11. A three-digit number is thought of, which with any of the numbers 543, 142, and 562, matches one digit in the same place, while the other two do not match. What is the number thought of?
11. If the first digit of the desired number is 5, then either the second digit is 4 or the third is 2 (since a match with the second number is required). Both cases lead to a contradiction: a match with either the first or the third number will be in two digits, so the first digit cannot be 5. Reasoning similarly, we can see that the second digit of the desired number is not 4, and the third is not 2. The only remaining possibility is that the desired number is 163.

87 The Pigeonhole Principle

