There are 4 persons A,B,C and D of which 2 are liars and 2 are knights. Each of these persons has a lucky and an unlucky number. All the 8 numbers are different and they are from 1 to 9 only. It is known that sum of lucky and unlucky numbers is same for all of these 4 persons and also sum of lucky numbers is greater than sum of unlucky numbers. Find the lucky and unlucky numbers for each of them if they made the following statements:

A:
C's lucky number is 7.
The missing number is 5.

B:
C's unlucky number is 4.
D's lucky number is 2.

C:
A is a liar.
B's lucky number is 6.

D:
The product of B's numbers is 24.
The maximum of all our numbers is A's lucky number.

Note: The missing number is the number from 1 to 9 which is not any one of these people's lucky or unlucky number.

The sum of all the lucky and unlucky numbers is divisible by 4, and 1+2+3+4+5+6+7+8+9=45, so the missing number is either 1, 5, or 9. If it is 1, then the sum for each person is 11. If it is 5, then the sum for each person is 10. If it is 9, then the sum for each person is 9.

Suppose A is a liar. Then, C is a knight, so B's lucky number is 6. If D was a knight, then B's unlucky number would be 4, so the sum for each person would be 10. Then, the missing number would be 5, but A is a liar. Therefore, D is a liar. Since there are 2 knights, B is a knight. Then, C's unlucky number is 4. If the sum for each person was 11, then C's lucky number would be 7 and A would be a knight. However, A is a liar, so the sum is 9. Then, B's numbers are (6, 3), and C's numbers are (5, 4). Since B is a knight, D's lucky number is 2. Then, D's unlucky number is 7. A's numbers are 1 and 8. 1+6+5+2<8+3+4+7, but 8+6+5+2>1+3+4+7, so A's lucky number is 8. Then, D's second statement is true, but D is a liar. We get a contradiction, so A is a knight.

Since A is a knight, the missing number is 5, so the sum for each person is 10. C's lucky number is 7, so C's unlucky number is 3. Therefore, B is a liar because B said C's unlucky number was 4. Also, C is a liar because A is a knight and said that. Since there are two knights, D is a knight. Therefore, the product of B's numbers is 24, so they are 4 and 6. Since C is a liar and said that B's lucky number is 6, B's lucky number is 4 and B's unlucky number is 6. Also, the highest number is A's lucky number, so A's lucky number is 9. Therefore, A's unlucky number is 1. The remaining numbers are 2 and 8. B is a liar and said that D's lucky number is 2, so D's lucky number is 8 and D's unlucky number is 2.

Solution:

A:knight, lucky=9, unlucky=1

B:liar, lucky=4, unlucky=6

C:liar, lucky=7, unlucky=3

D:knight, lucky=8, unlucky=2

Edited on January 8, 2011, 9:19 pm

Posted by Math Man on 2011-01-08 18:51:04