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Ladybird Box

Stage: 1 and 2 Challenge Level: Challenge Level:2 Challenge Level:2

Thank you for all the answers you sent to this problem, however only a few of you described what you did to work out an answer. Sarah from Welton Primary School had a good way of putting the ladybirds in the box:

In the first column you put one in the top box and one in the second box.
In the middle column you put one at the top so you have two in the first row. Then if you put one in the bottom box you've got one in the bottom row.
Then in the last column you've already got two in the top row so you can't put one in there, so you will have to put one in each of the other rows so you've got two in all the rows and columns.

Thank you for taking us through your thinking, step by step, Sarah. Here is a picture of the way Sarah put the ladybirds into the box:

first solution

Stephen found this same arrangement of ladybirds, but went about it in a slightly different way. He said:

I worked out that column 1 could have two ladybirds, so row 3 could have two ladybirds, one each at the bottom of column 2 and column 3.
Then I worked out where the last two ladybirds could go.

Yara, who goes to the British International School of Riyadh wrote:

I did it by trial and error.

That's a good way to have a go at this probelm, Yara. (Although I like to call it 'trial and improvement'!) This means that you start by putting the ladybirds in any way, and then swap them around so that it works.

Several children from the Bishop Harvey Goldwin School sent in well-explained methods too. Here is the first one:

I firstly put three bugs diagonally so that there was one bug in each row and column.
Then I put one bug under the middle bug, one to the left of the middle bug and the last bug two squares to the right of the top bug.
Finally I checked my work to make sure there were two bugs in every row and column.

Here is a picture of this solution (which is definitely different from Sarah's, isn't it?):

second solution

Joel from Carr Green and Holly from Blockley both found four solutions to this problem. (I like the way you looked for more than one answer.) Here are the four that Joel sent which use 'L' to stand for a ladybird. Holly sent the same four solutions, but used a circle for a ladybird which is just as good. It's important to find a way to write things down that helps you.

third solution

Do you notice anything about these solutions? How do they compare to the one from Bishop Harvey Goldwin School?

I would suggest that they are all in fact the same as each other - just turned (rotated) or reflected. However, you may have counted them as different, which is fine. If we do say they are different, might there be other ways to draw Sarah's solution?