Four different integers from 1 to 10 are chosen. Sam is given the sum and Pat is given the product. Sam and Pat take turns stating how many of the four numbers they can deduce:

Sam: I don't know any
Pat: I know one
Sam: I now know two
Pat: I now know all four

What could the four numbers be?

Tip: A spreadsheet is very useful in solving this problem.

(In reply to Probable solution by Rogerio)

I don't see why the number that Pat can identify at first has to necessarily be 7.

The combination 2,8,9,10 should work because on Pat's first guess, there will be 3 combinations that yield a product of 1440: 2,8,9,10 - 3,6,8,10 - and 4,5,8,9. From these choices, Pat can only deduce that 8 is one of the numbers.

If the product was 2160, the three possibilities would be 4,6,9,10 - 3,8,9,10 - and 5,6,8,9. Again, there would be only one common number (in this case it would be 9).

If I set up the spreadsheet right, both 2,8,9,10 and 4,6,9,10 should satisfy the conditons of the problem.


The problem I run into trying to use the sets that sum to 20 is that Sam states that he knows 2 numbers on his second round. The four possibilities that would sum to 20 at that point would be: 7,8,3,2 - 7,6,5,2 - 7,6,3,4 - and 1,4,6,9. Since these are sets that would not have been ruled out by previous comments, Sam wouldn't be able to state that he knew 2 of the numbers at this point.

Posted by Michael on 2006-01-31 06:29:43