# Mathematics behind word puzzle

So I was presented with this puzzler:

The answer to the puzzle is a 6 letter word with 6 different letters. Each of these 12 words have one - and only one - letter of the mystery word in the correct position (i.e. 1st, 2nd, 3rd, etc.)

MYRIAD
FUMBLE
CATKIN
RECKON
TOWARD
OUTAGE
STEADY
SWAYED
BOUNTY
TONGUE
STOLEN
UNREST


I couldn't find a very efficient algorithm for solving it - more on this later - so I decided to put my programming skills to use and just generate all possible solutions via brute force.

Imagine my surprise when there was only a single 6 letter combination that met the criteria, and it was the answer to the puzzle!

I don't want to give away the answer in case you want to solve it yourself, but as I'm not interested in the answer so much as the method of devising the problem and its solution, I have a mild spoiler: each letter of the answer appears exactly twice in its correct position.

Answer:

TURKEY

So my questions are basically:

1. Is having 2 words per letter-position a function of building a unique solution to such a puzzle?
2. What's the algorithm for doing this without a computer? One attempt I worked at noted that the words RECKON, OUTAGE, BOUNTY, UNREST, and SWAYEDdon't share any letter-positions so they contain 5 of the letters, but I couldn't finish the thought.

I think I'm missing something obvious about the puzzle's design and intended solving process. I usually crank through puzzle books in an hour or two, but this one just felt impossible!

• How do you add spoiler text? – Kyle Hale Jan 3 '16 at 3:30
• See here – 2012rcampion Jan 3 '16 at 3:51

## 1 Answer

My process:

My first thought was, "Let's look at the common/repeated letters." So I took all the letters from each place and found all the repeated ones, giving this:
MFCRTOSSBTSU: TTSSS
YUAEOUTWOOTN: TTUUOOO
RMTCWTEAUNOR: TTRR
IBKKAAAYNGLE: KKAAA
ALIORGDETUES: EE
DENNDEYDYENT: NNNEEEDDDYY
Just looking down the rows at that, TURKEY kinda jumps out. But that didn't have to be true- it could have been that some of the letters were triples and some were singles, and wouldn't be revealed by looking at the multiples. If that were the case, I would have done a crossword-type deal. Taking S as a given, eliminate all of the multiples that show up in the same word as one starting with S, and then figure out what combos sum the remaining multiples to 12 (because there are 12 letters from TURKEY in the list). That process would give us this: SO(T/R)K-T. (The T at the end is a freebie, because that's the only letter that appears only once as a terminal.) No matches, though, unless SORKAT is a word now.
Doing the same with T gives us T(U/-)(T/R/-)(K/-)(E/-)(N/Y), which means our word is either TURKEY or TUTKEY, or TURKEN with one of the four intervening letters swapped. And if none of those had worked, I would have assumed the first letter was neither S nor T and then continued in the same way.

• This is ultimately the algorithm I picked up on (that the sum of the matches has to be 12, and the largest multiple was 3 so it had to include at least 3 multiples ) - they clearly wanted you to just look for multiples, but the problem seemed much more expansive when I first took a look at it. – Kyle Hale Jan 3 '16 at 5:42