I have 9 coins; 3 coins each of 3 denominations.
Coins of the same denomination all weigh the same.
Each coin weighs an exact whole number of grams, and at most 9 grams.
I can weigh any subset of my coins together, to get their total weight accurate to the gram.

I want to determine how much a coin of each denomination weighs. Obviously I could do it in 3 weighings, just weighing one coin of each denomination separately.

How can I get the weights of the 3 denominations using only 2 weighings?

  • 1
    $\begingroup$ I hope this is original. I thought of it myself but it seems an obvious variation. $\endgroup$ Commented Jan 3, 2022 at 12:45

1 Answer 1


Calling the denominations (or rather their weights) a,b,c one scheme would be weighing:

3a+b and 3b+c. Call the resulting readings X and Y. Then b=X mod 3 and consequently 3X+c=Y mod 9. Because of the limited range of admissible values this last equation determines c. b and a immediately follow via back substitution.

  • $\begingroup$ My own solution is the same, though I had a less compact proof. Would it have been a better puzzle if there were for example 4 or 5 coins of each denomination? Clearly the same solution still works, as would a similar solution with larger numbers, but would that have made it less obvious? $\endgroup$ Commented Jan 3, 2022 at 14:31
  • 1
    $\begingroup$ @JaapScherphuis I think the 9 (grams) is more of a giveaway. Perhaps cutting that to e.g. 7 (assuming it doesn't create unwanted alternative solutions) could have made a good smoke screen. $\endgroup$
    – loopy walt
    Commented Jan 3, 2022 at 14:36
  • $\begingroup$ As is it the puzzle has 18 possible solutions according to my computer, so I don't think adding alternative solutions is a problem. Those alternative solutions are less neat than this one, though I think essentially the same proof works in each case. Maybe lowering the 9 introduces anomalous alternative solutions that don't have the same proof. I'll have to look into that. $\endgroup$ Commented Jan 3, 2022 at 14:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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