Hummer's die mystery is a mathematical magic trick that identifies a chosen face of a cube using at most three yes-or-no answers about which faces are visible. The performer does not see the markings on the cube (Gardner 1956, pp. 45-46; Numberphile 2021).
Give the six faces distinct labels, and let a spectator choose a label without revealing it. The chosen label remains fixed throughout the trick. The spectator views the cube from a direction in which exactly three faces meeting at a vertex are visible. Between questions, the performer may rotate the cube while keeping track of the positions of its faces. Each answer states truthfully whether the chosen face is visible.
The three visible faces meet at a vertex, and the three hidden faces meet at the opposite vertex. The first answer therefore leaves three candidates meeting at a vertex. Replace a candidate face by its opposite to obtain another triple meeting at a vertex, then rotate the cube to display this triple. Exactly two candidates are now visible. A negative second answer identifies the third candidate. A positive answer leaves two adjacent faces. Another rotation can make exactly one of these visible, so a third question identifies the chosen face. The performer can then turn it uppermost. Only positions are tracked, so the method is independent of the arrangement of labels.
Three questions are optimal in the worst case. Two yes-or-no questions can distinguish at most 
choices, fewer than the six possible chosen faces. The construction
above achieves the bound of three questions.
