Chess Board and rectangles


How many rectangles you can see ?

  • you have on the upper side of the chess board 8 indivisible squares ,  7 points belongs to the edge and to the frontiers separating the squares , plus 2 points at the corners .  then if you want define a side of a rectangle you need to choose 2 points out of the 9 points belonging to the upper side. the order doesn’t matter , we will use combination : :  finally we have 9! / ( 2! * 7!) combinations
  • you can apply the same reasoning on the left side of the Chess Board
  • at the end you have 9! / ( 2! * 7!) * 9! / ( 2! * 7!) rectangles and squares (which are technically rectangle) but if you want only and only rectangle you can subtract the  number of squares with the formula defined in the previous post about Chess Board and squares :

Chess board and squares

pascal rocaHow many squares do you see on this picture ?

  • Obviously 64 of size 1
  • 1 of size 8
  • Horizontally from top left corner or from top right corner, you can build a side of length 7. Identically it could be done vertically, meaning that 4 squares of length 7 are fitting in the chess board
  • Same reasoning applies to the other size

    Finally the number of squares fitting in a chess board is :  chessBoardSquaresFormula