The squares are 169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961.
Since 0 isn't used in any of the squares, it must be one of the four digits left in the bag.
Paul has to conclude knowledge of two digits from George's forehead.+---------------------------+------------------------------+
| If Georgeīs | Paulīs number |
| forehead shows | can be |
+---------------------------+------------------------------+
| 169 (or 196, 961) | 324, 784 |
+---------------------------+------------------------------+
| 256 (or 625) | 784, 841 |
+---------------------------+------------------------------+
| 289 | 361, 576 |
+---------------------------+------------------------------+
| 324 | 169 (196, 961), 576 |
+---------------------------+------------------------------+
| 361 | 289, 529, 729, 784 |
+---------------------------+------------------------------+
| 529 | 361, 784, 841 |
+---------------------------+------------------------------+
| 576 | 289, 324, 841 |
+---------------------------+------------------------------+
| 729 | 361, 841 |
+---------------------------+------------------------------+
| 784 | 169(...), 256(625), 361, 529 |
+---------------------------+------------------------------+
| 841 | 256(625), 529, 576, 729 |
+---------------------------+------------------------------+ From that list, only if Georgeīs forehead shows 256 (or 625) can Paul know three digits. 8, 4 must be on his head and 0, 1 , 3, 7, 9 may be in the bag. He doesn't know if 784 or 841 is on his head, therefore he doesn't know if 1 or 7 is in the bag. The digits he knows are in the bag are 0, 3, and 9.
George, of course, being equally intelligent, figures this out; figures he must have 256 on his head for Paul to answer "three."
And can piece together the digit puzzle from what he can see on Paul's forehead. He'll know all four digits in the bag : 0, 3, 9 and either 1 or 7 depending on what Paulīs forehead shows.
So, Georgeīs answer is "four."
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