The area of each of the four congruent L-shaped regions of this 100cm by 100cm square is $$\frac{3}{16}$$ of the total area. How many centimeters long is the side of the centre square?

The total area of the outer square is $$100 \times 100 = 10000$$ square cm, so each L-shaped region has area $$\frac{3}{16} \times 10000 = 1875$$ square cm, and all four together take up $$4 \times 1875 = 7500$$ square cm. The remaining area belongs to the centre square, $$10000-7500=2500$$ square cm, so its side length is $$\sqrt{2500} = 50$$cm.