Another beautiful question! Outline solution: It can be done in 4 weighings, by weighing each one against the other three. The key is to note that all four weighings should have the same parity. If there is a rogue weighing (by +/- 1gram) its parity will be different from the other 3. Now use the three 'correct' equations to substitute in the fourth 'incorrect' equation, and we can then easily identify (by going mod 4) whether the 1 gram error needs to be added or subtracted. Now we have 4 'correct' equations, and 4 unknowns. Done Merlin