Unbelievable - The Flash Mind Reader

[Kishan]

Just check this URL:

 

http://www.cyberglass.co.uk/assets/Flash/psychic.swf

 

I have never seen such an amazing site. Think about the logic behind this.

[vsdprasad]

this is tricky but simple.

every number you get (by adding the digits of a 2 digit number and subtracting this from the number)is a multiple of 9. and all of these multiples of 9 have the same symbol.

the symbol changes when you try again but all the multiples of 9 have the same symbol.

/images/graemlins/cool.gif

-Prasad.

[Kishan]

It's very simple now.

 

- Kishan

[gHari]

I first encountered this Modulus Nine Magic in grade six when our school principal demonstrated its practicality for our class. I've used it ever since, and it is likely the reason for my future success in mathematics; at least certainly for my success in arithmetic in grade school and then in exam calculations throughout academia.

 

I will present it briefly now for your kids (more details and tutoring available on request). It truly is magic and can tell you almost immediately if you have added, subtracted, multiplied or divided incorrectly. I seemed to be the only one who picked up on it in my class; but whoever does will have a distinct arithmetic advantage over their classmates. Perhaps it seemed too good to be true for others; indeed, I did not hear the proof for it until University.

 

Modulus means the lowest remainder left over when you divide the modulus into a number. Modulus(3) of 7 would be 1 since when you divide 3 into 7 the remainder will be 1. Modulus(9) however has a special feature because it is one less than the base 10 which we use universally (I guess because of our ten fingers). Modulus(9) has the special property that when you add the digits of a number together (no matter how big or small the number) the resulting total of the digits will always have the same modulus(9) as the original number. If the total is greater than nine, the digits can again be added together, again producing the same modulus(9) of the original number. Continuing to add multiple digit results will eventually yield a single digit from 0 to 8 (since a 9 becomes a 0, i.e. there is a zero remainder dividing 9 by 9). That is the actual modulus(9) of the original number. An example may clarify.

 

Original number: 123456

Adding digits we get 1+2+3+4+5+6 = 21

Adding these digits we get 2+1 = 3.

Therefore the modulus(9) of 123456 is 3.

You can verify that when you divide 9 into 1232456 you will have a remainder of 3.

 

There is even a shortcut whereby you can very quickly determine the mod(9) of a number. If any of the digits add to 9 (or 18), then they need not be added. Therefore in the example, 123456, we can disregard 3 and 6, as well as 4 and 5 since both sets of digits add to nine. Thus we needed to only add 1+2 quickly reaching 3. I used to "cast out the nines" (as my principal called it) on the car licences that passed my father's car on the highway. One can become very adapt at reaching the required calculation in mere seconds.

 

Well, this is cute, but you're still not impressed. Soon. Soon.

 

To verify additions we take advantage of the fact that when two numbers are added together, the modulus(9) of the answer will always be the sum of the mod(9)s of the two numbers. Yes, we need an example.

 

123456

345678 +

-------

469134

 

No calculator used, so let's see if we can prove it by mod(9) wizardry. We've already established that the mod(9) of the first number was 3. The second number's mod(9) can be calculated as follows: [Method 1] 3+4+5+6+7+8 = 33; 3+3 = 6 .... OR ...[Method 2 Faster] disregard the 3+6 and 4+5 leaving us with only 7+8 which is 15; which again adds to 6.

 

We now know that answer must be 3+6 = 9 which reduces to 0 since it is exactly divisible by 9 yielding a zero remainder. Again, now we look for the mod(9) of 469134: adding to get 4+6+9+1+3+4 = 27 then 2+7 = 9 which reduces to 0. Using the express method of casting out the nines makes it even easier since all digits are disregarded with 4+1+4=9 and 6+3=9 and 9=9 giving the total of zero. Since the answer is mod(9)0 which is to be expected when adding a mod(9)3 number and a mod(9)6 number, we can be happy. This illustration may make this more clear and memorable:

 

123456 ... 3

345678+...+6

------- ..--

469134 ... 9 ---> 0 . . This should match the answer.

 

Subtraction is much similar except that the mod(9)s are subtracted not added.

 

123 ... 6

108-...-0

--- . .. --

15 . . 6 . . . The answer should be mod(9)6.

 

NOTE: if you have to subtract mod(9)6 from mod(9)3 just like decimal mathematics you have to borrow and carry, but rather than ten you only borrow nine. So we add 9 to 3 and get 12 and then from 12 we subtract 6 arriving at 6.

 

Multiplication is about how you might imagine it if you've read this far. Just multiply the mods of the two numbers and that answer's mod should match the mod(9) of the result.

 

11 . . . 2

12x . . x3

--- . .. --

132 . . . 6 . . . . The answer should be mod(9)6.

 

Division is slightly more compicated since sometimes a remainder is involved. Here, the mod of the answer times the mod of the divisor, added to the mod of the remainder should equal the mod of the quotient.

 

2345 divided by 62 - again no calculator!

.... ___37___ . . . . . . Mod(9)37 = 3+7=10 = 1+0 = 1

62 )2345 . . . . . . . . . Mod(9)62 = 6+2 = 8

.. .. 186^

.. . . ---

. . . . 485

. . . . 434

. . .. . ----

. . . .. . 51 . . . . . . .. Mod(9)51 = 5+1 = 6

 

That is in the example above:

1 x 8 = 8

8 + 6 = 14 and 1+4 = 5 or mod(9)5

The mod of the quotient is indeed:

2345 => 2+3 = mod(9)5.

