by subtracting just 2 squares from each other.
I think a lot of people fail to see the power of this.

Another example: 9 X 15 = 12^2 - 3^2 = 144 - 9 = 135

How would you organise a system, without multiple images?

4

I plan to make another 100 pegs of Major and use it as 4 digit system like Wang Feng. Sorry, i have not done yet that can give the correct opinion. Hope this will work well.

5

Doing the “big table” is a ton of work…keep it going and please write here occasionally how it works.

6

a > b and b < a
a Ă— b = c

a - b = d
b Ă— d = e
b Ă— b = f
e + f = c

Example: 3 Ă— 7

7 > 3 and 3 < 7
7 Ă— 3 = 21

7 - 3 = 4
3 Ă— 4 = 12
3 Ă— 3 = 9
12 + 9 = 21

Example: 57 Ă— 114

114 > 57 and 57 < 114
114 Ă— 57 = 6,498

114 - 57 = 57
57 Ă— 57 = 3,249
57 Ă— 57 = 3,249
3,249 + 3,249 = 6,498

7

Hi David,

Nice way of calculating.
If I look through you somewhat cryptic post, you are basically saying that 3X7 = 3X4 + 3X3. True of course.
And 57X114 = 57X57+57X57.

If you memorized a lot of squares this is helpful.

8

I’m currently learning the squares at the moment, I didn’t think of using them in this way so that is really cool, thanks!!

9

Seamagu, knowing the squares is indeed very important!
They will turn up everywhere. Even in places where you would not expect them.

Example: 57 X 47 = 52^2 - 5^2
Or PhyberDragons way of 47^2 +10 X 47

Also; did you know that the squares close to 50 are quickly calculated like this:

2500 + (difference with 50) X 100 + (difference with 50)^2

So 52^2 = 2500 + 2 X100 +2^2 = 2704
And 47^2 = 2500 - 3 X 100 + 3^2 = 2209

1 Like
10

1dx2d multiplication is not difficult :confused:
2dx2d gets a bit harder :frowning:

You’ll need to learn different methods to master 2dx2d multiplication, for example:

23x69: here I use difference of squares
23x22: here i use base multiplication
43x47: here, I don’t know how this method is called but you can see that 3+7: 10 and they are in the same base
60x37: this is the same as 1dx2d, you only add one 0 to the answer
11x45: the 11 rule
25x64: the 25 rule
99x34: the 99 rule
55x65: PhyberDragons way :slight_smile:
19x19: this is a square. LOL

And many others. :slight_smile:

Indeed, learning the squares is very useful.
The good new is that some squares can be easily calculated :wink:

1 Like
11

There are some 2d tables that are worth learning from 1 to 9
I mean, the 36 table for example

36x1
36x2
…
36x9

why? Because you can convert km/h into mt/s and vice versa easily if you know them, also the 22 table, you can convert kg to lb and vice versa easily :slight_smile: and many others :slight_smile:

To calculate a 2d multiplication with those numbers, just add the products that you know :slight_smile:

1 Like
12

@chiguin can you please give examples for those methods of 2dx2d multiplication?

1 Like
13

@Kinma can you please tell me how to memorize times tables upto at least 20x20 using coding system/pegs/PAO system etc.?

14

I can not; for the simple reason that I don’t memorize these numbers.

For me, it is much more fun to actually calculate these numbers.

This is the way I do them.

23x69: There are a couple of ways to do this. One is starting with 23 X 70 = 1400 +210 = 1610. Then subtract 23 from 1610 = 1600 - 13 = 1500 + 13’s complement = 1587. See my post about complements if you want to know more about this.

Another way is realizing that 69 = 23 X 3. Then the calculation becomes 23 X 23 X 3.

If you must do the difference of squares: the average of 23 and 69 is 46, so 23 X 69 becomes 46^2 - 23^2. Incidentally, you might realize that 46 is 23 x 2, so 46^2 - 23^2 = 23^2 X ( 4 - 1 ), which takes you back to the previous calculation.

23x22: either do 23 X 11 X 2 = 253 X 2 = 506, or do 46 X 11 = 506.

Alternatively, start with 22: 22^2 +22 = 484 + 22 = 506

43x47: difference of squares is 45^2 - 2^2 = 2025 - 4 = 2021.
Criss cross leads to 40^2 = 1600, plus 40 X (3+7) = 2000 plus 3X7 = 2021.

60x37: Start with 60 x 40 = 2400 then subtract 3 x 60 = 180

25x64: = 100 X 16

99x34: = 3400 -34 = 3300 + 34’s complement = 3366

55x65: = 50X60 + (50+60) X 5 +25

19x19: = 20^2 - 2X20 + 1 = 400 - 40 + 1 = 361

3 Likes
15

Did u seen the best way how to learn multiplication tables?
http://Aztekium.pl/Master

1 Like
16

Please tell me how to learn tables 17 * 1
17 * 2
17 * 3
17 * 4
…
17*10

1 Like
17

17 = 10 + 5 + 2

Use distribution property :

17 Ă— 5 = 10Ă—5 + 5Ă—5 + 2Ă—5

= 50 + 25 + 10 = 85 :heavy_check_mark:

17 Ă— 8 = 80 + 40 + 16 = 136 :heavy_check_mark:

17 Ă— 9 = 90 + 45 + 18 = 135 + 18 = 145 + 8 = 153 :heavy_check_mark:

@Senkuu

17 Ă— 2 = 34 :heavy_check_mark:

17 Ă— 3 = 51 :black_circle:

17 Ă— 4 = 34 Ă— 2 = 68 :heavy_check_mark:

17 Ă— 5 = 85 = 170 Ă· 2 = (100 Ă· 2) + (70Ă·2) :yellow_circle:

17 Ă— 6 = 51 Ă— 2 = 102 :black_circle:

17 Ă— 7 = 17( 4 + 3 ) = 68 + 51 = 50 + 18 + 50 + 1 = 100 + 18 + 1 = 119 :red_circle:

17 Ă— 8 = 68 Ă— 2 = 136 :heavy_check_mark:

17 Ă— 9 = 17 Ă— 3 Ă— 3 = 51 Ă— 3 = 153 :black_circle:

Observe the pattern

Also learn left to right addition

589 + 642 = (500 + 80 + 9) + ( 600 + 40 + 2 )

( 500 + 600 ) + ( 80 + 40 ) + ( 9 + 2 )

= ( 500 + 500 + 100 ) + ( 40 + 40 + 40 ) + ( 11 )

= (1000 + 100) + (100 + 20 ) + ( 10 + 1 )

= 1231

3 Likes