Representations of Integers

This most recent Sunday, I was reading Bill Gates’ The Road Ahead, and came across his discussing of obtaining all integer wattages between 0 W and 255 W by having 8 light bulbs (each with its own light switch) having wattage 1,2,4,8,16,32,64,128. I mentioned this problem to my class on Monday and was asked whether any integer between 1,\cdots,\frac{n(n+1)}{2}, for some \in \mathbb{Z}^{\geq 1}, can be represented as the sum of distinct integers in the set \left\{1,\cdots,n\right\}. After a couple false proofs, I came up with one, and one of my students came up with a more elegant proof. The interested reader can find the proofs here. I have omitted the student’s name out of respect for his privacy.

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