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every integer can be written in the form \pm 1 \pm 4 \pm 9\pm ... \pm n^2

Source: 1991 ITAMO p3

January 31, 2020
number theorySumSum of Squares

Problem Statement

We consider the sums of the form ±1±4±9±...±n2\pm 1 \pm 4 \pm 9\pm ... \pm n^2. Show that every integer can be represented in this form for some nn. (For example, 3=−1+43 = -1 + 4 and 8=1−4−9+16+25−36−49+648 = 1-4-9+16+25-36-49+64.)
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