P&Cq9

[Saumyojit]

no of ways to choose two square from 8*8 chessboard so that two square have a common side.

64c2-->choosing every possible pairs


now I know that i have to subtract all the invalid pairs .


just visualize,

picking either one of 4 cornered square =13+48=61 invalid squares that "that cornered " square should not pair with as they don't have a common side .


picking either one of all the similar type of square present like in row1 column 2 (visualise) =60 invalid squares that "that type" square should not pair with.

picking either one of all the similar type of square present like in row2 column 2 =59 invalid squares that "that type" squares should not pair with.
Then what?





Did you understand my approach ?

 

[Dr.Peterson]

no of ways to choose two square from 8*8 chessboard so that two square have a common side.

First check the problem and show the original wording (or an image or a link). This makes no sense grammatically; Do they, or do they not, have a common side?

 

[pka]

no of ways to choose two square from 8*8 chessboard so that two square have a common side.
64c2-->choosing every possible pairs

There are eight columns and eight rows. There are eight vertical line segments, the same length as the side of each square, that separate each square in the first column from a square in the second column. Now there are seven such sets of eight such vertical line segments. That means that there are fifty-six of horizonal touching pairs. Now, Saumyojit. you tell us the number of vertical touching pairs there are.
HERE is a visual aid.

 

[Dr.Peterson]

no of ways to choose two square from 8*8 chessboard so that two square have a common side.

64c2-->choosing every possible pairs

now I know that i have to subtract all the invalid pairs .

just visualize,

picking either one of 4 cornered square =13+48=61 invalid squares that "that cornered " square should not pair with as they don't have a common side .

picking either one of all the similar type of square present like in row1 column 2 (visualise) =60 invalid squares that "that type" square should not pair with.

picking either one of all the similar type of square present like in row2 column 2 =59 invalid squares that "that type" squares should not pair with.
Then what?

Did you understand my approach ?

I'll assume you mean "number of ways to choose two (1x1) squares from an 8*8 chessboard so that the two squares have a common side."

You appear to be trying to count pairs that do not have a common side, which seems like the hard way to do it. Is there a reason for that choice?

Here is my attempt to correct your English so I can make sense of what you wrote:

pick one of the 4 corner squares: 13+48=61 invalid squares that that corner square should not pair with as they don't have a common side . [Where do these numbers come from? It seems much easier to count the 2 (adjacent) squares that do have a common side with it, out of 63 other squares.]​

​

pick one of all the edge squares, like in row 1 column 2: 60 invalid squares that "that type" square should not pair with. [Again, why not just say that there are 3 adjacent squares, so there are 63 - 3 = 60 that are not, if you must count that way? And how many of these edge squares are there?]​

​

pick one of the interior squares like in row 2 column 2: 59 invalid squares that "that type" squares should not pair with. [These squares have 4 adjacent squares, leaving 63 - 4 = 59 non-adjacent.]​

So I apparently do understand your approach, and you have your numbers are right as far as you went, though you are doing this the hard way. You haven't yet counted how many there are of each type, and therefore counted the total; and you will end up double-counting, since the second square might have been chosen first. A case-by-case approach like yours, but not using subtraction, would be a little easier, but still rather slow.

The much easier way, similar to pka's suggestion, is just to consider the two possible orientations of a touching pair, and count the places such a pair could be placed (by counting possible locations for the upper or left-most square in the pair).

 

[Saumyojit]

There are eight columns and eight rows. There are eight vertical line segments, the same length as the side of each square, that separate each square in the first column from a square in the second column. Now there are seven such sets of eight such vertical line segments. That means that there are fifty-six of horizonal touching pairs. Now, Saumyojit. you tell us the number of vertical touching pairs there are.
HERE is a visual aid.

thanks pka.
How i didn't see this !
Yes if we take the horizon and vertical line segments then 112 squares .