### 20110305: 6 AM QUIZ (V) Introduction to Chapter 4: Algebra... Crystal Ball Gazing...

Could you UNCOVER THE MYSTERY behind the crystal ball?
(in the Algebraic way!)

1. This is because : Let the tens value of the 2 digit number is a and the ones value be b. So,10a + b, den 10a+b-(a+b)
which is (10a-a)+(b-b) so the answer is 9a. The answer will be also a multiple of 9. Firstly, the chart's number which is multiple of 9 will be the same symbol. After that, the crystal ball just need to show the symbol which represent all the numbers that is multiple of 9. So, we will,sort of,get tricked, but this is actually because of algebra.

I do this before in primary school...so i am just lucky. :D
BTWFYI, I am Owen.
P.S. Jia Qi and Matthew will also know the 'secret'.

2. Say two digit number is mn
Value of the two digit number is 10m+n
Add together both digits is m+n
Subtract from the value is 10m+n-m-n=9m
The answer is 9m (m=1,2...9), ie 9, 18, 27...81
From the chart, all these numbers always have the same symbol.
That's the answer that crystal ball always gives.

3. Firstly,not all the numbers at there are even possible ,only multiples of 9.
Let the tens digit be X and the ones digit be Y,so the number you are thinking of is XY
next they ask you to minus away the 2 digit of your number which is
XY-X-Y (keep in mind that the X and Y are simply the digits not the actual number)
which then can be turn into this:XY-X-Y=X0-X(the X and zero is a digit so EG.X0 can be 70,60,10,)
We can then write it as:XY-X-Y=X0-X=X*10-X*1=X*(10-1)=X*9
thus ,we can conclude that the digit is a multiple of nine
Confusing? Let me show a actual example,the number i'm thinking of is 52 so using the calculation just now is should be:
Don't believe? 5+2=7 ,52-7=45

As for the symbols i realized they change each time that is why you always get a different symbol, so since your answer will be a multiple of 9,all the symbol for multiple of nine is the same.(you can go check)

4. Lets say the first digit of the number is X and the 2nd digit is Y.

So 10X+Y-X+Y=9X
So the answer has to be a multiple of 9. So the program makes all of the multiples of 9 except 90 and 99 to have the same symbol. It then shows that symbol tricking you into believing that it has read your mind.

It cannot be 90 and 99 as the subtraction is too much. It is due to the fact that the digits above 90 all have to be at least subtracted more than 9 and thus cannot become the digits 90 and 99.

5. 10x+y-(x+y)
=10x+y-x-y
=9x

Where x and y are both positive integers. For example: 98-(9+8)=81
And: 9=x, so 9*9 also=81

The symbol for all the multiples of 9 are the same, as we have the relationship shown above.

6. Let one number be a and the another be b.
So ab can also be written as 10a+b.
So 10a+b - (a+b) / 10 a+b-a-b
which is equal to 9a.
Thus the symbol that the crystal ball shows will always be a multiple of 9.

7. 1st explanation : While looking through the symbols , I realised that after every turn , the symbols would change but every turn , the multiples of 9 , except 90 , would have the same symbols . And some of the numbers are impossible. So the symbols of the multiples of nine are always the answer.
2nd Explanation : Say that a and b are the two digits and a is in the tens place while b is in the ones place . so 10a+b-a-b = 9a
thus , we can conclude that the answer the crystal ball gives is always the multiple of nine.

8. I found out that whatever number you choose,the answer will be always be in the multiples of 9, and as I observed the patterns of the symbol, the multiples of 9 willl always have the SAME symbol, except for some of the bigger numbers like 90,99 etc. This is because that these numbers are impossible to be your answer. THey may also inject some numbers of the same symbols just to lessen suspicion.

9. Lets say the number you choose is xy where x and y are the digits of the number. xy=x*10+y.
you subtract (x*10+y)-(x+y)
=10x+y-x-y
= 9x

Thus whatever is your final answer will be a multiple of 9. So at every multiple of 9 a specific symbol is shown, which is the symbol in the crystal ball.

