Robert M. Dickau - Shortest Paths (text)

# Shortest-path diagrams (text version)

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Here are some diagrams that represent the possible paths of length 2n from one corner of an n-by-n grid to the opposite corner. The number of paths are the central binomial coefficients

Binomial[2n, n] or (2n)!/(n!)^2,

central meaning they fall along the center line of Pascal's triangle.

The first few are 1, 2, 6, 20, 70, 252, ...

```1 x 1 grid, 2 paths:

#....................................  #####################################
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#                                   .  .                                   #
#####################################  ....................................#

2 x 2 grid, 6 paths:

#........................ #....................... #........................
#           .           . #           .          . #           .           .
#           .           . #           .          . #           .           .
#           .           . #           .          . #           .           .
#           .           . #           .          . #           .           .
#           .           . #           .          . #           .           .
#........................ #############........... #########################
#           .           . .           #          . .           .           #
#           .           . .           #          . .           .           #
#           .           . .           #          . .           .           #
#           .           . .           #          . .           .           #
#           .           . .           #          . .           .           #
######################### ............############ ........................#

#############............ #############........... #########################
.           #           . .           #          . .           .           #
.           #           . .           #          . .           .           #
.           #           . .           #          . .           .           #
.           #           . .           #          . .           .           #
.           #           . .           #          . .           .           #
............#............ ............############ ........................#
.           #           . .           .          # .           .           #
.           #           . .           .          # .           .           #
.           #           . .           .          # .           .           #
.           #           . .           .          # .           .           #
.           #           . .           .          # .           .           #
............############# .......................# ........................#

3 x 3 grid, 20 paths:

#.............. #............. #............. #............. #..............
#    .    .   . #   .    .   . #   .    .   . #   .    .   . #   .    .    .
#.............. #............. #............. #............. #####..........
#    .    .   . #   .    .   . #   .    .   . #   .    .   . .   #    .    .
#.............. #####......... ##########.... ############## ....#..........
#    .    .   . .   #    .   . .   .    #   . .   .    .   # .   #    .    .
############### ....########## .........##### .............# ....###########

#.............. #............. #............. #............. #..............
#    .    .   . #   .    .   . #   .    .   . #   .    .   . #   .    .    .
######......... #####......... ##########.... ##########.... ###############
.    #    .   . .   #    .   . .   .    #   . .   .    #   . .   .    .    #
.....######.... ....########## .........#.... .........##### ..............#
.    .    #   . .   .    .   # .   .    #   . .   .    .   # .   .    .    #
..........##### .............# .........##### .............# ..............#

######......... #####......... #####......... #####......... #####..........
.    #    .   . .   #    .   . .   #    .   . .   #    .   . .   #    .    .
.....#......... ....#......... ....#......... ....######.... ....######.....
.    #    .   . .   #    .   . .   #    .   . .   .    #   . .   .    #    .
.....#......... ....######.... ....########## .........#.... .........######
.    #    .   . .   .    #   . .   .    .   # .   .    #   . .   .    .    #
.....########## .........##### .............# .........##### ..............#

######......... ##########.... ##########.... ##########.... ###############
.    #    .   . .   .    #   . .   .    #   . .   .    #   . .   .    .    #
.....########## .........#.... .........#.... .........##### ..............#
.    .    .   # .   .    #   . .   .    #   . .   .    .   # .   .    .    #
..............# .........#.... .........##### .............# ..............#
.    .    .   # .   .    #   . .   .    .   # .   .    .   # .   .    .    #
..............# .........##### .............# .............# ..............#

```
Designed and rendered using Mathematica 3.0 for the Apple Macintosh.