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From the first two clues, we know that two of the digits are in the 1-4 range, and one is in the 5-8 range, so the fourth digit in the master number is either a 9 or a 0 (since three of the digits are accounted for and repeats are not allowed). Assume that neither 1 nor 4 is in the master number. Then from clue (3), 7 and 0 would both have to be in the master number (one gets an X and the other gets an O). This would mean that neither 6 nor 9 were in the master number. If that were true, there could not be an X in the grade to guess (4), as all four of those digits have been eliminated. Since this is not the case, our assumption is incorrect, and either 1 or 4 is in the master number. From clue (4), if either 1 or 4 is in, then 6 and 9 have been eliminated as possible digits, and if 9 is out then 0 must be in the master number as well. If 0 is not the last digit, then either 1 or 4 is in the right position and earns an X. But this cannot be correct, since that digit earns an X in clue (4). Therefore 0 must be the last digit of the master number, which rules out 4 being in the correct place in clue (4). Thus 1 is the third digit. Since 4 has been eliminated and the third and fourth digits of the master number are accounted for, the digit earning the X in clue (1) must be the 3. From clue (3), 7 has been eliminated since 0 and 1 account for the X and O, respectively, thus the final digit is either 5 or 8. Since this digit is the first digit in the master number, and it was in the wrong location as it appeared in guess (2), the first digit must be 8. The master number is therefore 8310. |
