We will refer to squares by rows A, B, C and D and columns 1, 2, 3 and 4. Suppose thegrid can be filled with two corners the same colour.

Suppose adjacent corners, say A1 and A4, are the same colour, say green.

Then A2 and B2 cannot be green becauseA1 is green. Also, A3 and B cannot be green because A4 is green.

But then the sub-square A2, A3, B2, B3 contains no green. This is a contradiction, so adjacent corners cannot be the same colour.

Suppose opposite corners, say A1 and D4, are the same colour, say green. Then one of C2 and B3 must be green to give the middle sub-square a green square. Since they create symmetrical diagrams, suppose, without loss of generality, B3 is green. Then C1 must be green so that the sub-square B1, B2, C1, C2 contains a green. Also, D2 must be green so that the sub-square C2, C3, D2, D3 contains a green. But then the sub-square C1, C2, D1, D2 contains two greens.

This is a contradiction, so opposite corners cannot be the same colour.

Since no colour can be in two corners, all four corners must contain different colours.

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Alternative solution

The whole grid has four 2 by 2 sub-squares so contains all four colours four times each.

Columns 2 and 3 have two sub-squares, so contain two of each colour. The centre sub-square has one of each colour, so A2,A3, D2 and D3 contain one of each colour.

Rows B and C have two sub-squares, so contain two of each colour.

Therefore the twelve squares excluding the corners contain three of each colour.

Therefore the four corners must contain one of each colour once each, so they are all different colours.

Alternative

The whole grid has four 2 by 2 sub-squares so contains all four colours four times each.

Each corner square is in one sub-square. Each edge square is in two sub-squares. Each centre square is in four sub-squares. Each colour needs to be in 9 sub-squares.

The total for each colour must be 9, which is odd, so must contain an odd number of corner squares. Therefore all four colours need to be in at least one corner square.

There are only four corners, so all four corners contain different colours.

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