Auntie's Teacups

In training this year we had the opportunity to try out some tactile mathematical problems. Faced with a room full of attractive mathematical delights, my attention was immediately drawn to the simplicity of a stack of cups and saucers and a sawn off piece of table cloth. They, bizarrely, reminded me of times when mismatching the plates and cutlery in my family home and re-matching them, occupied hours of my time. Upon closer investigation I recognised an 'old friend' of a problem: Auntie's teacups.

The problem:

Auntie loves a good tea party. Auntie has 4 sets of 4 teacups and saucers: a green set, an orange set, a blue set and a pink set. She also has a chequered tablecloth with 16 squares on it. Auntie is a little eccentric and she likes to set the table so that every possible combination of teacup and saucer is present on the table. Additionally, she likes to make sure that each row and each column contains no teacups or saucers of the same colour: Teacups 

Using the problem in the classroom:

As a training, or newly qualified, teacher, the full extent of the resource cupboard remains a mystery. Some departments might helpfully have materials to hand which make setting this problem up quite an easy task. Generally, however, you have no idea whether a handy set of teacups and saucers are lurking in the cupboard behind the wall of overgrown dice, and dignity does not always permit a trip to the Early Learning Centre to source you own. Thankfully, there are about as many ways to present the problem to the class as there are ways of solving it.

The benefit of having tactile objects when solving the problem is that changing the position of the cups and saucers becomes easier. It involves significantly less correcting fluid than trying to complete the task on paper. It encourages students to keep playing with the task, as it avoids the fear of the ruined exercise book! Tactile resources do not need to be physical cups and saucers. A fellow trainee used large and small disks of card. The disadvantage comes with recording the patterns. It becomes significantly more difficult to notice permutations and symmetries when all the patterns are not in front of you.

Whichever way you choose to present the problem to your students, be prepared to expect the unexpected. The task is almost unique in producing countless tangents and methods.

Solutions:

Libby quite frequently implores us to not 'spoil each other's fun'. In the spirit of this, I won't reveal the solutions, and encourage you to have a play with the task yourself. I suspect there are more systematic methods to solutions than I know. The 4 'groups' of solutions, up to rotation and permutation are presented in the following video, which we, as a graduating class of trainee teachers created for our end of year celebration, in Libby's honour:  

See the Video!