You are shown 4 coins, their labels specifying their weighs: 1,2,3,4.

You are told that the labels may be correct, but there is also a possibility of one coin not fitting its label.

Determine whether all coins are ok, and if not which is the faulty one, specifying is it heavier or lighter than its label.

Clearly the above task can be accomplished in 2 weightings using balance scale only.

Please describe the procedure and prove that there is only one solution.

There are only three possible weighings:

(a) 1 + 2 = 3

(b) 1 + 3 = 4

(c) 2 + 3 = 1 + 4

Using (b) and (c) fails, as it cannot differentiate between a heavy 3 and a light 4.

Using (a) and (c) fails, as it cannot differentiate between a heavy 3 and a light 1.

But 1 + 2 = 3 and 1 + 3 = 4 works

The possibilities are

Left Side (Light, Heavy or Even)

L, L -- 1 is light

L, E -- 2 is light

L, H -- 3 is heavy

E, L -- 4 is light

E, E -- All correct

E, H -- 4 is heavy

H, L -- 3 is light

H, E -- 2 is heavy

H, H -- 1 is heavy