I have a bag containing the digits 0 through 9, and used six of them to stick two different threedigit perfect squares on the foreheads of Paul and George. Both Paul and George know this fact, but each one can see only the other's number.
I ask Paul, "How many of the digits remaining in my bag can you exactly tell me?"
Paul replies, "Three."
If I now ask the same question to George, what should he reply?
(In reply to
re(2): George's answer by Bob Smith)
Sorry. I misunderstood.
Stated in more detail:
There are only 13 3digit perfect squares that have no repeated digits. Of these, 16, 196 and 961 share the same digits, as do 256 and 625, so there are only 10 sets of 3 digits. None of them have a zero, so it's already known that a zero is in the leftover pile.
The possible combinations for the two people are shown below, marked with the leftovers other than zero:
169 256 289 324 361 529 576 729 784 841
169 578 235
256 139 739
289 457 134
324 578 189
361 457 478 458 259
529 478 136 736
576 134 189 239
729 458 356
784 235 139 259 136
841 739 736 239 356
If the left column represents what Paul sees on George's forehead, then he would have been able to identify two of the digits (plus the zero, making three in all) left over, only if he saw 256.
So when George saw the 784 or 841 on Paul's forehead, he'd know precisely the four remaining digits: either 1, 3, 9 and 0 or 7, 3, 9 and 0, respectively.
Edited on August 4, 2005, 7:38 pm

Posted by Charlie
on 20050804 18:15:04 