Alice: Pick a 5-digit number.
Bob: Got it.
Alice: Now reverse it and subtract the lesser from the greater.
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