He was owner, publisher and editor of its predecessor, Recreational Mathematics Magazine, which appeared from 1961 to 1964.

[5] Gardner responded in the negative, including a box containing his correspondence and suggesting Madachy could use the addresses to promote the magazine.

[11] Longtime colleagues and co-authors include Martin Gardner, Harry L. Nelson, and Isaac Asimov, and Solomon Golomb (with pentominos).

[19] His recreational mathematics work included areas in chess, magic squares[20] and calculator art.

[5]: 272 Madachy is mentioned in the Jack Reacher novel series in the book Never Go Back, which uses perfect digit-to-digit invariant numbers in the plot: "Such numbers had been much discussed by a guy called Joseph Madachy, who once upon a time had been the owner, publisher, and editor of a magazine called Journal of Recreational Mathematics."