Map Folding

A general Formula giving the number of distinct ways of folding an $N=m\times n$ rectangular map is not known. A distinct folding is defined as a permutation of $N$ numbered cells reading from the top down. Lunnon (1971) gives values up to $n=28$.

$n$	$1\times n$	$2\times n$	$3\times n$	$4\times n$	$5\times n$
1	1	1
2	2	8
3	6	60	1368
4	16	1980		300608
5	59	19512			18698669
6	144	15552

The limiting ratio of the number of $1\times(n+1)$ strips to the number of $1\times n$ strips is given by

$$\lim_{n\to\infty} {[1\times(n+1)]\over [1\times n]} \in [3.3868, 3.9821].$$

See also Stamp Folding

References

Gardner, M. ``The Combinatorics of Paper Folding.'' Ch. 7 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, 1983.

Koehler, J. E. ``Folding a Strip of Stamps.'' J. Combin. Th. 5, 135-152, 1968.

Lunnon, W. F. ``A Map-Folding Problem.'' Math. Comput. 22, 193-199, 1968.

Lunnon, W. F. ``Multi-Dimensional Strip Folding.'' Computer J. 14, 75-79, 1971.