Problem 20 · 2017 Math Kangaroo
Stretch
Spatial & Visual Reasoning
tiling-tessellationsymmetry
A square floor is tiled with triangular and square tiles in grey and white. What is the smallest number of grey tiles that must be swapped with white tiles so that the floor looks the same from each of the four marked viewing directions?
Show answer
Answer: C — one triangle, one square
Show hints
Hint 1 of 2
Looking the same from all four directions means the pattern must be unchanged by a quarter-turn rotation.
Still stuck? Show hint 2 →
Hint 2 of 2
Find the fewest grey tiles to recolour so every quarter-turn maps grey onto grey.
Show solution
Approach: enforce 4-fold rotational symmetry with fewest swaps
- For the floor to look identical from all four sides, the grey pattern must repeat under a 90° rotation.
- Compare each tile to where it lands under the rotations and fix the mismatches.
- The smallest fix recolours one triangular tile and one square tile.
- So the answer is one triangle, one square (C).
Mark:
· log in to save