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Answer for the clue "Geometric toy whose sides change depending on how it's folded ", 8 letters:
flexagon

Word definitions for flexagon in dictionaries

Wiktionary Word definitions in Wiktionary
n. (context geometry English) A flat model made from folded strips of paper that can be folded, or flexed, to reveal a number of hidden faces.

Wikipedia Word definitions in Wikipedia
In geometry , flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front. Flexagons are usually square or rectangular ...