The genesis of this trick begins with a routine performed by John Henry Anderson, the Wizard of the North, who – as detailed in his Adelphi Theatre Program, 1841 (see The Linking Ring, Vol. 77 No. 4, Apr. 1997, p. 74) – did a sort of Cups and Balls sequence with large dice and five borrowed hats. Shells were involved. The combination of this trick with the Changing Caddy from Professor Hoffmann's Modern Magic, 1876, p. 348, evolved into the Sucker Die Box as we know it.
An early version, using a two-compartment cabinet with two front doors, is described as “A Trick with a Big Die” in Ponsin on Conjuring, 1854, p. 141 of the Sharpe translation. According to Ed Vyson in his Wizardry, 1906, p. 13, British magician and ventriloquist Alexander Davis invented the four-door sucker version and premiered it in the U.S. in 1886. Davis sold manufacturing rights to Otto Maurer, who marketed it in 1887 as the “Most Wonderful Dice Trick.” Davis's nephew, Prof. Davison, carried one back to England, where he performed it in 1889. All this predated claims of invention by Sidney Lee (c. 1896) and Julian Wylie (c. 1899). Carlton (Arthur Philps) released a marketed version with the now-standard presentation fully formed as “The Mysterious Cabinet and Travelling Die Illusion,” 1907 (see Ellis Stanyon's Magic, Vol. 7 No. 11, Aug. 1907, p. 87).