Four coloured lights
Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?
Age
11 to 14
Challenge level
Problem
Image
Imagine a machine with
four coloured lights and a keypad for entering numbers.
Each light responds to a
different rule. When a number is typed in, lights go on if their
rule is satisfied. If the number satisfies more than one rule, then
more than one colour will light up.
Your challenge is to find
the smallest possible number which will make all four colours light
up.
Here are some
data which have been collected about the responses from the
machine. Try to make sense of the information and figure out the
rules used by the machine.
A second machine works in
the same way but according to a different set of rules. Here are the
data from the second machine.
Send us your solution for a number which will turn on all the lights, with an explanation of how you arrived at your answer.
Getting Started
There is no number less than 1000 which will switch on all four colours.
Use four coloured pieces of paper or post-it notes to record the information gathered about each light.
Student Solutions
We had correct solutions from Jack Bird and Jack Manku of Isleworth, Syon and Vicknan from Colester Royal Grammar and from Oswald, Jimmy and Derek of Doncaster Gardens Primary School who sent us this:
The Green light is divisible by $11$ according to the clue at the bottom of page 2 whose numbers are all divisible by $11$.
Red is divisble by $3$, since at the top of page 1 it says that the only single digit numbers that do not light red are $8,4,2,5,7$ and $1$ which leaves out $3, 6, 9$, therefore mutiples of $3$.
Purple being the hardest one took some time to figure out. According to the information on the bottom of page 1 and the information on page 2, out of the numbers from $1$ to $9$ none light it up purple, yet $10$ to $99$ do. Also, the 4 digit and 6 digit numbers light it up such as $1234$ and $111111$ therefore, even digit numbers for example in this case, $2,4$ and $6$ digit numbers.
Orange is lit up by square numbers, referring to the information on page 2, saying $4,16,25$ and $36$ all light orange up.
The smallest number that lights up all lights is $1089$ it is equal to $33$ squared, is divisible by $3$ and $11$, and is a $4$ digit number.
Benjamin from Maidehnead sent us a partial solution for the second machine:
Red is divisble by $5$
Green is divisble by $11$
Blue lights up when the tens digit is odd
Lydia from Wellington School worked on both machines:
For each of the problems I wrote down all the numbers for each colour that I knew. I then looked for the patterns between the numbers.
Using this method I found that, in the first problem:
square numbers turned on the orange light,
multiples of 3 turned on the red light,
numbers with an even number of digits switched on the purple,
and multiples of 11 the green.
The lowest number that is a multiple of 3 and 11, that is square and has an even number of digits is 1089, and so this is my answer.
For the second problem I used the same method. I soon discovered that:
multiples of 5 turned on the red light,
multiples of 11 turned on the green,
numbers with a digital root of 8 turn on the orange light,
and numbers that had an odd number of tens switched on the blue.
935 is the smallest number which will make all four colours light up.
Teachers' Resources
Why do this problem?
Many standard questions give exactly the information required
to solve them, with one or two standard approaches signposted in
the question. This problem is different, in that learners are given
a large quantity of information to sort through and make sense of
for themselves in order to reach a solution. Along the way,
learners will have to make choices about how to proceed - the
opportunity to make such choices in problem solving is an important
part of every child's educational experience.
Possible approach
Begin the lesson 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
scientific enquiry, 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 which will not come up in
the main activity:
- Odd numbers
- Negative numbers
- Numbers which are not whole numbers
- Prime numbers
- Triangular numbers
- The sum of the digits is odd
- The numbers are always one more than multiples of 3
Once the class have tried the activity with a couple of rules
until all are reasonably convinced their hypothesis holds, move on
to the main task.
For the main activity, arrange the class into pairs or small
groups. Hand out a set of these cards (cut out in
advance) to each group, and introduce the task as it is described
in the problem.
Towards the end of the time working on the problem, leave some
time for the class to come together to discuss how they approached
the task, the decisions they made about how to organise themselves,
and the justifications for the conclusions they came to.
The second set of
cards could be used as a follow-up some time later, with
discussion afterwards focusing on whether they worked more
efficiently having attempted a similar problem before.
Key questions
How will you sift through the data?
How will you record your current thoughts?
How will you check your hypotheses?
Are there any cards that don't fit in with your hypotheses?Possible extension
The problem Charlie's Delightful
Machine offers a similar task based on an interactive
environment.
There is a follow-up problem, A Little Light
Thinking which would be suitable for the higher attaining
students.