Map Folding
A general formula giving the number of distinct ways of folding an 👁 m×n
rectangular map is not known. A distinct folding is defined as a permutation of 👁 N=m×n
numbered cells reading from
the top down. Lunnon (1971) gives values up to 👁 n=28
.
| 👁 k | OEIS | 👁 k×1 , 👁 k×2 , ... |
| 1 | A000136 | 1, 2, 6, 16, 50, 144, 462, 1392, ... |
| 2 | A001415 | 2, 8, 60, 320, 1980, 10512, ... |
The number of ways to fold an 👁 n×n
sheet of maps is given for 👁 n=1
, 2, ..., etc. by 1, 8, 1368, 300608, 186086600, ... (Lunnon
1971; OEIS A001418).
The limiting ratio of the number of 👁 1×(n+1)
strips to the number of 👁 1×n
strips is given by
![👁 lim_(n->infty)([1×(n+1)])/([1×n]) in [3.3868,3.9821].](https://mathworld.wolfram.com/images/equations/MapFolding/NumberedEquation1.svg){kind=link}
See also
Stamp FoldingExplore with Wolfram|Alpha
More things to try:
References
Gardner, M. "The Combinatorics of Paper Folding." Ch. 7 in Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 60-73, 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.Sloane, N. J. A. Sequences A000136/M1614, A001415/M1891, and A001418/M4587 in "The On-Line Encyclopedia of Integer Sequences."Wells, M. B. Elements of Combinatorial Computing. Oxford, England: Pergamon Press, p. 238, 1971.Referenced on Wolfram|Alpha
Map FoldingCite this as:
Weisstein, Eric W. "Map Folding." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MapFolding.html