3. Three statements are made about a four-digit number: "The digits 1, 4, and 5 appear in the number", "The digits 1, 5, and 9 appear in the number", "The digits 7, 8, and 9 appear in the number". Exactly two of these statements are true. Which digits appear in the number?
3. Answer: $1,4,5,9$.

Solution. One of the three statements can be false. 1) If the statement "The digits 1, 4, and 5 appear in the number" is false, then the statements "The digits 1, 5, and 9 appear in the number" and "The digits 7, 8, and 9 appear in the number" are true, and the digits in the four-digit number are: $1,5,7,8,9$, which is impossible because there are five digits.

2) If the statement "The digits 1, 5, and 9 appear in the number" is false, then the statements "The digits 1, 4, and 9 appear in the number" and "The digits 7, 8, and 9 appear in the number" are true, and the digits in the four-digit number are: 1,4,7,8,9, which is impossible because there are five digits.
3) If the statement "The digits 7, 8, and 9 appear in the number" is false, then the statements "The digits 1, 5, and 9 appear in the number" and "The digits 1, 4, and 5 appear in the number" are true, and the digits in the four-digit number are: $1,4,5,9$, which is possible because there are four digits.

## Grading Criteria.

Correct answer only: 1 point.

Only one case with the correct answer considered: 2 points.

Two cases considered: 3 points.