Problem:
Square ABCD has side length 1. Points P,Q,R, and S each lie on a side of ABCD such that APQCRS is an equilateral convex hexagon with side length s. What is s?
Answer Choices:
A. 32​​
B. 21​
C. 2−2​
D. 1−42​​
E. 32​
Solution:
Without loss of generality, assume that P and Q lie on AB and BC, respectively. Then AP=PQ=QC=s and PB=BQ=1−s, so PQ=2​(1−s). Because APQCRS is equilateral, it follows that s=2​(1−s), from which