Guess How I Do It!

1. Take abcd as a 4-digit number.
The value of abcd is 1000a+100b+10c+d.

If we jumble up the digits, there are a few possibilities of the result after subtraction of the numbers.

For a, it could be 1000a-1000a=0, or 1000a-100a=900a, or 1000a-10a=990a, 1000a-a=999a.
For b, it could be 100b-1000b=-900b, or 100b-100b=0, or 100b - 10b=90b, or 100b-b=99b.
For c, it could be 10c-1000c=-990c, or 10c-100c=-90c, or 10c-10c=0, or 10c-c=9c.
For d, it could be d-1000d=-999d, or d-100d=-99d, or d-10d=-9d, or d-d=0

All of the above are multiples of 9, hence the result of the subtraction is a multiple of 9. After adding every digit of the result of the subtraction, the answer should also be multiple of 9.

For example, if you type 870, 8+7+0=15, we need single digit 3 to make the total as 18 which is multiple of 9.

-Nina

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4. Take a as thousands place, b as hundreds place, c as tens place, d as ones place so 1000a+100b+10c+d.

This is a rule i observed: the rule of nine
1000a+100b+10c+d = (999a+99b+9c) +(a+b+c+d) = (Multiple of 9) + (Sum of digits) so the sum of the digits would be a multiple of 9. Other examples : 810 , all the multiple of 9.

So, when the digits get scrambled up, the sum of the digits would still be the same. The difference would also be the same , a multiple of 9. After canceling out one digits, and adding up the remaining digits, we just need to find the nearest multiple of 9 to the sum of the digit but the multiple of 9 must be more as.......

a+b+c+d = multiple of 9 , cancel one digit, lets take it as 'a' , so b+c+d= multiple of 9-a and thus we must take multiple of 9 - b-c-d to get a.

BTWFYI, I am Owen.
Once agn, did before in primary school so is just lucky

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6. Choose any 4 digit number, 1000w + 100x + 10y+ z.

The possible combinations when you jumble the digits are:

(a)1000w + 100x + 10z + y
(b)1000w + 100z + 10y + x
(c)1000z + 100x + 10y + w
(d)1000w + 100y + 10x + z
(e)1000y + 100x + 10w + z
(f)1000x + 100w + 10y + z

If you subtract each possible combination form 1000w + 100x + 10y+ z or subtract 1000w + 100x + 10y+ z from the possible combination, you get:

(a) (1000w + 100x + 10y+ z)-(1000w + 100x + 10z + y)
=1000w + 100x + 10y+ z-1000w - 100x - 10z - y
=9y-9z
=9(y-z)
(b) (1000w + 100x + 10y+ z)-(1000w + 100z + 10y + x)
=1000w + 100x + 10y+ z-1000w - 100z - 10y - x
=99x-99z
=99(x-z)
(c) (1000w + 100x + 10y+ z)-(1000z + 100x + 10y + w)
=1000w + 100x + 10y+ z-1000z - 100x - 10y - w
=999w-999z
=999(w-z)
(d) (1000w + 100x + 10y+ z)-(1000w + 100y + 10x + z)
=1000w + 100x + 10y+ z-1000w - 100y - 10x - z
=90x-90y
=90(x-y)
(e) (1000w + 100x + 10y+ z)-(1000y + 100x + 10w + z)
=1000w + 100x + 10y+ z-1000y - 100x - 10w - z
=990w-990y
=990(w-y)
(f) (1000w + 100x + 10y+ z)-(1000x + 100w + 10y + z)
=1000w + 100x + 10y+ z-1000x - 100w - 10y - z
=900w-900x
=900(w-x)

As 9, 99, 999, 90, 990 and 900 is a multiple if nine, the answer of the subtraction of the numbers is a multiple of nine. When you pick a digit and jumble the rest, the sum of all the digits is still a multiple of 9, thus it is divisible by 9. When you key in your final answer, the system adds the digits of the number you have given. The difference between the smallest number that is closest to the sum of the digits you have given and the sum of the digits you have given is the digit you are thinking of.

Lets take 910 for example. Lets jumble 910 to 109. The difference between them is 720. Lets circle the digit 2 in 720. We jumble the remaining digits and we get 70. The sum of 7 and 0 is 7. The closest multiple of 9 to 7 is 9. 9-7=2. Thus we get 2 as the answer

-Abhimanyu Arora

7. @owen:
Something I don't quite understand: Why is it that
1000a+100b+10c+d
= (999a+99b+9c) +(a+b+c+d)
= (Multiple of 9) + (Sum of digits)
so the sum of the digits would be a multiple of 9.
Q: How do you know that the sum of digits will sure be 9?

8. @Nina @Abhi:

Good attempt! Let's see if there are other "takers" to this question :)

9. The is a loophole to the question : If we take 101 , then jumble up it to become 110. We then take 110 minus 101 which we will get 9, a single digit number. But it asks us to circle one digit and not include that when we type it out, hence we have to choose 9 (the only number) and the flash player does not allow us to enter no digits. Loophole! :D

10. would someone able to explain the 'loophole' that jia Qi pointed out?

11. @ Jia Qi: For 3 digit numbers, a=0. It has the same result in my way. :)

12. After circling 9, 0 has to be entered into the blank.

13. 9-0=9, thus Fido will give you the answer as 9.

14. i cant really come up with an algebraic expression but i observed this,what ever your answer is for the,for example:(1000A+100B+10C+D)-(1000D+100C+10B+A),when you add all the single digits in that number together,you will get 18,so since we have to key in the other few numbers except the one which we circled,all the computer has to do is put 19-(A+B+C) if we circled D for example.

15. My 2nd way to explain,

4 or 3 digits number: abcd (3 digits, a=0)
Value of abcd is 1000a+100b+10c+d=(999a+99b+9c)+(a+b+c+d)

After jumble up become wxyz (formed by 4 digits of a,b,c,d)
Value of wxyz is 1000w+100x+10y+z=(999w+99x+9z)+(w+x+y+z)

a+b+c+d=w+x+y+z
abcd-wxyz=999(a-w)+99(b-x)+9(c-z)

Hence the result of subtraction is multiple of 9. After adding every digit of the result of the subtraction, the answer should also be multiple of 9.

For example, if you type 870, 8+7+0=15, we need single digit 3 to make the total as 18 which is multiple of 9.

If the result of subtraction is single digit 9, it has to be the circled number, 0 has to be entered into the blank. Fido will give the answer of 9.

Nina

16. Miss Loh

Initially, I do not really grasp the idea of why Fido can guess my answer so I did not post any explanation. I hope you will understand . Can I approach you to find out what is the explanation about?

17. @Mavis:
Sure, I'll explain to you... U may want to consider looking at Nina's explanation :)
Both explanations are pretty clear.

Cheers, Ms Loh

18. @Nina: Well Done!
One ninja for you :)