**Background**

A good thinking skills lesson comes derives from the problem of using 2 cubes to form a perpetual calendar. This is a problem that can take ages or be solved in a moment though so be prepared for either extreme. According to Professor Stewart’s Cabinet of Mathematical Curiosities the solution is enshrined in a patent from 1951 (so out of date now). It is used to produce calendars like that shown in the picture. The day of the week and month are irrelevant to this puzzle. We are concerned only with the 2 middle cubes.

**The challenge**

The challenge is to create a perpetual calendar for the dates of every month on 2 cubes. Each face of the cubes can have only a single digit number and the numbers 01-31 must be represented. The 2 cubes must be used so the leading 0 on a single digit number is required. So the question in a nutshell is what numbers have to be on the two middle dice for it to work as a perpetual calendar?

**The lesson**

In the solution you will see that a solution is quickly found if a certain idea occurs to the students. For this reason it is best to add a competitive element to the challenge so that they are less likely to call out the insight if they get it quickly.

There are two ways that I’ve equipped this lesson. The most straightforward is to issue each group of students with blank dice and a whiteboard marker. They can then draw directly on to the dice. Alternatively you can combine the challenge with using nets of 3d shapes. Each student makes a cube and then joins with one other to form a pair. Writing on a card net is difficult so in this case issue each group with some of the smallest size of stick on notes you can get. They can then write and erase on these before sticking them on to their cubes.

The least fiddly way of solving this problem is with a pencil and paper so ensure each student has one of these but leave them to choose their tools for the problem. Invariably they go for the dice.

As students say they’ve solved it ask them to count on their dice from 01 upwards. Quickly they’ll spot any mistakes. Most students start with a trial and error approach and will replace existing numbers with missing numbers and end up chasing an error all around the dice.

**The solution**

The key idea is to treat the 6 and 9 as the same numbers but turned upside down. Students who write their 9 with a curved tail are obviously more likely to spot this as those who write it straight have to change their style. A methodical approach can be used to show that this has to be done or the puzzle would not be possible.

If you show the numbers in a table you can talk through an efficient method for solving it and prove you are right.

Cube 1 |
Cube 2 |

1 | 1 |

2 | 2 |

3 | 7 |

4 | 8 |

5 | 9 |

0 | 0 |

Starting at first with what numbers we need two of we see 11 and 22 need to be represented. After this carry on listing the numbers. You might choose to put a 6 where the 0 is and carry on. The students should then realise this doesn’t work as you need a 0 on both dice to get 08 and 09 etc. So change the 6 to a zero and explain you will use the 9 as a 6. There are just enough faces to do it this way!

The perpetual calendar pictured above is one I purchased a long time ago an equivalent one that might go nicely in a clasroom is this one from amazon: Gisela Graham Tea Party Wooden Set Calendar. This one uses only the two relevant dice rather than the month as well.

Alternatively with a couple of plain wooden cubes, such as those available for teaching maths, it could make a nice craft project to make some with your class.