Strategy guides › Odd Coin
How to solve Odd Coin: count possibilities, split them in three
▶ Play Odd Coin🎯 Try the classic twelve coins
Twelve coins, one fake, and you don’t know whether it’s heavy or light. Most people start the obvious way: six on each side. The beam tips left, and now you have twelve coins that are all still suspects. Six could be the heavy one and six could be the light one. You’ve used a weighing and learned almost nothing.
Odd Coin is built so that this kind of start always costs you. There’s no hidden fake to get lucky with. The balance decides each result as you weigh, and it always gives the answer that leaves you the most work. The only way to make par is a plan that works whatever the scale says. That sounds harsh, but it makes the game very fair: everything below is a method, not a guess.
Count possibilities, not coins
The one idea that solves every puzzle in the game is to count possibilities, not coins. A possibility is a coin plus a direction: “Moon, heavier” is one, “Moon, lighter” is another. With twelve coins and no clue about the direction there are 24. If the card says the fake is lighter, each coin is just one possibility.
Every weighing has three outcomes: left down, right down, or level. So one weighing can split the possibilities into at most three groups, two weighings into nine, and three into 27. That’s the whole limit. Twelve unknown coins is 24 possibilities, which fits under 27, so three weighings might be enough. The game’s solver confirms it: par for twelve coins is 3.
Because the balance always picks the largest group, a weighing is only as good as its worst outcome. The goal of every weighing is to split the possibilities into three groups that are as equal as possible, with none bigger than the next power of three.
Why four against four is the only opening
Here’s what our solver found when it tried every possible first weighing for twelve coins. Out of six options (one against one up to six against six), exactly one keeps par: four against four.
Three against three looks careful, but a level result leaves the six coins you didn’t weigh, which is still twelve possibilities. Five against five fails the other way: a tip makes all ten weighed coins suspects. Four against four leaves eight possibilities in every case. If the beam tips, four coins could be heavy and four could be light. If it stays level, the four unweighed coins are still suspects in both directions, but the eight you weighed are now known to be genuine.
The same count works on every daily. With thirteen coins and the stamped silver weight, there are 26 possibilities and only one opening keeps par: five coins against four coins plus the weight. A tip leaves nine possibilities, a level result leaves eight. With nine coins and the fake known to be lighter, there are nine possibilities and the only opening is three against three.
After a tip, mix your suspects
Say the beam tips left after four against four. Coins 1 to 4 can now only be heavy, and coins 5 to 8 can only be light. On Easy and Normal the coins show this with H? and L? tags. On Hard you pencil the marks yourself with ✎. Eight possibilities and two weighings left means each outcome of the next weighing must leave at most three.
The trap is to keep the groups apart. Weigh 1 and 2 against 3 and 4 and a level result leaves all four light suspects: too many. The fix is to mix heavy suspects and light suspects across both pans, and use a known-genuine coin to make the pans equal.
This isn’t the only good second weighing. Our solver counts 24 that keep par out of 192 legal ones (pan sides counted separately), which means about seven in eight possible second weighings lose a weighing. All the good ones share the same shape: suspects from both groups on the pans, a few left off, and each outcome capped at three.
If the beam stays level
A level result after four against four leaves four fresh coins (8 possibilities) and eight genuine coins to help. The solver’s good moves all weigh three of the fresh coins against genuine ones: three fresh coins against three genuine ones, or two fresh coins against one fresh coin plus a genuine one. Weigh the four fresh coins against each other two against two and a tip leaves four possibilities with only one weighing left. That’s too many.
Don’t name a coin early
When you name the fake, the game checks one thing: is any other coin and direction still possible? If one is, the game shows you that coin and the round scores nothing. On Easy and Normal the notebook counts the possibilities left for you. Name the fake only when it reads one. On Hard, check your marks: every coin except one should be âś“, and that coin should have a single direction.
How long a puzzle should take
In 2,000 daily puzzles from the generator, 84% have par 3 (nine to twelve coins with the direction unknown, or eleven to thirteen with the silver weight) and 16% have par 2 (nine coins with the direction given). Easy is always par 2. Hard is par 3, except that 15% of Hard puzzles have thirteen coins and no silver weight. Those have par 4: the solver proves 26 possibilities can’t be split in three weighings without a genuine coin to start with.
Practice
The practice puzzle is the classic: twelve coins, direction unknown, par 3. Open with four against four. After a tip, mix the suspects as in the second figure. After a level result, weigh three fresh coins against three genuine ones. Before each weighing, count how many possibilities each outcome would leave.
