The array

contains dots, arranged in 5 rows and 5 columns.

We can view the array as consisting of a series of disjoint diagonals:

Counting the dots in the diagonals we get

which, looked at another way, says

.

Of course, there’s nothing special about 5: we can see by looking at an square array of dots that

and, more generally, if is any positive integer then

.

Seeing a square array of dots as a series of diagonals is key to understanding, and proving, that

for any positive integer .

This idea is a small, yet significant, step in awakening a developing mathematical brain.

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