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 2 years ago
The streets of a city are arranged like the lines of a chessboard. Let the dimensions be \(m \times n\). Find the number of ways in which a man can travel from the NorthWest corner to the SouthEast corner, going the shortest possible distance.
 2 years ago
The streets of a city are arranged like the lines of a chessboard. Let the dimensions be \(m \times n\). Find the number of ways in which a man can travel from the NorthWest corner to the SouthEast corner, going the shortest possible distance.

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karatechopper
 2 years ago
Best ResponseYou've already chosen the best response.0thts a lot of turns i can take x(

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1Kinda sounds like problem asked by FFM

karatechopper
 2 years ago
Best ResponseYou've already chosen the best response.0who know when i could get lost :/ @Diyadiya will drive me! then no need to worry!

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.0It is similar to FFM's problem.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1then it must be related to this http://en.wikipedia.org/wiki/Motzkin_number

karatechopper
 2 years ago
Best ResponseYou've already chosen the best response.0@Ishaan94 i have an answer

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339345895603:dw seems to be six int this case

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1yes...six..\({4\choose 2}\)

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1maybe I should write it \[{2+2\choose 2}\]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339346361572:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339346637952:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1so for 2x3 ...we have 10

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1so adding one segments ... \[ \binom {4 + 1}{2}\]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339346850772:dw

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1for your 3x2 you will have to go 3 units up and 2 units to the right in total to get to the upper right hand corner UUURR the number of ways to arrange these letters is \({3+2\choose 2}\)

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339346924974:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.12*4+2*10+6*3 = \( \binom{9}{2} = \binom{3 + 3 + 3}{2}\)

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1there are only 20 ways for a 3x3

goxenul
 2 years ago
Best ResponseYou've already chosen the best response.0Yeah, because you take m+n steps and you must decide which n of them will be right, or which m of them will be down (it's equivalent).

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1isn't there 36 steps in 3x3??

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1show me a path that requires 36 steps

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1something wrong with draw app here ... can't copy my own drawing!!

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1only 6 steps are needed to get to the bottom right

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339348259654:dw

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1there are 6 ways to do a 2x2 not 10

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1Oh .. i made mistake then ... it must be 3x2

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339348688431:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1shouldn't it be 2*3*3+2*4+2*6 = 38 ??

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1try and draw 38 minimum distance paths using that graph

siddhantsharan
 2 years ago
Best ResponseYou've already chosen the best response.1I'm getting ( m + n 2)!/(m1)!(n1)! ??????

siddhantsharan
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339384651187:dw

siddhantsharan
 2 years ago
Best ResponseYou've already chosen the best response.1Let each square formed be a unit square. Therefore if a man needs to get from NW to south east, He basically needs to cover "n1" units on the east line. And "m1" units south.  For the Shortest possible path when he decides to never turn back. Now in order to do this each street taken by him must be in the South or east direction. Every time he goes east Lets call that turn/ street taken E. Every time he turns south we call that S. So any path followed by him can be written as: SSESESESSSSEEE...........  Meaning the man goes first south then south again without turning at next intersection Then turns east at the next turn.......... Here number of S's = m1 No. of E's = n1. So the problem has become an equivilant of Finding the permutations of n1 E's and m1 S's. = ( m 1 + n 1)!  (m1)!(n1)!

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1339422624581:dw this is a 1x1 block so m=1 and n=1 using your formula \[\frac{(11+11)}{(11)!(11)!}=\frac{0!}{0!0!}=1\] but there are 2 paths to the end

siddhantsharan
 2 years ago
Best ResponseYou've already chosen the best response.1My formula deals with streets. => lines. => m = 2, n=2 in this caase

Zarkon
 2 years ago
Best ResponseYou've already chosen the best response.1I go by the number of blocks created and the number of steps to go from one corner to the other for a 1x1 block you have to go down 1 and to the right 1

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1should have never doubted @Zarkon http://math.stackexchange.com/questions/103470/howcanifindthenumberoftheshortestpathsbetweentwopointsona2dlatti
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