Dudeny dissection

What is the minimal number of parts an equi­lat­eral triangle should be cut into so that a square can be formed rear­ranging them? This problem was offered as a chal­lenge to the readers of the Daily Mail news­paper issued as of 1st and 8th February 1905. Among hundreds answers obtained, only one was right: four parts are enough.

Dudeney writes:

I add an illus­tra­tion showing the puzzle in a rather curious prac­tical form, as it was made in polished mahogany with brass hinges for use by certain audi­ences. It will be seen that the four pieces form a sort of chain, and that when they are closed up in one direc­tion they form the triangle, and when closed in the other direc­tion they form the square.

How can one guess such a dissec­tion? Take a triangle and a square of equal area and make two regular stripes, repeating each of the shapes. Putting the stripes above one another so that the maximum number of one's side midpoints matches the sides of the other, the desired dissec­tion is obtained. This is in a way a general method of finding dissec­tions of equal area poly­gons. Solving such prob­lems is the subject of Harry Lind­gren's ”Recre­ational prob­lems in geometric dissec­tions and how to solve them” book.

Other models in “Areas and dissections of shapes”