How many squares in 3x3 grid

A 4x4 square board has 30 squares, the smaller 16 plus 9 3x3's, plus 4 2x2's plus 1 4x4 one. Are you beginning to see the pattern?

What would your guess be for the number of squares on an 8 x 8 board? Can you derive a general expression for the answer? Here it is. As for how many rectangles there are in a square nxn squares big? We count only "rectangles", not the squares which are special cases of rectangles. Remember, only rectangles where the length is longer than the width. Again, the best way to creep up on a solution is to start out with the small size squares and see where it leads you.

For a 2x2 square, we have a total of 4 possible rectangles, each 1x2. For the 3x3 square, we can find 12 1x2 rectangles, 6 1x3 rectangles, and 4 2x3 rectangles for a total of For the 4x4 square, we can find 24 1x2's, 16 1x3's, 8 1x4's, 12 2x3's, 6 2x4's, and 4 3x4's for a total of 70 rectangles.

For the 5x5 square, we get 40 1x2's, 30 1x3's, 20 1x4's, 10 1x5's, 24 2x3's, 16 2x4's, 8 2x5's, 12 3x4's, 6 3x5's, and 4 4x5's. Did you notice anything as you looked through these numbers? Lets put them into a tabular form. Along side each nxn square will be the number of rectangles identified at the top of the column.

The number of 3 square wide rectangles in each case is equal to the number of 2 square wide rectangles in the previous case. The total number of rectangles in a square of nxn squares is equal to the sum of the 1 square wide rectangles for each rectangle from the 2x2 up to and including the nxn one being considered. Re: formula for grids A big thankyou to all who replied Cheers, Barry.

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Skip to content. Prove there are no other possibilities. The proof proceeds in 5 steps. Step 1: The magic sum is 15 By definition, every row, column, and diagonal has the same sum M. On the other hand, if we sum up all 9 elements, we must have the sum of the numbers 1 to 9. If a magic square exists, then each row, column and diagonal has to be Step 2: Combinations that sum to 15 Suppose you use the numbers 1 and 2.

Step 3: the center square is 5 The center square is involved in the middle row, the middle column, and both diagonals. Which number is involved in 4 sums of 15? There is only one! This is the number 5. Step 4: the corners are even; the edges are odd 1, 3, 5, 7 Similarly, each corner square is involved in a row, a column, and a diagonal sum.

By the same logic, the edge squares could be the odd numbers 1, 3, 5, and 7. Step 5: enumerating the 8 possibilities The number 1 can go in any of the 4 edges.

The number 9 will be forced to be opposite. I studied Economics and Mathematics at Stanford University. How many 3x3 squares are in a strip of squares that is 3 squares tall and 50 squares long?

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