This is in continuation of A Cup Of Coffee
You have a five cup mug, a three cup mug, a water supply, a sink with a drain, and a packet of instant coffee which when dissolved in one cup of water produces coffee of strength 100%.
The packet may be used at any time, but the entire contents of the packet must be dissolved into a single mug when it is used.
What integer values of c (from 1 to 25 inclusively) is possible if the task is to fix 4 cups of coffee at exactly c% strength? Prove that these are indeed the only possible values of c.
(In reply to re: Possible solution
Sorry, I had a rather gross problem with rounding errors, and 7% should have been 6.67%. I have now adjusted my earlier post.
I did try to post the mixing chains (from which my errors would have been obvious at once) but couldn't get the formatting right. However, using F as fill, T for a transfer and S as sink, the first coordinate for the 3 mug and the second for the 5, a possible representation is:
(0,0) (0,F5) T(3,2) (S3,2) T(2,0) (2,F5) T(3,4) for the first chain, and
(0,0) (F3,0) T(0,3) (F3,3) T(1,5) (1,S5) T(0,1) (F3,1) T(0,4) for the second.
But I see from your print out that other chains may also be possible; in particular, 5Fill 5Pckt 5T 5Dump 3T 3Fill 3T 5T 3T 5Dump 3T 3Fill 3T 5T 3 1 2: equivalently
(0,0) (0,F5) T(3,2) (3,S2) T(0,3)(F3,3) T(1,5) T(3,3) T(1,5) (1,S5) T(0,1) (F3,1) T(0,4) T(3,1);
where the last step is valid but unnecessary; but this seems to lead to a concentration of 3% (all 20% down to (F3,3), then (20%,8%)(12%,8%)(12%,9.6%)(12%,0)(0,12%)(0,12%)(0,3%)?
Edited on March 7, 2012, 1:58 am
Posted by broll
on 2012-03-06 23:35:51