00:01
Okay, so i don't know what theorem 14 .2 .3 is for you because it's not included in the question you've uploaded, but i'm going to hope that it's burnside's lemma because that's what we would usually use to do this sort of question.
00:12
If it's not and you need to use a different theorem to complete it, then please just re -upload the question and include a little picture of the theorem that we've got to use so that we can help you as best we can.
00:22
But hopefully this is right.
00:24
And burnside's lemma tells us that the number of distinct colourings is equal to the number of colourings fixed under the symmetries of the group, or under symmetries of the square, divided by the total number of symmetries of the square.
01:06
So the square has symmetry groups.
01:10
There are eight different symmetries, so the number of symmetries of the square is eight, and those eight different symmetries are a 90 degree rotation, a 180 degree rotation, a 270 degree rotation, a 360 degree rotation, which is just the identity, as in not doing anything to the square, it's also a zero degree rotation.
01:40
Then we have a reflection in the vertical line through the centre, in the horizontal line through the centre, in that diagonal and in that diagonal.
01:54
So by the vertical line through the centre i mean reflection about this axis here.
02:02
And we want to find the fixed colourings under that.
02:05
So if you rotate, so if i just do a little colouring like this, if you rotate the square by 90 degrees, then you'll get this colouring...