I'm thinking of 4 real numbers, $a,b,c$ and $d$ (they need not be integers, or even rational). There are 6 products you can make by multiplying pairs of these numbers, namely, $ab,ac,ad,bc,bd$ and $cd$. I'll tell you what five of those products are, in no particular order: $$ 2,3,4,5 \text{ and }6 $$ What is the sixth product?

Small hint:

You can do this without needing to figure out what $a,b,c$ and $d$ are.

This is a fun puzzle I found on the xkcd puzzle wiki.

  • $\begingroup$ Is it 1 or 7??? $\endgroup$ – ʇolɐǝz ǝɥʇ qoq Mar 28 '15 at 0:23
  • $\begingroup$ Do a,b,c,d have to be integers? $\endgroup$ – Rand al'Thor Mar 28 '15 at 0:24
  • $\begingroup$ They need not be, question has been clarified, though you solved it anyway :) $\endgroup$ – Mike Earnest Mar 28 '15 at 0:43

The six numbers can be split up into pairs such that the product of the two numbers in each pair is the same across all three pairs: $(ab,cd),(ac,bd),(ad,bc)$. The only two pairs of numbers among $2,3,4,5,6$ which have the same product are $(3,4),(2,6)$. So the sixth number must be


Nice quick puzzle :-)

  • $\begingroup$ would it be possible to find the four real numbers a,b,c and d? $\endgroup$ – Craftsman Jul 4 '20 at 22:18
  • $\begingroup$ @Craftsman That could make a good follow-up puzzle! $\endgroup$ – Rand al'Thor Jul 5 '20 at 3:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.