I have a bag containing the digits 0 through 9, and used six of them to stick two different three-digit 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 3-digit 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 2005-08-04 18:15:04