--THE 7TH GUEST-- --CANS PUZZLE SOLUTION-- by: quickbrownfox42
In The 7th Guest, the cans puzzle is solved by rearranging the letters
YYY YYYYY YYYBC GHLLLM PPPRR RS SS SSTTT
to form
SHY GYPSY SLYLY SPRYLY TRYST BY MY CRYPT
- With the goal of 'fastest solution' in mind, the object should be to rearrange the letters using the fewest exchanges possible.
Since the solution really depends on each can being in the correct location, the puzzle can actually be broken down into 33 conditions which must be met - the position of each can. The quickest way to solve would be if the puzzle began in the same order as the solution, so we'll first compare all the cans that satisfy that condition. The format of words, including spaces, will be disregarded in the interest of simplicity
01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Y Y Y Y Y Y Y Y Y Y Y B C G H L L L M P P P R R R S S S S S T T T S H Y G Y P S Y S L Y L Y S P R Y L Y T R Y S T B Y M Y C R Y P T
The cans satisfying this condition are in positions 3, 5, 8, 11, 18, and 33. (0 moves)
Removing those cans from the set that still needs to be solved leaves
01 02 04 06 07 09 10 12 13 14 15 16 17 19 20 21 22 23 24 25 26 27 28 29 30 31 32 Y Y Y Y Y Y Y B C G H L L M P P P R R R S S S S S T T S H G P S S L L Y S P R Y Y T R Y S T B Y M Y C R Y P
- The next fastest possible way to solve is to switch two cans so that both cans will be in their correct locations. These are found by locating two locations whose starting and solved letters are each others' inverse (6 moves)
01 26 06 22 07 28 10 17 20 32 23 30 Y <-> S Y <-> P Y <-> S Y <-> L P <-> T R <-> S S Y P Y S Y L Y T P S R
Removing these cans from the set that still needs to be solved leaves
02 04 09 12 13 14 15 16 19 21 24 25 27 29 31 Y Y Y B C G H L M P R R S S T H G S L Y S P R Y R T B M C Y
- A map of the remaining substitutions is illustrated
L -> B -> R Y -> S -> C Y -> G -> s -> M Y -> H -> P -> R -> T ^ / ^ / ^ / ^ /
Note that C and M are interchangeable as both need an S in their place and must be moved to a location where a Y was.
The letters in the above map are somewhat ambiguous - for example, while there are several Y's, each must be exchanged with different letters. More specifically, the exchanges are as follows
12 25 16 09 29 13 04 14 27 19 02 15 21 24 31 B <-> R <-> R Y <-> S <-> C Y <-> G <-> S <-> M Y <-> H <-> P <-> R <-> T L B L S C Y G S M Y H P R T Y
This leaves each letter in the correct position (11 moves, 17 total)
Appendix: solution summary
I have tried to devise a shortcut which may make the solution easier to remember and/or interpret. Each letter of note is the one capitalized in each section
Leave: shY, gYpsy, gypsY, slYly, sprYly, crypT Exchange: Shy-bY, gyPsy-trYst, gypSy-mY, sLyly-sprYly, Tryst-cryPt, trySt-cRypt Exchange: By-spRyly-slyLy, Crypt-slylY-Slyly, Gypsy-Spryly-My-sprylY, sHy-sPryly-tRyst-trysT-crYpt
