## December 2005

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In my December diary I posed the following brain-teaser.

Given any four numbersa,b,c, andd, you can form six "pairwise products":ab,bc,cd,ad,ac, andbd.

I have four particular numbers in mind. Five of their pairwise products (not necessarily in the order I just listed) work out to 2, 3, 4, 5, and 6. What does the sixth pairwise product work out to?

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*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
√10. I
didn't ask for them, though!