For a larger chessboard such as a 10 x 10 board your answer would be because you add all the square numbers from 1 - I get squares because you know where the top corners of each square are to fit the size eg. So using this you can work out that there is 64 1x1s, 49 2x2s, 36 3x3s ect. You can see the pattern of square numbers appearing and then you can add them all up to get a final total of We found that there were squares altogether on the chessboard. This is how we did it.
First of all we counted how many 8x8 squares there were which wasn't very hard as there was one. Next, we conted how many 7x7 squares there were; which there were 4, one in every corner. We did this until we finished counting the 1x1 squares. We also found out that the nxn was a square number as how many squares of that size fitted into the chessboard there were. Thank you for reading this comment. I started with the 10x10 chessboard as suggested but I think it'd be easier to start with 2x2 and then 3x3, getting bigger.
This will help find the general rule. So, for 2x2 chessboard:. Now go on to a 3x3 chessboard. Express each answer in relation to the size of the original chessboard. That sounds like a great strategy.
I wonder if anyone can develop it into a rule for working out the number of squares on ANY size chessboard? Let's start by using this strategy and listing out how many squares there are in 1x1, 2x2 etc:. From this, we can see that we are adding on square numbers, 4 and then 9.
But how to explain this? Well, imagine a 4x4 square. Since it is a chessboard, the measure of how many you can fit onto each row and column needs to be squared to see exactly how many you can fit in total. Since we are squaring this, the result will always be square, hence the reason why square numbers are being added on.
Well, it can be explained by the increasing size of the square itself. For example, take a look at a 3x3 vs a 4x I therefore have proved that as the size of the square increases, the square numbers added on increase. The sequence of number of squares therefore follows the pattern: 1,5,14,30,55,91,,, We can see that the 8th number in the sequence is , so is how many squares an 8x8 will have.
But a general rule? Well, if we allow n to be the length of the side of the big square, the number of squares is equal to:. This is the nth term for this particular sequence, and so serves as a way of quickly working out the number of squares in any chessboard, such as squares in a x I agree that there are 64 1x1 squares but I got more than you for some of the others. Did you have a systematic way of counting them? Hi there, I have a solution on a word document, so is there an email address I can send it to?
I don't want to use the link above as I don't have outlook configured on my PC. Writing Tutorials. Performing Arts. Visual Arts. Student Life. Vocational Training. Standardized Tests. Online Learning. Social Sciences. Legal Studies. Political Science. Welcome to Owlcation. Related Articles. By Eric Caunca. By Linda Crampton. By Jule Romans. By precy anza. By Alianess Benny Njuguna. By Jason Ponic. This shows that my rule is correct. What have I noticed? From these cutouts I have noticed that the diagonal difference of a 2 x 2 cutout is 10 and that the grid length is Want to read the rest?
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Join over 1. Page 1. Save View my saved documents Submit similar document. Share this Facebook. How many squares in a chessboard n x n. Extracts from this document In this case m and n have Decreased by one, m-1 n
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