Ladybird box

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

Age

5 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

Some toy ladybirds are kept in this box which has 9 little square compartments.

Can you place six ladybirds into the box so that there are just two ladybirds in every column and every row?

You could try this interactivity.

Printable NRICH Roadshow Resource.

Student Solutions

We had just a few submissions for this activity but they show a variety of solution systems.

Liva and Maya from the International School of Latvia sent in the following:

We solved the ladybird box problem. We both solved it without knowing the other one did too. Then we got together and talked about it.

Liva: So first I read the instructions. I tried some ways online, but it didn't work so I read the instructions again and it didn't say that the ladybirds can't be three diagonally, so I put all six randomly.

Then I put all the ladybirds that fit only the first instruction - two ladybirds in every column.

Then I tried to put them so they fit the second instruction.

After that I put only five - top right corner, top left corner, exact middle, bottom left and bottom right corner.

Then I saw that in the middle row was only one ladybird so I put the sixth ladybird next to the middle one and then I moved left bottom one space to right, because I knew about now they fit in every row and column.

Then I counted... Six! And it fits the instructions!!!

Maya: So first I read the instructions. It said that I have to put them in with only two in each column and row but it never said diagonally, so I started on the first column and put two going down.

Then I went to the second column and put two on the bottom going up.

Then the last two spaces that would work were the top and bottom squares on the last column.

Then I looked at the last words. 'Can you find different ways of doing it?' Yes! Just put it upside down! So that's how I solved the ladybird problem!

Jimmy from the Crescent Head Public School in Australia sent in the following:

I used counters to figure it out. I made a few mistakes at the start but once I got the hang of it I didn't need the counters.

Thomas spoke to his Granny who wrote down what he said:

"I put two in the first row then two in the second row. I saw one column with one ladybird and a row with one so I moved the last ladybird until there were two."

David from Histon Junior School said at first he couldn't do this problem. But when he'd found one way to place the ladybirds, he kept finding more and more!

Well done!

Thank you for these. I hope you'll send in solutions to other activities soon.

Teachers' Resources

Ladybird Box

Some toy ladybirds are kept in this box which has 9 little square compartments.

Can you place six ladybirds into the box so that there are just two ladybirds in every column and every row?

You could try this interactivity.

Printable NRICH Roadshow Resource.

Why do this problem?

This problem requires both spatial and number understanding. It fits well with work on pairs and counting by twos. To understand the problem, the children will need to be comfortable with the idea that the $3$ by $3$ grid represents a top view of the nine-hole box. They will also need to be familiar with the meaning of rows and columns. The problem provides a good opportunity for developing positional language and utilising ordinal numbers. For example: "I put a ladybird in the middle of the second row" or "Go across three then down one". It is also a good context in which to discuss what makes one solution different from another.

Possible approach

You could place some ladybirds in the box in some way using the interactivity and invite children to describe the positions of the ladybirds. This will give an opportunity for the class to come to an agreement about good ways to describe each 'cell' and to check the meaning of words such as 'row' and 'column'. You could then introduce the problem and encourage the children to work on it individually or in pairs. They could use a prepared $3$ by $3$ grid and six counters or the grid and ladybirds from this sheet .

When someone finds a solution quietly ask him/her to explain why it is correct, then have him/her draw the solution onto a grid. Invite him/her to find another way to do it. This could lead to a plenary discussion comparing the different solutions. You can encourage learners to decide what makes a solution different from another one - how about rotating the grid and mirror reversals of solutions?

To extend the opportunity for developing positional language, the $3$ by $3$ grid could be marked out with chalk on the floor or playground, and children used as the ladybirds. Children could take turns in giving instructions for the movement of the ladybirds until a solution has been formed.

Key questions

How many counters/ladybirds have you in this row? In this column?

Which row and which column can still have a second counter/ladybird?

Possible extension

Children could be encouraged to find all possible solutions and explain why they are sure they have found them all.

Possible support

You could encourage children to start by placing just four counters then seeing which row and which column have less than two counters in them.

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Number and algebra

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For younger learners