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One coin is heavier or lighter and you have three weighings on a balance to work out which one it is.
One coin among the pile is counterfeit; it is either heavier or lighter than the rest and you are not told which. Tap a coin to put it on the left pan, tap it again to move it to the right pan, and once more to take it off. Both pans must hold the same number of coins. Press Weigh to see which side goes down or whether they balance, and you get three weighings in total. When you have worked it out, press Name the fake, choose a coin and confirm. A balance answers in three ways rather than two, so splitting the suspects into three even groups cuts them to a third each time, while splitting them in half throws one of those three answers away. Coins that have balanced are cleared and can be used as padding to compare groups of different sizes. Each round adds one more coin and shortens the clock. Three wrong answers ends the run, and your score is the number of fakes you correctly name.
The single idea that decides this puzzle is that a balance answers in three ways, not two. Left down, right down, level. Most people reach for halving because that is what searching feels like everywhere else, but halving only ever uses two of those three answers, and the cost is enormous.
The numbers here are not estimates. Every possible case can be worked through exactly, and doing so gives a clean result: dividing the suspects into three even groups identifies the fake among thirteen coins in three weighings every single time. Thirteen is the proven ceiling — with fourteen coins no strategy can guarantee it, and the best possible play drops to about nine times in ten. Against that, halving the suspects lands the answer around one time in ten, and comparing coins one against another manages roughly three times in ten. That is the whole difference between knowing this one fact and not knowing it.
So the first weighing should almost always be a third of the coins against another third. With twelve coins that is four against four: whichever way it goes, you are left with four suspects when the pans move and four when they balance. With nine coins it is three against three. If the count does not divide evenly, aim for groups as close to equal as you can rather than reaching for a half.
The second idea is that information about direction is worth as much as information about identity. When the left pan goes down, you have not learned that a specific coin is fake — you have learned that either a coin on the left is heavy or a coin on the right is light. Both live on as separate possibilities, which is exactly why the answer for thirteen coins is thirteen and not twenty-six. Track suspects as pairs of possibilities, not as coins, and the arithmetic suddenly makes sense.
The last trick is padding. Coins that have already balanced are known to be genuine, and genuine coins are tools: put them on a pan to make up the numbers when you want to compare a group of five against a group of three. Without padding you are stuck comparing equal-sized groups only, which is often the difference between a clean split into thirds and an awkward one.