Dudeney 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.

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.

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”.