Interesting question! At first, one might suppose that the answer is obvious - 8x8 is 64, of course! But the answer is not so simple. One must ask, how many squares can be made in total - including the squares that can be made out of the other squares.
To answer this question, we definitely want to systematize our approach. It would be way too complicated to just make random squares here and there, and to try to count them up on our fingers. In mathematics, we often see patterns develop that can help us to determine complicated questions by first finding the solution to very simple ones. In some ways, this is like the inductive approach to proving a mathematical statement or equation - first prove n=1, then assume n, and show that n+1 is true. Let us try to solve this problem by considering the most basic case: a 1x1 square. Obviously, there is only one way to form a square out of this.
Next, consider a 2x2 square. In this case, we have the 4 individual squares, plus the one bigger one made up of the smaller ones. So in total we have 4+1=5.
Then, consider a 3x3 square. We have the 9 little squares. Then, if we count the number of 2x2 squares, we find 4. Finally, we have only 1 3x3 square. So the total is 9+4+1.
Next, consider the 4x4 square. We have 16 little squares, 9 2x2 squares, 4 3x3 squares, and 1 4x4 square, for a total of 16+9+4+1=30.
We definitely see a pattern emerging! The pattern is as follows:
1x1
1x1+2x2
1x1+2x2+3x3
1x1+2x2+3x3+4x4
...
1x1+2x2+3x3+..........nxn, where n is the number of sides of the square.
This can be proven fairly simply by induction.
It is interesting - induction can be used to prove a whole series of problems having to do with geometric patterns. Number theory and geometry (the most abstract and the most concrete topics in mathematics!) combine beautifully in order to determine the solution to these problems. Wherever there are patterns, there is often a mathematical description to the pattern.
Lovely solution and reflection, Etienne!
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