  In 1959, Goodman determined the minimum number of monochromatic triangles in
a complete graph whose edge set is two-coloured. Goodman also raised the
question of proving analogous results for complete graphs whose edge sets are
coloured with more than two colours. In this paper, we determine the minimum
number of monochromatic triangles and the colourings which achieve this minimum
in a sufficiently large three-coloured complete graph.
