How to count total number of squares in a given square?

Question

Respected/Dear Mathematical friends, I'm not dealing with mathematics hence poor in it. I got a book in which a square with 4 rows and 4 columns are drawn saying that totally they count 30. I'm not getting more than 26 by counting. Is there any formula or method to count or any trick? Please help this poor mathematical friend or your student.

Answer 1

There are 16 small squares. But look for the larger squares to get your total. For example, every 2x2 block has a square around it as well as the 3x3 blocks. There is also a square around the entire 4x4 block.
Here is the breakdown:
4x4 squares = 1
3x3 squares = 4
2x2 squares = 9
1x1 squares = 16
total = 30

The number of total squares for an nxn square is always going to be equal to the sum of 1^2 + 2^2 + 3^2 +.....n^2.

Answer 2

The way that I solved this problem was by getting some graph paper and then I drew a seven by seven square. I would put my pencil on the point where the 7x7 square would end and say that there is only one way to do a seven by seven square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7x7 x shows the bottom right corner that the . . . . . . . 7 x 7 square can fit in . . . . . . . . . . . . . x Then I would find the point in the 7x7 square where a 6 by 6 square would fit in and put a dot everywhere where i could put the 6x6 square in it. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6x6 x shows the bottom right corner that the . . . . . . . 6 x 6 square can fit in . . . . . x x . . . . . x x Then I repeated this way a few more times for the 5x5 square and the 4x4 square and I found a pattern that helped me solve the rest. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 x 5 x shows the bottom right corner that the . . . . x x x 5 x 5 square can fit in . . . . x x x . . . . x x x If I had five dots on the out side of my "x-ed" surface then I would add two to it and then I would add those two numbers together and get n answer. And my answer adding up 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140 squares. for 6 * 6 squares 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2= 91 squares

Answer 3

In the 4x4 square, you have 16 squares that are sized 1x1. Then you have 9 squares that are 2x2, 4 squares that are 3x3, and finally the whole 4x4 grid itself is one big square. So the sum of all of these is 30.

There are couple of methods to doing this to make sure you count all of the squares. Here's one way:

1) Start with the smallest squares. If your square is N by N, then there are N*N of these mini-squares.

2) Now consider 2x2 squares. Make one out of the 4 mini-squares in the upper left corner, then slide this over to the right, then right again, and so on until you come to the end of the row. Then move the square down a row and repeat again from lef to right. Notice that there are only (N-1)(N-1) places where the center of a 2x2 square could be, because if you remove the border of an NxN square you're left with an (N-1) by (N-1) square.

3) repeat this process for 3x3, 4x4 etc. squares up until you get to the biggest size, which is one the giant NxN square.

You'll notice that the answer is:

1² + 2² + 3² + ...(all the way up to)... + N²

In fact, there's also a shortcut to this formula. It's always equal to N(N+1)(2N+1) / 6

Answer 4

If you'll draw 5 vertical and five horizontal lines you'll get a table of order 4X4 i.e. 16 small squares.

Now, inorder to get 1 square you require 2 horizontal and 2 vertical lines which are equally spaced apart from each other.

If you observe you'll find 16 smallest possible squares, then 9 squares of order 3X3 i.e. consisting of 4 smallest square each, this will continue till order is 1X1 i.e. the biggest square, hence total number of squares are

4X4 + 3X3 + 2X2 + 1X1=16 + 9 + 4 + 1=30

Answer 5

There are 16 of size 1 and 1 of size 4. That's the easy part.

There are also 9 of size 2.

And there are 4 of size 3.

Probably, you forgot the ones of size 3.

Answer 6

Multiply the first row with the first column. and you'll get the total number of squares in the square

Answer 7

the total no. of squares in this case are 30, and it is correct.the method to count this is: 4 sq.+3sq.+2sq.+1sq. =16+9+4+1=30

this method is applicable to all the squares.