14 X 14 squares will make howmany rectangles.?

Answer 1

Start with the upper left square. Using that square, you can make 14^2 = 196 rectangles (of course a square counts as a rectangle) since any other square can be used as the bottom right end of the rectangle. Then move to the next square to the right. Using that square, there are 14*13 = 182 rectangles that do not include the first square. Then move to the next square to the right; there are 14*12 = 168 rectangles that do not include the previous two squares. And so on: 14*11 = 154, 14*10 = 140... down to 14*1 = 14 with the top right square. So, in total, there are 14*(14+13+12+...+1) possible rectangles that can be made from the squares in the top row.

Then move to the next row. Starting with the leftmost square in the second row, you can make 13*14 = 182 rectangles starting with that square that does not include any squares in the first row. Then move to the right one square. With that square, you can make 13^2 = 169 rectangles that don't include the first row or the first square in the second row. And so on: 13*12 = 156, 13*11= 143, and so on down to 13*1 = 13. So there are a total of 13*(14+13+12+...+1) rectangles that can be made from the squares in the second row that do not include the squares in the first row.

So, in general, if you are in row k, then there are k*(14+13+...+1) rectangles that can be made that do not include the previous rows. The last row will have 1*(14+13+12+...+1).

So, in total, there are
14*(14+13+12+...+1) + 13*(14+13+12+...+1) + ... +
1*(14+13+12+...+1)
= (14+13+...+1)(14+13+...+1)
= (14+13+...+1)^2 rectangles.

You may remember that the sum of the first n whole numbers is equal to n(n+1)/2. So the sum of the first 14 whole numbers is 14+13+...+1 = 14(14+1)/2 = 105. So, in total, there are 105^2 = 11,025 rectangles that can be formed from the squares of a 14 by 14 grid.

In general, if you have an n by n grid, there are {n(n+1)/2}^2 rectangles that can be made.

Answer 2

first of all, it wasn't specified on how many 14 x 14 squares are at hand, but the answer is 2 squares to each rectangle you care to make

Answer 3

98. 14 x 14 squares=196 then you can split the squares into 2 triangles so 196/2=98.....so 98 is the answer

Answer 4

depends on how many squares you have, but as long as you pair 2, or more, squares together in one direction, you'll have a rectangle; also as long as the tops or sides are different lengths.

Answer 5

Depends on the size rectangles you want.

Answer 6

0. Duh. Unless you fold them in weird ways.