# Four-digit targets

## Problem

*Four-digit Targets printable sheet*

You have two sets of the digits from $0$ to $9$.

The idea is to arrange these digits in the five boxes to make four-digit numbers as close to the target number as possible.

You may use each digit once only.

You might like to use this interactive to try out some of your ideas:

*This activity has been adapted from one of BEAM's free Maths of the Month resources, which unfortunately are no longer available. **You can play it as a game against a friend using* this sheet *from BEAM.*

## Getting Started

Try using cards numbered from $0$ to $9$ or even pieces of torn paper.

Where is the best place to put $9$ when you are aiming for the lowest number?

Where is the best place to put $9$ when you are aiming for the highest number?

## Student Solutions

There were a lot of solutions sent in to this challenge. They showed that there had been a lot of trial and improvement, as well as calculation and thinking about place value.

The number of correct solutions was much smaller because most of the solutions sent in did not follow the instruction that said "You may use each digit once only". There were 20 digits to use and 5 solutions needed each using 4 digits, so all the 20 digits needed to be used but none of them twice. So, I'm displaying here the ones that followed the full instructions and gave some explanation of what they did (although scroll to the bottom for a spreadsheet of all the different solutions we received):

Ashton, Emily, Jamie, Lizzie, Ben at Crofton Hammond Junior wrote these very thorough solutions. Thank you for your good explanations:

On the first question I got confused but I realised how to do it. I had to give it a nine to make it one of the biggest odd numbers, then 875 to make it high but still having big numbers. On the largest even I gave it 987 so it was high and then 6 so it was even:

9875; 9876; 6330; 1105; 4422

The reason I did 9753 as the largest odd number was because I wanted to save one of the 9s and the 8s for other number. I did 9876 for the largest even number because I couldn't use the 9 in any other number:

9753; 9876; 8652; 1230; 4410

On my first question I looked down the list and saw that there were two other questions that said "largest" on them. I decided to share them out equally, so the numbers would all be rather big. For the smallest multiple of 5 I knew that a multiple of 5 has to end in a 5 or a 0, so I put the 0 at the end and at the beginning it was a very small number. For the last question I just used my left overs and it made a number close to 5000:

9767; 9886; 5532; 0110; 4432

For my first question I used a mixture of small and big numbers. Going down the list of questions I got really confused. I worked my way through the second question by using all of my sevens, this made it harder. I rubbed out some answers and replaced them with different numbers. I used all of my fives, but I also had to have four digits for the smallest multiple of 5:

9783; 8716; 6847; 0110; 4952

Maryam at St Mary and St Pancras School sent in the following:

For the largest odd number I wrote 9887 because it has four digits and I used the 9 for the thousands because it asked for the LARGEST odd number.

Largest even number 9762 because of the remaining digits I had, this was the largest possible number I could make which was also even.

For the largest multiple of three 6540 worked because the sum of its digits were 6. It is also quite large.

For the smallest 1150 for of the remaining digits 1150 was the smallest I could make.

Finally for closest number to 5000 4332 because that was the closest you make with the numbers I had left!

Dhruv and Harrison at Middleton Primary School wrote:

Largest odd number: 9987

Largest even number: 8766

Largest multiple of 3: 5544

Smallest multiple of 5: 1010

Closest number to 5000: 3322

We used the largest numbers for the odd number and the largest for the even number. We used the next largest numbers and divided it by 3 . We did the same thing to find the multiple of 5. We used the last numbers to find the closest number to 5000 .

Mali at St Philips Primary School wrote; My solution is: 9873, 9866, 7443, 1125, 5002

I cut the numbers in small paper squares, and organised them according to the rules. Then, I moved the numbers hoping that it would improve the result.

Ethan from Osidge School sent in the following:

I aimed to get the lowest cumulative difference for all five boxes. To do this I realised that the first three boxes must be in the 9000 and 8000 region. Then using the highest remaining numbers I matched the requirements.

However as the final box is nearest to 5000 I held the 5 back.

Then I tackled the lowest multiple of five which was 0110 but this left 5233 for the final box. However by swapping the 2 from 5233 with the second 1 in 0110 this reduced my net difference.

9865, 9746, 8742, 0120, 5133 also 9987, 8766, 5433, 0110, 5422

Luca, Luke, Finn, Daisy, Dina, Frankie, Freya and Bea of St Aidan's Primary School sent in a very clear account of what they did, which you can read here: LucaANDothers.doc

I've included all the correct solutions received up to date here.xls so that some groups of pupils could think about all the variety of different answers that we received. Sometimes there's only one correct answer, but in this case pupils have found solutions that obeyed all the rules but still differed.

I wonder whether we could judge whether some solutions are 'better' than others? How might you do this, do you think?

Some last minute solutions came from Nicholas, Arya, Alex, Lauren, Jodie, Matin, Zara, Patrick, Samuel & Ross, Grace & Kit, Maddie, Grace, Seven, Alice, Rosalyn From Hayes School

## Teachers' Resources

**Why do this problem?**

This problem would fit in well when members of the group are partitioning, rounding and ordering four-digit whole numbers. It requires considerable understanding of how the number system works. Playing the game, and discussing it afterwards, can really help to develop a firm concept of place value.

### Possible approach

### Key questions

Why have you put a $5$ here?

Where is the best place to put $9$ when you are aiming for the lowest/highest number?

### Possible extension

Learners could make up their own criteria for a new game possibly using four five-digit numbers.

### Possible support

Children could use this simpler version of the problem.