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.

Thanks for the tip about the spreadsheet.

Rogerio is correct that if the product reveals only one number then the product must not be unique (that is, more than one combination must result in this product, and that it must have 7 as a prime factor, since any other possible prime factor is also a factor of another possible member.

If this information gives Sam only one other number, his sum must be one which has only two groups which include 7.

So we examine the Sums, and find that only one sum is derived from combinations of which only two contain 7: 1,2,5,7, and 1,3,4,7.. So Sam has revealed that his sum is 15, and Pat's product must be 84, so the group we are looking for is:

1,3,4,7

 

Posted by Robert Fulton on 2006-01-18 13:32:22