Scale Puzzles 301 - Second Weighing: Balanced this time Back to Top of Puzzle * * Please help us keep our puzzles free! * * Back to <www.MAZES.com>

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You have twelve coins (A, B, C, D, E, F, G, H, J, K, L & M). One of the coins is counterfeit, but you don't know if it is heavier or lighter than the others. You are allowed to use a balancing scale three times to find the counterfeit. Find a strategy that will allow you to find the counterfeit in three weighings, no matter which one is fake.

First Weighing:    ^_(A+B+C+D)_ /\ _{_(E+F+G+H)_}
(the right side of the scale went down)

This told us that either

• The fake is lighter than a real coin and in the group (A+B+C+D), or
• The fake is heavier than a genuine coin and in the group (E+F+G+H).
• We also know that (J+K+L+M) are all genuine coins.

We created three new groups of coins by

• taking (B+C+D) off the left side of the scale (which includes the fake if scales balance second time)
• moving (F+G+H) from the right to the left side (which includes fake if scale tilts other way second time)
• moving (K+L+M) onto the right side of the scale
• The fourth group of three coins is (A+E+J), which did not move from their previous position. If the scale does not change its tilt, the fake coin is in this group.

What does this tell us?

• The fake coin was ON the scale during the first weighing
• The fake coin was NOT on the scale during the second weighing
• The fake coin must be one of the three coins that we removed from the scale between the weighings
• We removed (B+C+D) from the scale after the first weighing, so
• The fake coin must be one of the coins in the group (B+C+D)
• The fake coin must be lighter than a genuine coin, because (A+B+C+D) is lighter than (E+F+G+H)

Third Weighing: We know that the fake is LIGHTER and it is in the group (B+C+D). Can you use the balance scale one more time to find the fake coin. If you need more hints in order to develop your strategy, please review what we did in Scale Puzzles 101, when we had three coins, one of which was a heavier fake.

You are now done with this set of possible results.

Click here to return to the Scale Puzzles 301 Top Page and try another set of possible results.

There are nine different "third weighing" pages, like this one, that you can eventually reach.