Four-letter word
- What on earth are you doing?
- I am trying to fit these words into a 4×4 grid of cells: MPMM, OMPM, OMPP, OOMO, OOMP, POOM, and POPM.
- How does it go?
- I find it hard.
- Could it be because a four-letter word is missing?
- Oh, yes! You are right.
- What is the word?
Problem source: Joe Konhauser via Math Central.
January 26th, 2010 at 6:20 pm
If OOMP is the 2nd row, the 4th column is either a) MPMM or b) the missing word. If it is MPMM, then the other rows are either i) OMPM, POOM and POPM, or else ii) two of those and the missing word. Suppose case a)i). Then the 1st 3 columns are anagrams of, respectively, OOPP, MOOO and PMOP. Thus they are the missing word, OOMO and OMPP respectively. Checking the rows against the given words, we see that the 1st, 3rd and 4th rows must be POOM, OMPM and POPM respectively, giving
POOM
OOMP
OMPM
POPM
the missing word being POOP.
Hm. Seems I managed to find a solution while searching only a fraction of the search space. I could describe some of the dead ends I followed, but wouldn’t that just be boring?
January 29th, 2010 at 12:40 pm
You don’t need to try a lot of dead ends to prove that it is the only solution.
Suppose MPMM is vertical (solution can be reversed with a diagonal symmetry). If MPMM is on the 1st column, then 3 horizontal words must begin with M. But we miss two words (there is MPMM, which is used and maybe the missing word), therefore MPMM is not on the first column.
MPMM cannot be in the second column either, as we would need 3 words with a 2nd letter M and 1 word with a 2nd letter P. But we only have 2 words with a 2nd letter M and no with P. The missing word isn’t enough to complete both missing M and P.
If MPMM is in 3rd column, we need both OOMO and OOMP to match 2 of the 3 M. The missing word must be horizontal and provides the 3rd M.
Then, OOMO and OOMP give twice a double O on first and second column, but we only have one remaining word with two O : POOM (and the missing word is used). Therefore, MPMM should be in the 4th column:
. . . M
. . . P
. . . M
. . . M
OOMO cannot be horizontal.
OOMO cannot be in the 3rd column (only POOM and maybe the missing word have a 3rd letter O)
If OOMO is in the 1st column, then the 3rd line must be the missing word, and the first line one of OMPM, OMPP, OOMP. But these have a M starting a column and we don’t have any word beginning with M (the missing word being used.)
Therefore, OOMO is in the second column:
. O . M
. O . P
. M . M
. O . M
1st and 4th lines match only POOM, POPM and maybe the missing word.
a) If the missing word is on the 1st line, then last line is POOM or POPM and there’s no M but in the end, which doesn’t leave any place to put the other POOM/POPM -> IMPOSSIBLE
b) If the missing word is on the 4th line, then the first line begins with P and so do the 1st column.
c) If the 1st and 4th line are POOM and POPM, then the first column begins with P and must be the missing word.
b) and c) both tell that the 2 P words and the missing words are on the X:
xOxM
xO. P
xM. M
xOxM
Filling the blanks with remaining words, we must have OMPM in the 3d line and OOMP in the second line, leaving only OMPP to fill the 3rd column:
xOOM
OOMP
OMPM
xOPM
POPM must be on last row. POOM must then be on 1st row. Missing word is POOP on 1st column.