Visualising the sum of the first $n$ positive odd integers

Question

Using the fact that $1+2+\cdots+n=\frac{n(n+1)}{2}$, we can deduce that sum of first $n$ positive odd integers is $n^2$. However, is there a way of finding the sum of $1+3+5+\cdots+(2n-1)$ visually?

Answer 1

Here is a ‘proof’ I once found in a book for young children. It is not a real proof in the mathematical sense, but rather a convincing example that any mathematician feels could be transformed into a rigourous proof:

Imagine wooden cubes stacked in rows, with the basis containing, say, $7$ cubes, the row above, $5$ cubes, the row still above, $3$ and the last row $1$, like this:

It is a geometrical evidence that, moving the grey squares from the bottom right corner to the top left corner, one recreates a square with sides equal to the number of rows, i.e. $4$ units, hence we have $16$ of them for the sum of the $4$ first odd numbers.