## 4. Imagined Number

Grga thought of a three-digit number, and his friends are trying to guess it.

These are their attempts:

Boris: $218 \quad$ Robert: $571 \quad$ Marko: $732 \quad$ Darko: 853

Grga told them: "One of you has guessed all the digits, and the others have only one each, but none of the guessed digits are in the correct position." To this, his friends replied: "Based on this information, we cannot determine which number you thought of because there are multiple such numbers." Determine the sum of all such possible numbers.
## Solution.

Assume Boris guessed all the digits.

Then the digit 2 is neither in the hundreds place nor in the units place (since Marko did not guess the exact position of any digit) so it must be in the tens place.

The digit 1 is neither in the tens place nor in the units place (since Robert assumed that 1 is in the units place) so it must be in the hundreds place.

The digit 8 is neither in the units place nor in the hundreds place (Darko's assumption) so it must be in the tens place.

However, the tens place cannot have two digits (2 and 8) so this is impossible.

Boris did not guess all three digits.

Assume Robert guessed all the digits. Then similarly, we conclude that: 5 is neither in the hundreds place nor in the tens place so it must be in the units place, 7 is neither in the tens place nor in the hundreds place so it must be in the units place, which is impossible.

Assume Marko guessed all the digits. Then we conclude:

7 is neither in the hundreds place nor in the tens place so it must be in the units place, 3 is neither in the tens place nor in the units place so it must be in the hundreds place, 2 is neither in the units place nor in the hundreds place so it must be in the tens place. This is only possible if the intended number is 327.

Assume Darko guessed all the digits. Then:

8 is neither in the hundreds place nor in the units place so it must be in the tens place, 5 is neither in the tens place nor in the hundreds place so it must be in the units place, 3 is neither in the units place nor in the tens place so it must be in the hundreds place. This is only possible if the intended number is 385.

The sum of all possible numbers is $327+385=712$.