Try to get the kid in question to come up with the following diagram and understand it:

X o o o o o o
x X o o o o o
x x X o o o o
x x x X o o o
x x x x X o o
x x x x x X o
x x x x x x X
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The number of x's (either capitalization is your answer. The number of points in the square is n*n. The square is evenly divided between x's and o's with the exception of the n X's on the diagonal. Therefore the number of x's is (n*n-n)/2 (half the number off of the diagonal) plus n (the number on the diagonal). That factors as follows:

(n*n - n)/2 + n = n*n/2 - n/2 + n
                = n*n/2 + n/2
                = n*(n+1)/2
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IIRC this explanation appears in Polya's excellent book, "How to solve it!" (I wouldn't bet too heavily on that particular memory though.)

Incidentally if the kid shows a mathematical inclination (lack of formal schooling notwithstanding), then I can highly recommend picking up The Mathematical Experience by Davis and Hersh. It is a hodgepodge of different topics at different levels. While they are mostly written at a level that can be understood by laypeople, by and large the topics are genuine research topics of interests to professional mathematicians. And what is communicated is truly, as best as is possible without years of training, the range of human experience within mathematics.