Nine keys, one is a lightweight

Question

You have nine keys that look identical. Eight are solid; one is a hollow fake that weighs a little less. You have a two-pan balance and no weights. Find the fake key using only two weighings, and explain why one weighing cannot guarantee it.

Accepted Answer

The key idea is to divide into three equal groups, not two, so that a balanced result still teaches you something. Weighing 1. Split the nine keys into three groups of three: call them A, B, C. Put A on the left pan and B on the right pan; leave C off the scale. - If the pans balance, neither A nor B holds the light key, so the fake is somewhere in C. - If one pan rises, that lighter group of three contains the fake. Either way, after one weighing you have narrowed the fake down to a specific group of three. Weighing 2. Take those three suspect keys. Put one on the left pan and one on the rig…

Nine keys, one is a lightweight

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