 

The method is not 100%, since if we really mess up the arithmetic there are many possible mistaken answers that will match our mod(9)5 requirement, but the odds are greatly in our favour. I can't remember ever not catching an error using this method. The secret is in quickly 'casting out the nines', then in only a few seconds one can verify even the largest of calculations.

 

I realize that this has been very complex and the chances that I have explained it well enough are mighty slim. If you have grade school kids, you would do well by them to pass this on to them. I will happily cooperate. As I mentioned, the other kids in my class must have thought it was too outrageous to adopt it, but I got nothing but 100s on my tests because of it.

 

Here is one last outrageous example (this time I will use the calculator). I will use the fast method of casting out the nines and try to depict that silent activity with highlighting:

 

12<font color="blue">3</font color><font color="red">45</font color><font color="blue">6</font color>7 . . . . mod(9) is 1

123<font color="blue">45</font color>6 x . . . mod(9) is 1+2 = 3

------------ . . . . 3 x 1 = 3 and therefore the mod(9) of our answer should be 3

15<font color="blue">2</font color>4<font color="red">1</font color>4<font color="blue">7</font color><font color="yellow">0</font color><font color="red">3</font color>5<font color="red">5</font color>2 . . . mod(9) is 1+2 = 3 !!!!

 

Therefore we can have a high level of confidence that the answer of 152414703552 is indeed correct.

 

gHari

[Gauracandra]

Now release the real Ghari nice and slow and there won’t be any trouble /images/graemlins/smile.gif

[vsdprasad]

Fascinating, Jim!!

Did Ramanujan by chance reincarnate as Gary? /images/graemlins/smile.gif

 

-Prasad.

[theist]

Why didn't you tell me right at the top that division would be involved.

[Avinash]

Take a positive integer. If it has more than one digit, then add the digits. If the resule again has more than one digit, then again add the digits. Keep doing that till you get a single digit number. Let as call that digit as d.

 

Take any two positive integers. Call the larger as X and the smaller as Y.

 

Let diff = X-Y

 

d of diff = (d of X) - (d of Y)

if d of X is not equal to d of Y

 

Otherwise

d of diff = 9

 

Try it with any combinations of X and Y. It works.

 

[theist]

Yeah its all clear now. /images/graemlins/confused.gif

[Avinash]

Let there be two positive numbers x and y. Let z = x + y

Calculate the sum of digits in x. Keep doing it till you get a single digit (the same you did to the multiples of 9). Do the same with y. Now you have two digits. Add the two digits. If you get a two digit figure, then add them to get a single digit. You will find that this is the sum of digits of z.

Example:

Let x = 155 (sum of digits = 11, sum of digits of 11 = 2)

y = 75 (sum of digits = 12, sum of digits of 12 = 3)

 

z = 155 + 75 = 230 (sum of digits = 5 which is same as 2 + 3, calculated above).

 

Now I will explain why the sum of digits of any multiple of 9 is 9. The sum of digits of 10 is 1. 9 is 1 less than 10 and 0 is 1 less than 1 (sum of digits of 10). So, in this way 9 behaves somewhat like 0. You multiply any number to 0, you get 0. Similarly, you multiply any number to 9, you will get a no. whose sum of digits is 9. You add 0 to any number, that number is not changed. Similarly, you add 9 to any number, the sum of digits of that number is not changed.

 

Example:

35 + 9 = 44

 

Sum of digits of 35 = 8

Sum of digits of 44 = 8

 

I talked about positive numbers. What about negative ones? We know that adding zero to any number does not change the number. We have found a similarity between 0 and 9. So, if you have a negative number, then add keep adding 9 to it till you get a positive number. In other words, add a multiple of 9 to get a +ve no. Then you will find that it shows the behaviour I have described for positive numbers above.

 

Example:

 

34 - 14 = 20

 

34 - 14 is same as 34 + (-14).

34 is already a +ve no.

-14 is a -ve no.

 

-14 + 9*2 = -14 + 18 = 4 (+ve)

 

Instead of doing 34 + (-14), let us do 34 + 4. We get 38. Sum of digits = 11. Sum of digits of 11 = 2.

 

Sum of digits of 20 (34 - 14) is also 2.

 

Now let us do the reverse i.e.

 

14 - 34

 

We get -20

 

14 - 34 = 14 + (-34)

14 is +ve and -34 is -ve.

-34 + 9*4 = -34 + 36 = 2

 

Instead of 14 + (-34), let us do 14 + 2. We get 16. Sum of digits = 7

 

14 - 34 = -20 (-ve)

-20 + 3*9 = -20 + 27 = 7

So, both are 7.

 

I have described here addition and subtraction. You will find such kind of behaviour holding true for multiplication and division also. Of course, in case of division, the division must finish. The result can be fraction but we must know the exact digits after decimal.

Example:

25 * 3 = 75

Sum of digits of 25 = 7

7 * 3 = 21, sum of digits = 3

 

Sum of digits of 75 = 12, again sum of digits = 3

 

Example:

17/2 = 8.5

Sum of digits of 17 = 8

8/2 = 4

Sum of digits of 8.5 = 8 + 5 = 13, whose sum of digits = 4