Alice: Pick a 5-digit number.
Bob: Got it.
Alice: Now reverse it and subtract the lesser from the greater.
Bob: Done.
Alice: Now pick one of the five digits of the result and circle it (but don't choose a zero). Tell me the other digits, in some random order.
Bob: Sure; they are 1, 1, 0, and 8.
Alice: Then the circled digit is....

Alice does indeed correctly give the circled digit. How did she do it?

Let the bigger of two reversals be written nas a concatenation (ABCDE), a lesser (EDCBA). The difference is 9999(A-E)+999(B-D).

This difference, consisting of 3,4, or 5  digits, is clearly divisible by 9, so the erased non-zero digit  is easily recovered by complementing the sum of the revealed digits to the  nearest multiple of 9, e.g, 1245==>12==>18-12=6.

The trick will  not work on a palindrome , but then Alice cannot pick a non-zero digit.

Edited on November 3, 2012, 11:22 am

Posted by Ady TZIDON on 2012-11-03 11:14:42