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 The Remaining Digits (Posted on 2005-08-04)
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?

 See The Solution Submitted by pcbouhid Rating: 4.3333 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re(3): George's answer | Comment 5 of 20 |
(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 841169             578                 235256                                 139 739289                 457     134324 578                     189361         457         478     458 259529                 478             136 736576         134 189                     239729                 458                 356784 235 139         259 136841     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

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