## Task B-1.4.

Three friends, Ante, Bojan, and Vinko, are guessing an unknown six-digit number composed of the digits $1,2,3,4,5,6$, with no repeated digits. Ante said the number is 123456, Bojan said 245163, and Vinko said 463215. None of them guessed the exact number, but Ante correctly guessed the positions of 3 digits, Bojan also correctly guessed the positions of 3 digits, and Vinko correctly guessed the position of 1 digit. Determine the unknown number.
## Solution.

Notice that Ante, Bojan, and Vinko together correctly guessed the exact position of 7 digits, which means that some pair must have correctly guessed the position of the same digit. Since only on the third position did a pair (Ante and Vinko) assume the same digit, the number 3, the third digit must be 3.

From this, it follows that Bojan placed the wrong digit, 5, on that position. The number 5 is not on the sixth position either, because the digit 3 was the only correct digit for Vinko, so the number 5 must be on the fifth position, as Ante assumed.

Next, we observe the fifth position in Bojan's number, where the digit is 6. The digit 6 cannot be on that position (since the number 5 is there), nor can it be on the second position because Vinko did not guess that position correctly, so it must be on the sixth position, as Ante assumed. The positions of the other digits were correctly assumed by Bojan, i.e., 2 is on the first position, 4 on the second position, and 1 on the fourth position. Therefore, the sought number is 243156.