Solution to "How Many Rectangles?" on Page XX
Slow and easy method: Consider a 4 X 1 grid, that is 4 rows and only one column. This has one rectangle with 4 cells, another with 3 cells, another with 2 cells and lastly a single cell rectangle (which is also a square). Notice that counting the top 3 or the bottom 3 gives us the same 3 X 1 rectangle. Thus we have 4+3+2+1 = 10 rectangles.
We also see that an n X 1 grid contains n + (n-1) + (n-2) + ... + 1 = n(n+1)/2 rectangles.
We are going to use n(n+1)/2 often, so let us use a special notation for it. From now on, if we say n^*, we mean n(n+1)/2.
Now consider a 4 X 2 grid. It has two 4 X 1 grids and then some extra. Those two "columns" give us 20 rectangles. How many are extra now? If we were to consider each rectangle of width 2, we have exactly as many width-2 rectangles as we counted earlier! So we get 30 rectangles totally now.
For n X 2 grid, we get 3.n^* rectangles.
Now add another column. By now we know what is going on, so let us do it directly for n. We get n^* from that column, n^* width-2 rectangles and n^* width-3 rectangles. Thus we get 3.n^* more rectangles making a total of 6.n^* rectangles.
Without detailed calculation we can see that adding the next column will result in a total of 10.n^* rectangles. We are building the progression 1, 3 = (1+2), 6 = (1+2+3), 10 = (1+2+3+4). But this is going to again give us 1+2+...+m = m(m+1)/2 = m^* as a factor!
Thus, when we consider a grid with n rows and m columns, we have m^*.n^* rectangles.
For the 4X6 grid we get 10 times 21 = 210 rectangles.

Quick and elegant method: In mathematics, we always look for elegant proofs, which are short and sweet.
A grid with n rows and m columns has n+1 horizontal lines and m+1 vertical lines. To define any rectangle within the grid, we must choose 2 among the horizontal lines and 2 among the vertical lines. There are "n+1 choose 2" ways of doing the first, and "m+1 choose 2" ways of doing the second. But "n+1 choose 2" = n^*. Thus we get a total of n^*.m^* rectangles.

Bonus: Now consider when m = n, so we are considering a square grid, and we want to count only the number of squares inside. How many squares do you find?
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