# Simple number tricks?

1. Aug 24, 2006

### phoenixy

Hi,

I'm trying to get my 8 years old cousin into math. I taught him a couple math tricks that he can workout in his head, so that he can impresses his classmates and teacher(in "I can do this in my head faster than you with a calculator" kind of way).

Here the ones that I had off the top of my head:
11*XY = X (X+Y) Y; 11*45 = 4 (4+5) 5 = 495
(X5)^2 = (X*(X+1)) 25; 75^2 = 7*8 25 = 5625

Anybody knows any similar number games that can be done using only your head?

2. Aug 24, 2006

### quasar987

Simply (X+y)²=X²+2Xy+y²; 54²=50²+2*50*4+4²=2500+400+16=2916

with perhaps the combined knowledge that (X0)²=X²00 to get the first square quick.

3. Aug 24, 2006

### Robokapp

He's 8 and knows how to square a number? that's pretty impressive...I didn't until Middle School...

4. Aug 24, 2006

### phoenixy

he only knows times table. He can pop 12X12 off the top of his head though. I didn't use the word "square"; I just say 75X75. He can now do it in about 3 seconds. My effort definitely payed off when he announced "now I like math"

This one might work. But kids do have short memory retention, so the addition could be a problem.

5. Aug 25, 2006

### metacristi

I don't know if this is suited for small kids but here is a variant of the usual way to multiply numbers (my father taught it me once when I was 11):

Let's say that we have to multiply numbers having 2 digits, for example 97x86. The result can have at most 4 digits (because 100x100=10000). Let now the result be ABCD.

97x
86
---------
ABCD

The steps involved to find the result are:

1. 7x6=42 ---> D=2 and we have a carry R1=4

2. (9x6)+(7x8)+R1=54+56+4=114 ---> C=4; R2=11

3. (9x8)+R2=72+11=83 ---> B=3 and A=8

The number seeked is 8342. The method is a variation of the usual way to do multiplications but is easier to use for mental calculations, especially for numbers having 2 or 3 digits (at least in my case).

It can be extended for a greater number of digits too. For the 3 digit numbers multiplication, which is still in the reach of normal people, the strategy is:

784x
529
----------
ABCDEF

The result can have at most 6 digits (1000x1000=1,000,000 has 7 digits)

The steps involved:

1. 4x9=36 ---> F=6 and a carry R1=3

2. (4x2)+(8x9)+R1=8+72+3=83 ---> E=3 and R2=8

3. (4x5)+(7x9)+(8x2)+R2=20+63+16+8=107 ---> D=7 and R3=10

4. (7x2)+(8x5)+R3=14+40+10=64 ---> C=4 and R4=6

5. (7x5)+6=35+6=41 ---> B=1 and A=4

The seeked number is therefore 414736

The method does not seem easy but in reality is not so, once you learn the principles and get some practice it becomes much more friendly...Has this method been ever of real help for me? Well yes...when I was in the army...I was doing such mental calculations during the long boring nights when I was on guard :-)

Last edited: Aug 25, 2006
6. Aug 25, 2006

### metacristi

When (X+Y) is greater than 9 the method can be extended as:

11*XY ---> the result N is of the form: [X+1][X+Y-10][Y]

or alternatively [X + the decimal digit of (X+Y)][the units digit of (X+Y)][Y]

X=7, Y=9 ---> 11*79=[7+1][6][9]=869

Step by step:

[7][7+9][9]
[7][16][9]; Carry1=1 (the decimal place of 16)
[7+Carry1][6][9]
[7+1][6][9]
[8][6][9]

Likewise for 3 digit numbers:

11*XYZ = [X + the decimal digit of {(X+Y) + the decimal place of (Y+Z)}][the units digit of {(X+Y) + the decimal place of (Y+Z)}][the units digit of (Y+Z)][Z]

11*789 = [7+1][6][7][9] = 8679

Step by step the process can be written as:

[7][7+8][8+9][9]
[7][7+8][17][9];
[7][7+8+Carry1][7][9]=[7][7+8+1][7][9]=[7][16][7][9]; where Carry1=1 (from 17)
[7+Carry2][6][7][9]; where Carry2=1 (the decimal place of 16)
[7+1][6][7][9]
[8][6][7][9]
8679

This method can be generalized for any other number (multiplied by 11).

Last edited: Aug 25, 2006
7. Aug 25, 2006

### metacristi

Some other easy tricks which I remember now:

Multiplication by 21: multiply the number by 2, multiply the result by 10, finally add the initial number to the previous result (an easier way to put N*[2*10+1]).

Ex. N=29

N*2=58
(N*2)*10=580
21*29=580+29=609

Multiplication by 31: multiply the number by 3, multiply the result by 10, add the initial number.

Ex. N=32

N*3=96
(N*3)*10=960
31*32=960+32=992

The same holds (obviously) for multiplication by 41 (first muliply by 4), 51 (first muliply by 5) etc. But of course it's much more difficult to do such calculations in mind...

PS: I forgot 11 (multiply by 1 first).

Last edited: Aug 25, 2006
8. Aug 25, 2006

### Alkatran

Multiplying by 9 can take time, until you realize adding a zero and subtracting once is easier.
9*x = 10*x - x
example: 9*473 = 4730 - 473 = 4227

When you multiply two numbers you can do x*y or find the midpoint (m) and difference (d) and do:
x*y = (z-d)(z+d) = z^2 + d^2
example: 13*15 = 14^2 - 1 = 195
this makes memorizing squares more useful.

9. Aug 25, 2006

### Integral

Staff Emeritus
http://en.wikipedia.org/wiki/Mental_arithmetic" [Broken] has some tricks. You might want to do some searches on Jakow Trachtenberg. He was an engineer imprisoned during or just after WWII who while locked in a cell with no paper or pencil developed a mental arithmetic system.

Last edited by a moderator: May 2, 2017
10. Aug 25, 2006