Paper Folding And Cutting Sangaku
The following three problems are from the 1743 book Kanyjia Otogi Zoshi by Nakane Genjun (17011761) [Fukagawa and Rothman, pp. 7677, pp. 8587]. In an English translation, Collection of Interesting Results in Mathematics, it sounds almost like one of D. Wells' books (The Penguin Dictionary of Curious and Interesting Geometry, The Penguin Dictionary of Curious and Interesting Numbers and The Penguin Dictionary of Curious and Interesting Puzzles). And indeed it gives us samples of the Japanese recreational mathematics of the 18^{th} century. (Fukagawa and Rothman's book is devoted to a peculiar mathematical art of sangaku but they also took pains to highlight the historical background of Japan during the Edo period and along the way outline the biographies and work of many contemporary mathematicians, Nakane Genjun in particular.

References
 H. Fukagawa, A. Rothman, Sacred Mathematics: Japanese Temple Geometry, Princeton University Press, 2008
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Copyright © 19962018 Alexander Bogomolny
The first problem comes with two solutions which the diagram below make quite obvious.
A solution to the second problem which is the least trivial of the three is presented by the following diagram where the light dotted lines indicate folds and the dashed lines indicate the cuts.
The diagram begs for an explanation. First fold the rectangle along the long axis LM. Then fold the corner D into J on LM. This will generate point K and divide angle at A into three equal angles of 30°. Indeed, in the right ΔAJM,
Let's check the lengths. From ΔAKD, DK = √3/3 = JK.
Here is the same decomposition with colored pieces.
Solutions to the problems with five squares are shown by the diagrams and require little explanation if at all:
(a) 

(b) 
(c) 
The latter is nothing but a proof of the Pythagorean theorem.
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Copyright © 19962018 Alexander Bogomolny