10. Lets assume the 1st digit is x and the second digit is y. So the any digit number is xy. As x is in the tens digit place, we take x times 10, which we will get 10x+y. In the end we minus the sum of the digits, which is 10x+y minus x+y then we will get 9x.
Almost all the multiples of 9 in the website has the same symbol, so we will get that symbol as we get 9x in the end.

@Owen hehehe

11. The digits you choose is w and c.
Since w is in the tens place so it would be 10 *w=10w but, c is in the ones place so it is only c.Therefore your number would be 10w+c.
Now they want u to subtract your answer from your original number which will look like this: (10w+c)-(w+c)=9w
Hence, they your answer will be a multiple of 9 and all multiples of 9 that are <90 have the same symbols.

12. Explanation : Let the "tens" place of the 2 digit number represent "a" and the "ones" place represent "b". So , 10a + b, then (10a+b)-(a+b)
which is (10a-a)+(b-b) thus the answer is 9a. Thus, the answer will be a multiple of 9. Firstly, the chart's number which is multiple of 9 will be the same symbol.Next the ball just needs to show a symbol that represents a multiple of 9. If you observe closely, only the multiples of 9 have non-repetitive symbols :)
By the way, Owen this one is a bit different but with the same logic.

13. Oops, Correction-- The symbols for the multiples of 9 are all the same it is just that they keep on changing :0

14. Also, 90 and 99 cannot be possible in the answer so they put different symbols for them.

Part 2
BTWFYI, Owen here.

15. Hm... arising from some of your responses:

Is AB same as "10A+B"?
#iwonder

16. @ms loh yes it is the same,unless it is a(b) which would mean axb.we times a by ten because it is in the tens place while b doesnt need to be multiplied because it is in the ones place

Example:
I thought of the number:79
(10*7+9)-(7+9)=9*7

Another pattern that we can observe is that
anything from 90 onwards ans will be 81
anything from 80 onwards ans will be 72
and then the pattern goes on...

also,the pattern above can prove that no multiple of 9 above 81 can be found as an answer.to do so the original number will have to become a three digit number like 100+ but the crystal ball has a limit to 2 digit numbers only...

Lovy Lim
Milo

17. also to add in,the chart actually changes after every round,so you wont be able to notice that all the multiples under 81 are the same symbols,i got a clue on the answer from this observation too :)

18. Firstly,the symbols representing the multiples of (including 9 , except 90)is the same.While the rest of the number's representing symbol is random.

10x+y-(x+y)=9x

The second digit in the number is in the tens place so the value of 'x' must multiple by 10.
eg: x=6
10x=10x6
=60
We will arrived at the answer 9x.The answer will be a multiple of 9 so the common symbol will always appear in the crystal ball to trick people.

19. 10x+y-(y+x)
=9x
Therefore, the answer is a multiple of 9 e.g. 9,18,27,36,45,54,63,etc.
e.g. I choose the number 91.90=10(9).10(9)+1-(9+1)=91-10=81.So,81 is a multiple of 9.Then, the crystal ball will choose a common symbol for all the multiples of 9 and that is how it tricks people

20. cuz every time after we are done with calculating with the expressions,the answer would always be a multiples of 9 and i noticed that for every multiples of 9,the symbol is the same though it kept changing from time to time

21. The symbol of the multiples of 9 are all the same so the computer only need to show the symbol of multiples of 9. At the same time, when we finish the calculation, the results are also always multiples of 9. So that is the answer to the mystery~~~

22. @Ryan Tan: You mentioned that it doesn't work with numbers greater than 90. Good to test it out :)

23. This comment has been removed by the author.

24. Mark :) :( :&#39;( (^^^) :D :O :42: O:) :p 3:)3/07/2011 02:28:00 PM

The symbols of the multiples of 9 are al the same. And since when you subtract and get the answer, it is always a multiple of nine, the system only needs to guess the symbol of 9 and its multiples, therefore it is always correct.