## Wednesday, March 9, 2011

### Chapter 4: Algebra... Crystal Ball Gazing...

Could you UNCOVER THE MYSTERY behind the crystal ball?

1. I think this is why...
Let the first digit of our number be a and the second digit be b.
(a) our 2 digit number will be 10a+b
(b) the two digits added together will be a+b
(b) subtracted from (a) is (10a+b)-(a+b)=9a
all possible values of a are: 1, 2, 3, 4, 5, 6, 7, 8 and 9
hence, all the possible values for 9a are: 9, 18, 27, 36, 45, 54, 63, 72 and 81
I realised that the symbols for the above numbers are the same as the symbol that appears in the crystal ball.
Hence,the crystal ball just shows us the symbols for all the multiples of 9 (excluding 90 and 99) and it would be correct, unless we have done the steps wrongly.

2. This is what i think:
All the numbers that you think of subtracted by the two digits added together is divisible by nine.
e.g.35-8=27
95-14=81
78-15=63
No matter how many times they randomize the symbols, the simbols representing nine and its multiples are always the same so there is no chance of the 'crystal ball' getting the wrong symbol.

3. This should be how it works:
12 - 3 = 9
24 - 6 = 18
35 - 8 = 27
45 - 9 = 36
55 - 10 = 45
75 - 12 = 63
98 - 17 = 81
See that? Whatever two digit number that minus the sum of its two digits added, will get a multiple of 9 until 81 because thats is the maximum you can get with two digit. Look below
99 - 18 = 81
99 is the biggest two digit number you can get and if you minus the sum of its two digits, you can only get 81.
And if you see the symbol of all the multiples of 9 until 81, all the symbol is the same and after seeing, click on the crystal ball. It will be that symbol.

4. No matter what number you use, the answer will always be in multiples of 9. The symbols, no matter how randomly they are arranged, all the numbers which are multiples of 9 have the same symbol assigned to them, that is until the number 81 because 99 is the largest 2 digit number and that will be the largest possible answer.

To take the largest 2 digit number;99.
99-(9+9)
=99-18
=81

Some random number;64, 47
64-(6+4)
=64-10
=54

47-(4+7)
=47-11
=36

5. This is my solution:
No matter what double-digit number you use, you will have a final answer which is a multiple of nine. And when you look up the chart, all the multiples of nine have the same symbols.

Example:
10: 1+0=1 10-1=9
11: 1+1=2 11-2=9
50: 5+0=5 50-5=45
69: 6+9=15 69-15=54

They all end up as multiples of 9, which are represented by the same symbols. There is, therefore, no way the crystal ball will get the wrong symbol.

6. Your final answer will always be a multiple of 9. Also, these numbers always seem to have the same symbol. Therefore, the crystal ball will always show the correct symbol (your answer).