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.
Copyright © 1996 by Robert M. Dickau.
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