Pairwise Products

In my December 2005 diary on National Review Online I posed the following math problem.

From any four numbers a, b, c, d, you can form six "pairwise products":  ab, bc, cd, ad, ac, bd.  I have a particular four numbers in mind.  I know that five of their pairwise products (not necessarily in the order given above) are 2, 3, 4, 5, and 6.  What is the sixth pairwise product?   

Solution:

If you take the pairwise products in pairs like this:

ab, cd

ac, bd

ad, bc

then the product of the two numbers on each line is abcd.  All three products are therefore equal.  However, only one line contains the unknown pairwise product.  The other two lines, composed entirely of known pairwise products, multiply to the same result, abcd.  The only possibility with the numbers given is

2, 6

3, 4

giving abcd = 12.  The third line must be

5, x

(x standing for the unknown pairwise product); and this must also multiply to 12.  Therefore x = 12/5.

The numbers a, b, c, d themselves are actually irrational — multiples of the square root of 10.  I didn't ask for them, though!