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UKMT Mathematical Olympiad for Girls, 2021

Combinatorics Difficulty 3.0 AMC 10/12 Prove it United Kingdom

Consider a $4 \times 4$ grid numbered $1$ to $16$ left to right then top to bottom. Tile A or Tile B is placed onto the grid so that it covers three adjacent numbers. Tile A is a straight tromino covering three cells in a row (or column). Tile B is an L-shaped tromino covering three cells arranged in an L shape.

(a) If Tile A is placed onto the grid (the orientation of the tile may be changed), can the total of the uncovered numbers be a multiple of three? (3 marks)

(b) In how many different ways can Tile B be placed onto the grid (the orientation of the tile may be changed) so that the sum of the uncovered numbers is a multiple of three? (7 marks)

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