Light the Lights
2 out of 3
Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?
Problem
Here is a machine with four coloured lights. Each light responds to a rule.
When you type in a number, lights go on if their rule is satisfied.
If the number satisfies more than one rule, then more than one colour will light up.
Type in some numbers and see which lights you can switch on.
Do you get any surprises?
What are the four rules? How do you know?
Is it possible to light up all four lights at the same time? Why or why not?
Getting Started
How will you keep track of what happens with each number you try?
You might find a hundred square useful.
Student Solutions
Well done to everybody who worked out which rules made the lights turn on. We were sent some solutions which just described the rules for each light, but we've focused here on the solutions that explained in more detail what was going on.
Anya from GSS explained how they approached this activity:
There are 4 different types of lights. They are yellow, red, blue and green.
There are different types of rules for each light, which causes it to light up:
Yellow- When the number is bigger than the number 100
Red- When the number is divisible by the number 5
Blue- When the number is odd
Green- When the number is bigger than 4
There were some numbers with multiple lights, some with none and even some with all the lights lit up.
No Lights- 2,4
All Lights- 105,115,125,135,145
Multiple Lights- All the rest!
Solution-
The way I did it was after seeing all 150 numbers, I went to the table and compared each light with the number it matches with. I noticed lots of numbers being lit up, with each having its very own special rule!
I also used the line graph to see all the numbers on a line, and saw lots of yellow and green lights.
Lamu from Headstart International School in Phuket, Thailand noticed some patterns in the numbers, and found a number that would light up all the lights. This image can be clicked on to enlarge it:
Polina from Headstart International School found some similar rules in their lights:
Josh from Patterson Elementary in the USA explained why all four of the lights would never be on in their interactivity:
Four rules:
1. Yellow is always on
2. Red is on for odd numbers
3. Blue is on for even numbers
4. Green is on for all multiples of 10.
It's not possible to have all 4 of them on because a number can never be both odd and even!
We received a lot of solutions from the students at Ganit Kreeda in Vichar Vatika, India. Ruhi also found some cases where the lights could never all light up at the same time:
It certainly seems like the odd and even rules often make it difficult to light up all the lights at once. Avic also found an example which involved both odd and even numbers, and they gave a similar explanation:
Y: No. above 50
R: Even no.
G: Odd no.
B: No. under 5
It can not come because there is not a no. that can be an on and even no. at the same time.
Avic then found another example in which it was possible to light up all the lights at the same time:
Y: Even no.
B: All no. before 30
R: All no. before 50
G: Multiples of 5
Yes, at 0 because it is under 20 and 60, it is an even no. and it is in the table of 5.
Have a look at Avic's full solution to see more of their ideas about this.
Aarav found a different reason why their lights wouldn't all light up at the same time:
Reyansh noticed something interesting about prime numbers:
Q.1: What are the four rules? How do you know?
Yellow: table of 6, red: even, blue: table of 7, green: prime
Q.2 : Is it possible to light up all four lights at the same time? Why or why not?
No because 6 is not a prime number so then prime numbers and a table of 6 can not go together.
It certainly looks as if lights which light up on prime numbers and lights which light up on composite numbers can never light up together. I wonder which of Reyansh's lights could light up at the same time?
Thank you as well to the following children who sent in similar solutions to this activity: Charlie from Kilmore in Ireland; Simon, Thep and Dodo from Headstart International School; Mrunmayee, Kanna, Ishaan, Advaya and Ananya from Ganit Kreeda; and Nina, Maya, Archie, Xave, Al and Sas from Clifton Hill Primary School in Australia.
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?
This problem has the potential to surprise children and make them curious about the 'machine', thereby providing a motivating context in which they use their knowledge of number properties. Learners will have to make sense of information and work in a systematic way.
Possible approach
This activity will require children to have access to computers or tablets, ideally in pairs.
You could start by dividing the board into two columns, one headed with a tick and the other headed with a cross. Ask learners to suggest numbers, and write each suggestion in the appropriate column according to a rule of your own choice. Make it clear to the class that the activity is designed to model working like a scientist - there is no-one to say whether they have the right answer or not. So they can come up with a hypothesis for your rule, but you will not confirm their hypothesis, you will only place numbers in the appropriate column.
Here are some suggestions for rules:
- Odd numbers
- Numbers bigger than 20
- Multiples of 10
Once the class has tried the activity with a couple of rules and everyone is reasonably convinced their hypothesis holds, move on to the main task.
To introduce the main task, show the interactive and demonstrate entering a couple of numbers to see what lights up. Make sure learners understand that more than one light may be lit up at once, and that each light is governed by its own rule. Learners should work in pairs at a computer, and their challenge is to try to light up all the lights.
While the class is working, watch out for any moments of surprise and ask those children to try to explain why they were surprised. As you wander round the room, note any particularly good ways of recording or working systematically, and highlight them to the rest of the class.
Towards the end of the time working on the problem, leave some time for the class to come together to share what they have done. As well as discussing the number that lit up all the lights, encourage children to explain how they approached the problem. What are the four rules that govern the four lights?
Key questions
Which numbers have you tried? What happened?
Which number would be good to try next? Why?
How will you remember which lights each number lights up?
Possible support
Some children might find it helpful to use a hundred square to record their findings.
Possible extension
Learners could be challenged to create some rules so that it would be impossible to light all four lights. If appropriate, they might also have a go at Light the Lights Again which uses slightly higher-level mathematics.