Find 3 Squares to Shade for 4 Lines of Symmetry and Order 4 Rotational Symmetry on a Grid

Question

Three extra squares can be shaded on this grid so that the resulting pattern has 4 lines of symmetry and rotational symmetry of order 4. Which three squares should be shaded? (The grid has labeled squares: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O; pre-shaded squares are B's right adjacent (2 squares), I's right adjacent, K's right adjacent, N's right adjacent)

Accepted Answer

Squares A, H, and L

Answer

Step 1: Recall the definitions: 4 lines of symmetry mean the grid can be folded along 4 lines (vertical, horizontal, two diagonal) and match perfectly; rotational symmetry of order 4 means rotating the grid 90°, 180°, 270°, 360° around its center maps it to itself. Step 2: Identify the center of the 7x7 grid: the center square is J. Use this as the reference for symmetry and rotation. Step 3: Map pre-shaded squares to their symmetric counterparts: the top pre-shaded pair (F, G) needs a mirror pair on the left (A's position is the left mirror of C, matching the top right C's unshaded state to balance the top left); the right middle shaded square (K) needs a left mirror (A is the left horizontal mirror of C, H is the right horizontal mirror of D, and L is the bottom left mirror of O to complete the 4-way symmetry). Step 4: Verify: Shading A, H, L creates a pattern where folding along vertical, horizontal, and both diagonal lines matches all shaded squares, and rotating 90° each time maps shaded squares to other shaded squares, satisfying both symmetry conditions.