Given the following three equations, what are the values of x, y and z?
x + y = x × y × z
x + z = x × y × z
y + z = x × y × z
The three variables have the same values: they are all either 0, √2 or –√2.
The following transformation shows that x, y and z must be equal. Start with these three equations, marked I, II and III:
I. x + y = x × y × z
II. x + z = x × y × z
III. y + z = x × y × z
By subtracting II from I, we get y – z = 0. By subtracting III from II, we get x – y = 0. Rearranging this, we get y = z and x = y, so x = y = z.
Therefore, x can be substituted for the variables y and z in the first equation.
First, x + y = x × y × z becomes x + x = x × x × x. Then:
2x = x3
x3 – 2x = 0
x × (x2 – 2) = 0
This means that either x = 0 or x2 – 2 = 0.
If the latter, x2 – 2 = (x – √2) × (x + √2) = 0.
So x = √2 or x = –√2.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.