Finding the fake coin with a balance scale
This guide explains how to find a single counterfeit coin using a balance scale and a strict limit on how many times you can weigh. You will learn why splitting coins into three groups beats splitting into two, how to read the balance's three possible answers, and what makes the twelve-coin puzzle one of the hardest logic puzzles around.
The puzzle and its three answers
You have a pile of coins. All of them are genuine except one, which weighs the wrong amount. You also have a balance scale, but you can only use it a fixed number of times. After your last weighing, you must name the fake coin.
The balance gives exactly three results for each weighing. The left pan goes down, meaning the left side is heavier. The right pan goes down, meaning the right side is heavier. Or both pans stay level, meaning the coins on each side weigh the same. That level result is just as informative as either tilt, and learning to use it is the key to the whole puzzle.
The number three matters deeply here. Each weighing produces one of three outcomes, so two weighings can distinguish at most nine different things - three times three. Three weighings can distinguish at most twenty-seven. The puzzle sizes are built on these exact limits.
The three puzzle sizes
The Puzzle dropdown at the bottom offers three levels. The easy puzzle gives you eight coins and two weighings, and you are told the fake is heavier. The normal puzzle gives you nine coins and two weighings, also with the fake known to be heavier. The hard puzzle gives you twelve coins and three weighings, but you do not know whether the fake is heavier or lighter.
Eight coins in two weighings is a warm-up. Nine coins in two weighings is right at the mathematical limit - two weighings with three outcomes each can distinguish nine things, and you need to find one coin out of nine. There is no slack at all.
Twelve coins in three weighings with an unknown direction is the classic form of this puzzle. Three weighings can distinguish twenty-seven things. Twelve coins times two possible directions - heavier or lighter - gives twenty-four possibilities, which fits inside twenty-seven but only just. One wasted weighing and the puzzle becomes impossible.
Controls and how to weigh
Tap a coin once to place it on the left pan. Tap it again to move it to the right pan. Tap it a third time to take it off entirely. The coins show their current position - an L for the left pan, an R for the right pan, or nothing if they are sitting in the main pile.
Press the Weigh button when you have placed the same number of coins on each side. Both pans must hold exactly the same count. If they do not match, the game reminds you that uneven pans tilt simply because one side holds more coins, which tells you nothing about the fake.
After a weighing, all coins return to the pile and you set up the next one from scratch. The balance tips, stays level, or tips the other way, and the game updates the Still Possible count. When your weighings are used up, tap the coin you suspect and press Accuse.
Why three groups beat two
The most common beginner strategy is to put half the coins on one side and half on the other. This seems efficient because it eliminates half the coins in one step. But it wastes a third of the balance's power.
A weighing has three outcomes, so it should split the coins into three groups of roughly equal size - one group on the left pan, one on the right pan, and one left off. If the pans tilt left, the fake is among the left group or the right group depending on its weight. If the pans balance, the fake is among the coins you did not weigh at all.
With nine coins and two weighings, the first step is to put three on the left, three on the right, and leave three off. If the pans balance, the fake is in the three you left off, and one more weighing of one against one from that group identifies it. If the pans tilt, the fake is among the six on the scale, but you now know which direction it is - and one more weighing of three against three narrows it to one.
Split in half and you get two groups of four or five. The balance tells you which half, but the remaining half is still too big for one weighing. Split in thirds and each outcome leaves you with a group of three, which is exactly what one weighing can solve.
The still-possible counter
The game tracks a number called Still Possible. It counts how many coins could still be the fake given every weighing result so far. This count is worked out from the logic of the weighings alone, not from knowing the answer.
Your goal is to drive this number down to one. At that point you know exactly which coin is the fake, and pressing Accuse with that coin selected is a formality. If the number is still two or three when your weighings run out, you are guessing and may get it wrong.
Watching this counter teaches you more than any strategy guide can. After each weighing, check how much the number dropped. If it barely moved, your weighing was badly designed - it did not separate enough coins. If it dropped by two thirds, your weighing was ideal.
Strategy for the easy and normal puzzles
On easy and normal, you know the fake is heavier. This simplifies things because a tilt always means the heavier side contains the fake. When the pans balance, the fake is among the coins you left off, because none of the weighed coins are the odd one out.
For nine coins in two weighings, put coins 1 through 3 on the left and 4 through 6 on the right. If the left goes down, the fake is 1, 2 or 3. If the right goes down, the fake is 4, 5 or 6. If the pans balance, the fake is 7, 8 or 9. In each case you have three suspects left.
For the second weighing, take two of the three suspects and weigh one against the other, leaving the third off. If one side goes down, that side holds the fake. If they balance, the one you left off is the fake. This two-step method solves every easy and normal board perfectly.
The hard puzzle and swapping coins between sides
The twelve-coin puzzle is a different beast. You do not know whether the fake is heavier or lighter, so a tilt does not immediately tell you which side the fake is on. It could be a heavy coin on the down side or a light coin on the up side.
The first weighing is the same: four coins left, four right, four off. If the pans balance, the fake is among the four you left off and you know the eight weighed coins are all genuine. You can use some of those known-good coins in later weighings as anchors.
If the first weighing tilts, things get interesting. You now have eight suspect coins and you need to tell them apart in two more weighings. The secret is to swap some coins between pans and bring in known-good coins from the pile. If the tilt changes direction after a swap, the coin you moved is the fake. If the tilt stays the same, the coins you did not move are the suspects.
This swapping technique is what makes the twelve-coin puzzle famous. It is hard to work out for yourself, but once you see why it works, the logic is beautiful.
Mistakes that waste a weighing
- Putting different numbers of coins on each side. The heavier side is just the one with more coins. The result tells you nothing about the fake.
- Splitting in half instead of thirds. Two groups waste the level outcome. Three groups use all three answers and cut the suspects faster.
- Weighing the same coins twice without swapping any. If the same coins stay on the same sides, the second weighing gives the same result as the first and you learn nothing new.
- Forgetting to use known-good coins. After a balanced weighing, the coins that were on the scale are proven genuine. Use them as anchors in later weighings to isolate the suspects.
- Guessing when Still Possible is above one. If the counter says three coins are still possible, you have a one-in-three chance. Use your remaining weighing instead.
The mathematics behind the sizes
Each weighing produces one of three results. With two weighings, you get three times three equals nine possible outcome sequences. With three, three times three times three equals twenty-seven. These limits set exactly how many coins the puzzle can contain.
On the easy and normal puzzles, each possible fake is a single coin with a known direction. Nine coins fit into nine outcome slots, so nine coins in two weighings is solvable but has no room for error. Eight coins is slightly easier because there is one spare slot.
On the hard puzzle, each possible fake is a coin paired with a direction - heavier or lighter. Twelve coins times two directions gives twenty-four possibilities, which fits inside twenty-seven. But three of those twenty-seven outcome sequences are unusable because they mean all coins balanced every time, which contradicts the existence of a fake. That leaves exactly twenty-four usable slots for twenty-four possibilities. The fit is perfect.
No data leaves your device
The puzzle runs entirely in your browser. The coin positions, the balance results, and your score all stay on your machine. There is no server involved and no account to manage. The page works offline once it has loaded, so you can practise the twelve-coin puzzle anywhere without needing a connection.