All Change
Problem
Watch the video below (you do not need sound).
What do you notice?
What do you want to ask?
What are the 'rules' to this challenge, do you think?
How do you complete the challenge?
The rules are hidden below in case you would like to check them.
The aim of this challenge is to completely fill the grid with counters.
You will need a copy of the grid, 25 counters or buttons (or anything else that you have), a 1-6 dice, pencil and paper.
Throw the dice and place that number of counters anywhere on the grid.
Repeat this over and over again.
Each time you throw the dice, make a record (for example by keeping a tally) so that you know how many times you have thrown it so far.
Continue until you have completely filled the grid.
Make a note of the total number of throws that it took to fill the grid.
Now it is your turn! Have a go at the challenge for yourself (you may like to print off this sheet of the grid).
How many throws did it take to fill the grid completely?
Have some more goes to see if you can do it in fewer throws of the dice.
What is the smallest number of throws you did it in?
Do you think it would be possible to complete the grid in even fewer throws if you kept on trying? Why or why not?
Now take a look at two more videos, each one demonstrates a slightly different version of the challenge. (Once again you do not need sound.)
What is the same compared with the first version?
What is different?
Have lots of goes at these versions of the challenge.
How many throws did it take to complete the grid each time?
Which version of the challenge needed the fewest number of moves? Will that always be the case, do you think? Why?
Getting Started
If you aren't able to use the interactivity, you could print off this grid to use with counters and a 1 to 6 dice.
In the first version, what happens if you throw mostly high numbers? What happens if you throw mostly low numbers?
In the second version, how can you try to make sure you'll be able to go with your next throw?
Student Solutions
We've had a number of responses describing your thoughts on the problem:
Xing Yu from Catholic High School Singapore and Noor-Ul-Ain from Westfield Middle School both deduced that the later games will take more throws than the earlier games, because there are more rules and more restrictions. However, Xing Yu pointed out that this is only a general trend: "It could vary. If we get all ones for the first challenge, and get all sixes and fives for the other two, the second and third would take less throws." Edward from The Catholic School of St Gregory the Great, Freya from Simon Marks JPS and Precious from Bexley Grammar School all agreed that they thought the biggest factor was luck.
Maisie, Elle, Tahlia, Ryan and Finley from Moorgate CP School suggested that you could use your number bonds to 5 to work out what you need from the dice throws.
Cooper from Tuckahoe and Jade from Garland Junior School both said that all three games can be solved in just five rolls, if a 5 was rolled on each turn - however, Jade pointed out that this is very unlikely!
We'd like to hear if you have any more ideas!
Teachers' Resources
This activity has been inspired by Doug Williams' Poly Plug resource. However, you do not need sets of Poly Plug to have a go at it.
Why do this problem?
Possible approach
Key questions