To find the number of tiles required to cover the path, we need to calculate the area of the path.

The length of the garden with the path is 24 m + 2 * 1 m = 26 m. The width of the garden with the path is 14 m + 2 * 1 m = 16 m.

The area of the path is (26 m) * (16 m) - (24 m) * (14 m) = 416 m².

To convert the area to square centimeters, we multiply by 10,000: 416 m² * 10,000 = 4,160,000 cm².

The area of each tile is (20 cm) * (20 cm) = 400 cm².

The number of tiles required is 4,160,000 cm² / 400 cm² = 10,400 tiles. Answer: \boxed{10,400}.