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Let a be the amount of movement before the tilt, and b be the tilt.
∵ each dot is placed perpendicularly above the dot under, we find the side of a square is the hypotenuse of a right angled triangle, where a and b are the other sides.

Using pythagoras’s theorem, the side of a square is √a^2 + b^2
∴ the area of the square must be (√a^2 + b^2)^2 , which is equivalent to a^2 + b^2. We see the pattern of the areas of squares with tilt of 1 to be 1 more than a square number. This is because b^2 = 1, so the formula for the area is a^2+1