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 the 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:
{ C 1 := { 0 , 1 , 2 , 3 , 4 , 5 } C 2 := { 0 , 1 , 2 , 6 9 , 7 , 8 } {\displaystyle {\begin{cases}C_{1}:=\{0,1,2,3,4,5\}\\C_{2}:=\{0,1,2,{\tfrac {6}{9}},7,8\}\\\end{cases}}}
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.
{ C 1 := { e , g , j , o , r , y } C 2 := { a , c , f , n u , s , v } C 3 := { b , d p , l , m , u n , t } {\displaystyle {\begin{cases}C_{1}:=\{e,g,j,o,r,y\}\\C_{2}:=\{a,c,f,{\tfrac {n}{u}},s,v\}\\C_{3}:=\{b,{\tfrac {d}{p}},l,m,{\tfrac {u}{n}},t\}\\\end{cases}}}
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:
… excerpt ends here. Continue reading the full article.
