Mixed-Up Socks
2 out of 3
Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?
Problem
Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair.
Now try it with four pairs of socks. Is there more than one way to do it?
Getting Started
You could print out this sheet of six socks and colour them different colours, then group them in pairs.
Remember each pair must be different so you can't have, for example, one blue and one green in one pair and then another blue and another green in a second pair.
Student Solutions
Many of you solved this problem for three pairs of socks. Katarina explained:
1. First, choose three colours of socks.
2. Then, reduce the colours to just their initials, eg. Blue - B. It also makes it easier!
3. Next, have a play around with the letters!
B + G = pair
B + O = pair
G + O = pair
You have to make sure that you haven't repeated a pair or a colour.
What good advice, thank you, Katarina. I agree that checking your solution is very important as it's easy to make a mistake in this problem.
Bafiar, Yunus and Alper (from Private IRMAK Primary and Secondary School Maths Club, Istanbul, Turkey) said :
We discussed your Mixed-up Socks question in our Math club hour and we found the answer using coloured pens. We made a combination using three socks. When we made pairs with red and blue socks, the other green and red socks, so the third pair must be green and blue. We found only one way:
Lucy from Bishop Ramsey School, Priya from Loughborough High School and Katarina all then went on to introduce another pair of socks of a different colour. They each foundtwo ways to mix up the socks this time. However, Bafiar, Yunus and Alper found more:
For four pairs of socks we found three ways. For example if the fourth pair is purple:
1. Green-Blue, Green-Purple, Red-Purple, Red-Blue
2. Red-Green, Green-Purple, Purple-Blue, Red-Blue
3. Green-Blue, Red-Purple, Purple-Blue, Red-Green
I wonder if you can explain how we know we have got all the different ways?
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
Very young children can work on this problem, as all that is needed is an awareness of same and different, and the idea of making groups of two. The problem also fits in with activities about pairs and counting by twos.
Possible approach
Ask the children to display and compare their solutions. One way of doing this would be for half the class to stay in their seats with their solutions and the other half to move around, talking to those who are sitting about what they did. They could then swap over. In this way, learners get a chance to question others' solutions and there is a purpose to explaining what has beeen done. You could then invite a few children to draw attention to someone else's solution for a particular reason. It may be that it is recorded very systematically, for example, or that their way of finding an answer seemed particularly efficient or understandable. You may also wish to discuss the number of different solutions the children found.
Key questions
I wonder how you can tell if you have found all the different ways of doing it?
Possible extension
For an extra challenge, try the problem using gloves instead of socks. This adds the complication of there being a left and right glove. How does this affect the total number of possible solutions?
Possible support
Suggest that learners use paper socks starting with just two pairs. Those with difficulties in recording could paste the different pairs of socks onto a sheet of paper.
Handouts for teachers are available here (Word document, pdf), with the problem on one side and the notes on the other.