A square $666\times666$ has been covered using tiles $5\times1$. A single square $1\times 1$ hasn't been covered. Find in how many places the not covered square can be.
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Hint: Can you colour the individual squares with $5$ different colours such that any $5\times1$ tile covers one square of each colour? If so, what can you say about the colour of the uncovered square? Can you do this colouring in two similar but different ways? If so, the uncovered square must have the right colour in both colourings. Can you exhibit for each of the squares with the right colours an explicit covering that leaves that square uncovered?