Largest Even
Problem
The video below introduces this challenge:
You can have a go yourself using this interactivity:
To start with, the interactivity gives you one random digit.
Your task is to find the largest possible two-digit even number which uses the computer's digit, and one of your own.
Have a go several times.
How are you deciding which digit to choose?
Can you decribe a strategy that means your first 'guess' is always correct?
Clicking on the purple cog allows you to change the settings.
For a follow-on challenge, choose an odd two-digit target number (select 'Even False' in the settings).
How are you deciding which digit to choose now?
Can you decribe your new strategy so your first 'guess' is always correct?
You can explore the settings further to change:
- the number of digits in your target number
- the number of digits provided by the computer.
Let us know how your settings change your strategy.
Getting Started
What do you know about even numbers?
What do you know about the ones digit of even numbers?
Where could the digit that the computer has given us go - in the ones or tens? Why?
How can you make a large number?
How do you know that's the largest even number you can make?
Student Solutions
We had a few interesting solutions sent in for this activity - thank you to everybody who sent in their thoughts.
Jayden from the British Vietnamese International School in Ho Chi Minh City said:
To find an even number the ones in the number must be the digit: 0, 2, 4, 6, 8
For a two-digit number: if someone gives you an even number like 6 then you can put it at the end because 99 is the largest two digit number and with 6 you can make 96 which is a really high number
If it is an odd number like 5 then for two digits you put the number in the tens because if you put it in the ones it will be an even number. So you get 58 because 59 is an odd number.
Minh, also from the British Vietnamese International School, used the same strategy:
The strategy that makes my first answer always correct is first seeing if the number is even or odd. If it is an odd number, I will put it in the tens. In the ones, I will put the number 8, the highest single-digit even number. On the other hand, if it is an even number, I will put the number in the ones and put the number 9, the highest single-digit number in the tens.
Skyler from Westridge also looked at whether the number given is odd or even:
If the number you get is even, put a 9 in the tens place, and if the number is odd, use it as the tens place, and put an eight in the ones place.
Well done to all of you for working this out! Skyler also thought about whether or not there will always be a possible answer:
I think that every problem has a possible answer. Why? Because you can always make the largest even number. Even if the digit is 1. 18 is the largest number. Even if the largest number is small, it is still going to be the largest possible answer.
This is a good point, Skyler. How might this change if you were given two digits and asked to find the largest two-digit even number?
Dhruv from Pict Model in India made a flowchart to describe his strategy for making either the largest even number or the largest odd number. Thank you for sending this in, Dhruv - a flowchart is a very clear way of explaining what the best strategy is for this problem.
Lots of children from Twyford School in the UK worked hard on this task, and they started thinking about how they would find the largest three-digit or four-digit even number. Ben sent in the following method:
If you have an odd number and 2 squares you need to put it in the tens column. But if you have an even number you need to put it in the ones and put a 9 in the tens or hundreds.
So if you have an even number, you need to put it in the ones and put 9 in the tens and hundreds (if you have hundreds).
Let's say you have an odd number you need to put it in the tens or hundreds (if you have them) or even both if it is a 9.
And then you put the biggest even number 8, in the ones. When you have an odd number under 9 you need to put it in the tens and add a 9 in the hundreds and 8 in the ones.
Tessa sent us a picture of a similar method, which explains the different possibilities for each digit depending on whether the computer gives an odd or even number. This picture can be made larger by clicking on it.
Well done to both of you! I can see that both of these solutions will always give the largest possible even number.
Eva noticed that if the computer gives an odd digit, it can always be placed in the tens column to give the largest possible even number:
If the number given by the computer is even (0,2,4,6,8), then that number goes in the units column. Then 9 goes in the tens column. This gives you the largest 2 digit number. If you have a 3 digit number, you also put 9 in the hundreds column. If you have a 4 digit number, you also put 9 in the thousands column.
If the number given by the computer is odd (1,3,5,7,9), that number goes in the tens column. Then the largest number you can put in the units column is 8. That then gives you the largest 2 digit number. If you have a 3 digit number to guess, you do the same but then in the hundreds column you put 9. If it is a 4 digit number, you also put 9 in the thousands column.
This way your first guess is always right.
This is really clearly explained, Eva. Can you see how your method could be easily extended to even larger numbers?
Teachers' Resources
Why do this problem?
This problem encourages children to work together to develop a method for finding a solution which will always work. It will also help to reinforce understanding of odd and even numbers.
Possible approach
You could introduce the challenge using two sets of 0-9 digit cards. Hold up a 3 and ask the class which digit they could choose to put with the 3 to make a two-digit even number. Take a few suggestions and encourage learners to find all the possibilities (30, 32, 34, 36, 38), listing them on the board as you go. How do they know that each of these is even? What do they notice about these numbers? Encourage the class to explain why the 3 must go in the tens column if we are making an even number.
Repeat this, perhaps by holding up an even digit, for example 6, this time. Now they are many more possibilities as the 6 can go in the ones column or the tens column. Again, you can list them on the board as they are suggested by the class.
Then, introduce the interactivity, explaining that now we are being given a digit by the computer and our challenge is to find the largest possible even number with another digit of our choice. Children could write the answer to the computer-generated problem on individual whiteboards, or they could discuss with a partner before offering their suggestion. If the computer does not agree with the class' suggestion, take some time to discuss why. You will find that the interactivity does not allow a zero to be used as the tens digit of a two-digit number. If this comes as a surprise to the class, encourage them to consider why.
Ideally at this point, each pair of learners would have access to the interactivity on a computer or tablet so they can have a go at some more examples. Invite them to develop a strategy for always finding the highest even number on their first attempt. If you do not have access to multiple devices, you could continue working as a whole class using the interactivity displayed, or you could ask pairs to use digit cards to model what the interactivity does.
Bring everyone together after a suitable length of time to share ways of working. Encourage a few pairs to articulate how they make sure their first 'guess' is always the highest possible two-digit even number. You could compare and contrast the different approaches.
As a final challenge, give everyone time to consider how their strategy would change if they were asked to create the highest possible two-digit odd number. You can change the settings of the interactivity by clicking on the purple cog in the top right corner, and selecting 'Even False'. The observant amongst your class may notice that the interactivity now offers an extra button which says 'I don't think there is such a number'. When will this be useful? Why?
Key questions
What do you know about even numbers?
What do you know about the ones digit of even numbers?
Where could the digit the computer has given us go - in the ones or tens? Why?
How can we make a large number?
How do we know that is the largest even number we can make?
Possible support
Some children might benefit from having equipment to help them check whether a number is even, for example multilink cubes or counters that can be put in pairs.
Possible extension
Change the settings on the interactivity (click on the purple cog) to challenge children to extend their strategy for three-digit and four-digit numbers. They can choose to be given one, two or three digits by the computer (but the number of digits provided must be fewer than the target).
The problem Dozens includes a similar interactivity but with a more challenging starting point and the scope to explore multiples of any number, not just odd/even numbers. You could take a look at the Teachers' Resources of Dozens to see how that interactivity might be used in the classroom.