Seven Flipped
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Problem
You have seven hexagonal-shaped mats in a line.
These mats all have to be turned over, but you can only turn over exactly three at a time.
You can choose the three from anywhere in the line.
A mat may be turned over on one move and turned back over again on another.
What is the smallest number of moves you can do this in?
Try with other numbers of mats.
Do you notice any patterns in your findings?
Can you explain why these patterns occur?
Click here for a poster of this problem.
Getting Started
Student Solutions
Joe from Bishop Ramsey School looked at the seven mat problem. He said:
In this problem it doesn't matter where you start on the diagram. My solution is written in a number of stages:
1) pick any 3 hexagons (they will turn blue)
2) pick 2 more hexagons and then pick one that you chose in the first go. You should now have 4 hexagons blue.
3) finally pick the three remaining red hexagons.
Many of you answered this part well, including Alistair of Cottenham Primary School.
Kahlia and Amy from Ardingly College Junior School then looked a bit further and tried with other numbers of mats:
If the number of tiles is a multiple of 3 it will divide equally into the number of tiles
Amelia and Kathryn, also from Ardingly College Junior School, investigated many different numbers of mats in a very systematic way:
6= 2 moves
7= 3 moves
8= 4 moves
9= 3 moves
10= 4 moves
11= 5 moves
12= 4 moves
13= 5 moves
14= 6 moves
15= 5 moves
16= 6 moves
17= 7 moves
18= 6 moves
19= 7 moves
Kahlia and Amy identified a pattern:
If there is a number of tiles 1 more than a multiple of 3 you add 1 to the answer of the multiple below it eg: 18 tiles = 6 turns; 21 tiles = 7 turns
Jeff and Raphael from Zion Heights Junior High School relate this back to the strategy for flipping the mats:
For numbers with one remainder after dividing by three, you follow the strategy for 7 mats stated above.
So, thinking about this like Kahlia and Amy did, we could say that if the number of tiles is 2 more than a multiple of 3, you add 2 to the answer of the multiple below it.
Well done to everyone who tackled this problem - it wasn't easy at all.
Teachers' Resources
Seven Flipped
You have seven hexagonal-shaped mats in a line.
These mats all have to be turned over, but you can only turn over exactly three at a time.
You can choose the three from anywhere in the line.
A mat may be turned over on one move and turned back over again on another.
What is the smallest number of moves you can do this in?
Try with other numbers of mats.
Do you notice any patterns in your findings?
Can you explain why these patterns occur?
Why do this problem?
This challenge is quite tricky, but it is a motivating context in which children can develop a logical, systematic approach.
Possible approach
It would be useful to introduce the problem on an interactive whiteboard so that the whole group can be involved with deciding which mats to turn. Alternatively, cardboard mats coloured differently on each side could be used and pinned to a board. This initial whole group work will familiarise the children with the "rules" of the problem so that they will be confident to find the smallest number of moves in pairs.
It will be important for them to devise a recording system that they are happy with, and this is something that can be addressed in the introduction, for example by asking whether they would be able to repeat the moves they made. Encourage them to think about odd and even numbers of flips, and when they come to investigate other numbers of tiles, you might expect them to generalise for multiples of three at least.
Elise Levin-Guracar kindly shared this sheet with us which includes a chart for tracking moves. She found it was was helpful to point out to students where they were trying the same thing over and over again, and where they could be trying something new. The last page of the document is a chart by number of dots and the minimum moves to flip. There are also blank moves charts. Thank you, Elise!
Key questions
How will you keep track of what you have tried?