Two-cube calendar
A two-cube calendar is a desk calendar consisting of two cubes with faces marked by digits 0 through 9. Each face of each cube is marked with a single digit, and it is possible to arrange the cubes so that any chosen day of the month (from 01, 02, ... through 31) is visible on the two front faces. A puzzle about the two-cube calendar was described in Gardner's column in "Scientific American". In the puzzle discussed in "Mathematical Circus" (1992), two visible faces of one cube have digits 1 and 2 on them, and  three visible faces of another cube have digits 3, 4, 5 on them. The cubes are arranged so that their front faces indicate the 25th day of the current month. The problem is to determine digits hidden on the seven invisible faces. Gardner wrote he saw a two-cube desk calendar in a store window in New York. According to a letter received by Gardner from John S. Singleton (England), Singleton patented the calendar in 1957, but the patent lapsed in 1965. A number of variations are manufactured and sold as souvenirs, differing in the appearance and the existence of  additional bars or cubes to set the current month and the day of week. Solution of the problem. Digits 1 and 2 need to be placed on both cubes to allow numbers 11 and 22. That leaves us with 4 sides of each cube (total of 8) for another 8 digits. However, digit 0 needs to be combined with all other digits, so it also needs to be placed on both cubes. That means we need to place remaining 7 digits (from 3 to 9) on the remaining 6 sides of cubes. The solution is possible because digit 6 looks  like inverted 9. Therefore, the solution of the problem is: formula_1 If the problem is based on another given set of visible digits, the last three digits of each cube could be shuffled between the cubes. Three-cube variation for the month abbreviations. A variation with three cubes providing English abbreviations for the twelve months is discussed in a "Scientific American" column in December 1977. One solution of this variation allows displaying the first three letters of any month and relies on the fact that lower-case letters "u" and "n" and also "p" and "d" are inverses of each  other. formula_2 Polish 3-letter month abbreviations (informal but commonly used for date rubber stamps - sty, lut, mar, kwi, maj, cze, lip, sie, wrz, paź, lis, gru) are also feasible, both in lower and upper case: formula_3 Four-cube variation. Using four cubes for two-digit day number from 01 to 31 and two-digit month number from 01 to 12, and assuming that digits 6 and 9 are indistinguishable, it is possible to represent all days of the year. One possible solution is: formula_4 The formula_5 could be any digit. The last three digits of each cube could be  shuffled between the cubes such that each digit from 3 to 9 is placed on at least two different cubes. With assumption that 6 and 9 are distinguishable characters, it is impossible to represent all days of the year because the necessary number of faces would be 25 and four cubes have only 24 faces. However, it is possible to represent almost all days in the year. There is a family of the best solutions which excludes only one day, namely Nov 11, e.g.: formula_6 The last four digits of each cube could be shuffled between the cubes such that  no cube has two identical faces (especially the 2's).