Beads
2 out of 3
Three beads are threaded on a circular wire and are coloured either red or blue. Can you find all four different combinations?
Problem
Three beads are threaded on a circular wire and are coloured either red or blue.
Can you find all four different combinations?
You now repeat the following actions over and over again:
- between any two beads of the same colour put a red bead
- between any two beads of different colours put a blue bead
- then remove the original beads.
Describe what will happen to each of the combinations.
Getting Started
Why not use multilink or counters to help you?
Can you find an efficient way to record what happens?
Student Solutions
V. Rolfe sent the following very clear solution to the Beads problem:
1) All three beads are red - RRR
When more beads are added, they will also be red as all the beads are the same colour. The pattern therefore remains the same.
2) Three blue beads - BBB
Three red beads are added as the beads are all the same colour, we then have the scenario described above, with three red beads and the pattern remains red.
Thus BBB becomes RRR
3) Two blue beads and one red bead - BBR (The order is not significant in the circle)
There are two sets of beads of different colours next to each other therefore two blue beads are inserted. The two blue beads lead to the insertion of one red bead.
Thus two blue beads and one red bead are added so the pattern remains constant, although the ring does 'rotate'.
4) Two red beads and one blue bead - RRB
This is the scenario above with two different coloured beads next to each other and one pair of the same colour. Two blue beads and one red bead are therefore inserted. When the original beads are removed the pattern becomes that described above, BBR.
Thank you - you have gone through the possibilities very systematically.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.