Balance puzzle

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A balance puzzle or weighing puzzle is a logic puzzle about balancing items—often coins—to determine which holds a different value, by using balance scales a limited number of times. These differ from puzzles that assign weights to items, in that only the relative mass of these items is relevant.

Known	Goal	Maximum Coins for n weighings	Number of Weighings for c coins
Whether target coin is lighter or heavier than others	Identify coin	${\displaystyle 3^{n}}$	${\displaystyle \lceil \log _{3}(c)\rceil }$
Target coin is different from others	Identify coin	${\displaystyle {\frac {3^{n}-1}{2}}}$[1]	${\displaystyle \lceil \log _{3}(2c+1)\rceil }$
Target coin is different from others, or all coins are the same	Identify if unique coin exists, and whether it is lighter or heavier	${\displaystyle {\frac {3^{n}-1}{2}}-1}$	${\displaystyle \lceil \log _{3}(2c+3)\rceil }$

For example, in detecting a dissimilar coin in three weighings (n = 3), the maximum number of coins that can be analyzed is 3³ − 1/2 = 13. Note that with 3 weighs and 13 coins, it is not always possible to determine the identity of the last coin (whether it is heavier or lighter than the rest), but merely that the coin is different. In general, with n weighs, you can determine the identity of a coin if you have 3ⁿ − 1/2 - 1 or less coins. In the case n = 3, you can truly discover the identity of the different coin out of 12 coins.