Next: R-BY-C CROZZLE IS      NP-COMPLETE Up: Crozzle: an NP-Complete Problem Previous: Crozzle: an NP-Complete Problem


The R-by-C Crozzle problem, introduced at the 1996 ACM Symposium on Applied Computing [2], is a generalization of the Crozzle problem found in The Australian Women's Weekly. A Crozzle is a word puzzle played on a 10x15 grid. Words from a supplied list are placed on the grid subject to the following rules:

Not all of the words need to be placed.
All placed words must fit completely on the grid.
The intersection of two words must be at a shared letter.
No two words may be adjacent (unless the adjacent parts are covered by Rule 3) or placed end-to-end.
The words must form a single connected unit.

A Crozzle is scored as follows: 10 points for each word placed plus points for each letter that appears at the intersection of two words. Letters have the following point values:

\fbox{ \begin{tabular}{l l \vert l l}
a,b,c,d,e,f & 2~~~~~~~ & s,t,u,v,w,x & 16 \\
g,h,i,j,k,l & 4 & y & 32 \\
m,n,o,p,q,r & 8 & z & 64 \\

Figure: Two solutions (one with a high score, one with a low score) for a Crozzle with input words active, assert, movie, deduction. (The symbol `.' is used to represent a blank.)
\begin{figure}\par\begin{verbatim}.....d......... .........a.....
...}\begin{verbatim}score 48 score 66\end{verbatim}\vspace{-0.33em}

Figure 1 shows two solutions to a simple Crozzle. Algorithms for automatically finding good solutions to Crozzles have appeared in the literature [5,3].

The R-by-C Crozzle problem, introduced by Gower and Wilkerson to study the complexity of the original, generalizes the Crozzle problem as follows: in addition to a list of words, an instance of the R-by-C Crozzle problem has as input $R$ and $C$, the number of rows and columns in the grid. Thus the original Crozzle problem is R-by-C Crozzle with $R=15$ and $C=10$.

Gower and Wilkerson argue that R-by-C Crozzle is NP-Hard, but not in NP. Unfortunately this says nothing about the complexity of the original problem as it is possible that restricting R to 15 and C to 10 would place it in NP. Section 2 demonstrates that R-by-C Crozzle is in fact in NP and thus an NP-Complete problem. This implies that the original (more restrictive) Crozzle problem is also NP-Complete.

One technical note: the words supplied as part of a Crozzle are normally English words. There are a finite number of English words; thus, one could, in theory, precompute all possible Crozzle solutions giving a constant time bound to the problem. To study its complexity, we generalize the input to include arbitrary words taken from some finite alphabet.

Next: R-BY-C CROZZLE IS      NP-COMPLETE Up: Crozzle: an NP-Complete Problem Previous: Crozzle: an NP-Complete Problem

Copyright © 1997 Bradley M. Kuhn, David W. Binkley.

Verbatim copying and distribution of this entire paper is permitted in any medium, provided this notice is preserved.