15. Nine cards numbered from 1 to 9 were placed facedown on the table. Aleksa, Bart, Clara and Deindra each picked up two of the cards. Aleksa said, "My numbers add up to 6." Bart said, "The difference between my numbers is 5." Clara said, "The product of my numbers is 18 ." Deindra said, "One of my numbers is twice the other one." All four made a true statement. Which number was left on the table?
A 1
B 3
C 6
D 8
E 9
SolUtion
Since all four made a true statement, the numbers on Aleksa's cards were 1 and 5 or 2 and 4 ; the numbers on Bart's cards were 1 and 6,2 and 7,3 and 8 or 4 and 9 ; the numbers on Clara's cards were 2 and 9 or 3 and 6 and the numbers on Deindra's cards were 1 and 2, 2 and 4,3 and 6 or 4 and 8 .
Suppose the numbers on Aleksa's cards were 2 and 4. Then the numbers on Clara's cards could only be 3 and 6 . However, this would not leave any possible pair of cards for Deindra. Therefore the numbers on Aleksa's cards were 1 and 5.
Now suppose the numbers on Clara's cards were 2 and 9. Then the numbers on Bart's cards could only be 3 and 8 . However, this would again leave no possible pair of cards for Deindra. Therefore the numbers on Clara's cards were 3 and 6 . Finally, suppose the numbers on Bart's cards were 4 and 9. Again this would leave no possible pair of cards for Deindra. Therefore the numbers on Bart's cards were 2 and 7, leaving the cards numbered 4 and 8 for Deindra. Therefore the number left on the table was 9 .