Identical copies of one regular polygon are to be fitted edge to edge around a single point, filling the space completely with no gaps and no overlaps. Which of these regular polygons canNOT be used in this way?

The copies meeting at the point must have interior angles adding to exactly 360°, so the interior angle has to divide 360° a whole number of times. The equilateral triangle's 60° goes in six times, the square's 90° four times and the regular hexagon's 120° three times. The regular pentagon's interior angle is 108°, and 360/108 = 3⅓ — three pentagons leave a gap of 36° and four overlap, so the pentagon is the one that fails. In fact these three are the only regular polygons that tile the plane on their own.