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# Largest Even

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Age 5 to 11

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

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?