Nine coloured cube blocks fit exactly in the bottom of a square based box.
If 1 cube is red and 8 are white, then there are only 3 different ways that the 9 cubes can be put in the box. We show these patterns below. This is because if you put a cube in one corner you can rotate the box so it is in any corner. The same happens with the middle side cube.
In how many different ways can 2 red cubes and 7 white ones be put into the box? (Remember to think about what happens when you rotate the box.)
Solution
Let’s start with one red cube in the corner, and find all the different places for the second red cube.
Now let’s start with a red cube in the middle side position and look for patterns we haven’t made yet.
.gif)
Finally let’s start with the red cube in the centre. But a quick check shows we already have these patterns.
There are a total of 8 different patterns.
Extension
How many possible arrangements are there with 3, 4, 5, 6, 7, 8 and 9 red cubes?