## Saturday, 20 February 2016

### 6 AM Quiz: Let's Test FIDO out!

In this round of 6 am quiz, you will attempt to uncover the mystery behind Fido's Mind-Reading ability.

There are several ways to explain Fido's trick, using mathematical reasoning.
Make an attempt to use ALGEBRA to explain the "trick"?

Have Fun!

1. This comment has been removed by the author.

2. It does not work

1. 3. This comment has been removed by the author.

4. It did't work for me..

5. let the bigger number be 'abc'. This breaks down to:
100a + 10b + c

The reversed number is 'cba'. This becomes:
100c + 10b + a

When you subtract, the 10b terms cancel out and you get:
99a - 99c

Since 'a' is greater than 'c' (and all digits are different), this will be a positive number. (If a and c were the same, you would end up with zero. But this is why they tell you to pick a "completely random number with all the digits different".)

You can factor out a 9:
9(11a - 11c)

So this proves that no matter what 3 digit number you pick, the result will be a multiple of 9.

The same is true for a 4 digit number 'abcd':
1000a + 100b + 10c + d
The reverse is 'dcba'
1000d + 100c + 10b + a

Subtracting:
999a + 90b - 90c - 999c

Again you can factor out a 9:
9(111a + 10b - 10c -111c)

You could actually do this with 2 digit numbers, 5 digit numbers, 6 digit numbers, etc. With 2 digit numbers, you are liable to end up with only a single digit (9), so that wouldn't be too exciting and the trick might be apparent. With more digits, you are likely to make a subtraction error and ruin the trick. So they settled on 3 or 4 digits.

As we learned in school, the digits of a multiple of 9 also add up to a multiple of 9.
So when you leave out a digit, the program can easily figure out what number is missing to make it add up to a multiple 9. For example, 1089... If we tell the program everything but 1, the remaining digits (0,8,9) add up to 17... the next multiple of 9 is 18. The difference is 1. Or if we pick 8, the remaining digits (1,0,9) add up to 10. The next multiple is 18 and the difference is 8.
If we pick 9, the remaining digits (1,0,8) add up to 9. The *next* multiple of 9 is 18, so the difference is 9. Now notice they told you *not* to circle a zero (giving some excuse like "it is already a circle"). The reason they did this is that subtracting 0 or 9 will always make another multiple of 9, and it wouldn't be clear if the number that was removed was a 0 or 9. If they disallow you from picking a zero, this ambiguity goes away. Thus the program always looks for the digit that will get to the next multiple of 9.

6. 7. it works btw it is mind-blowing

8. 4321 rearranged to 1423
4321 minus 1423 = 2898
I choose to circle the 9, leaving the numbers 2, 8 & 8 to be typed in.
The only number that may be added to the sequence 2+8+8 and be divisible evenly by 9 is 9 (the digit I circled). [2+8+8+9=27]

1. They all can be perfectly divided by 9

9. The trick has to do with divisibility by 9. Try the first part of the
experiment again. Take any number (it doesn't really have to be 3 or 4
digits long, I imagine that's just so the computer has it a bit easy).
Mix up the digits and perform the subtraction.

Now, divide the result by 9. You should come out with a whole number.
In other words, the resulting difference should be divisible by 9.
This works no matter what original number you pick. If you are
familiar with number theory, the proof is simple enough. If not, just
take my word for it that the difference will always be divisible by 9.

So, we have a number divisible by 9. One property of numbers divisible
by 9 is that the sum of the digits of such numbers is also divisible by 9.

For example: 4059 is divisible by 9. 4+0+5+9 = 18. 18 is divisible by
9, so 4059 is divisible by 9.

4057 is NOT divisible by 9. 4+0+5+7 =16. 16 is NOT divisible by 9, so
4057 is NOT divisible by 9.

Let's use 4059 for now. If you pick 5, you enter 409 into the program.
The program adds the digits up: 4+0+9 = 13. It finds the smallest
number that can be added to 13 to get a multiple of 9. In this case 5:

18 - 13 = 5

So, 5 is the missing number.

That's why you are not allowed to pick 0. If the criterion is simply
divisibility by 9, then the program is unable to tell the difference
between 0 and 9.

For example:

4059 --> 459 ---> 4+5+9 = 18

The smallest number that can be added to 18 to get a multiple of 9 is
0. The other number that can be added to get a multiple of 9 is 9
itself. So, both 0 and 9 are possible choices. But since you are not
allowed to choose 0, the program will give you 9.
Try it out. Enter 459 into the machine. It will not give you 0, which
is the number we chose above. It will give you 9 instead.

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