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There is a method to solve multiplication arithmagons that works every time. You first multiply all of the numbers in the red bordered boxes together, and then find the square root. This will find the product of all the numbers in the purple bordered boxes. This is because all of the numbers in the purple bordered boxes are multiplied together at some point, but they are multiplied two times. So you find the square root.

To find the numbers in the purple bordered boxes you divide one of the red-bordered boxes from the number that you got in the previous operation. This will find the number in the purple bordered box that is not connected to the red one.

You can then find the rest of the numbers by dividing.