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Lucky and Unlucky Numbers (Posted on 2009-02-01) Difficulty: 2 of 5
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:

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

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

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

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.

No Solution Yet Submitted by Praneeth    
Rating: 4.6667 (3 votes)

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Solution Solution | Comment 1 of 8
Since the sums of lucky and unlucky numbers are the same for all players, the missing number can only be 1,5 or 9 with sums of respectively 11, 10 or 9. We consider these 3 cases separately.
* missing number = 1.
This means A is a liar (he lies about the missing number), implying C is a knight. B's lucky number must be 6, B's unlucky number (11-6)=5. 6x5 != 30 implies D is a liar, which makes B a knight. Then C's unlucky number must be 4, which makes his unlucky number 7, which is what A said. This gives a conflict with A being a liar.
* missing number = 5
This means A is a knight, and C is a liar. The lucky number of C is 7, the unlucky number of C is 3. This makes B a liar and D a knight. If the product of B's numbers is 24, and 3 is already taken by C, B's numbers must be 4 and 6. As C is a liar, B's unlucky number must be 6, his lucky number 4. As D is a knight, A's lucky number must be 9, his unlucky number 1. D has number 2 and 8; with B being a liar D's lucky number must be 8 and his unlucky number 2. This gives a valid solution.
* missing number = 9.
This means A is a liar, C a knight. B's lucky number must be 6, his unlucky number 3, which makes D a liar about the product of B's numbers. B must be a knight, making C's unlucky number 4 and his lucky number 5. This gives a conflict with what A says about this number.

The only solution is:
lucky number A: 9
unlucky number A: 1
lucky number B: 4
unlucky number B: 6
lucky number C: 7
unlucky number C: 3
lucky number D: 8
unlucky number D: 2

  Posted by Robby Goetschalckx on 2009-02-01 14:27:18
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