Imagine you’re the best mathematician in the kingdom, but you’ve been thrown into the dungeon for criticizing the emperor’s tax laws. One day, you’re brought before the emperor, who is furious because one of his 12 governors has used a fake coin to pay taxes. This counterfeit coin is now mixed with the real ones in the Treasury. The emperor offers you a chance to earn your freedom by identifying the fake coin.
In front of you are 12 coins that look exactly the same, and you have a balance scale. You know that the fake coin is either slightly lighter or heavier than the others. However, you can only use the scale three times to find it, or you’ll be sent back to the dungeon. With only the coins, the scale, and a marker, how can you figure out which coin is fake?
To solve this puzzle, you can’t weigh each coin individually. Instead, you need to weigh several coins at once by dividing them into groups. Here’s how you can do it:
Use the marker to keep track of your results. If the scale balances, mark the eight real coins with a zero. Then, weigh three of these marked coins against three unmarked ones:
Next, weigh two of the newly marked coins against each other:
If the first two groups you weigh don’t balance, mark the heavier side’s coins with a plus and the lighter side’s coins with a minus. The remaining four coins can be marked with zeros since the fake is among the ones on the scale.
Now, you need to use your last two weighings wisely. Here’s a strategy:
Now, you have three scenarios:
With your clever strategy, you identify the counterfeit coin, and the emperor is impressed. The guilty governor is sent to the dungeon, and you earn your freedom!
Imagine you are the mathematician in the story. Use a set of 12 identical objects, like bottle caps, and a simple balance scale (or a makeshift one using a ruler and a pencil). Try to identify the counterfeit coin by following the steps outlined in the article. This hands-on activity will help you understand the logic behind the solution.
Form small groups and discuss different strategies to solve the counterfeit coin riddle. Each group should present their strategy to the class, explaining why they think it would work. This will help you develop critical thinking and collaborative problem-solving skills.
Take turns playing the roles of the mathematician, the emperor, and the governors. As the mathematician, explain your reasoning and steps to identify the fake coin. This role-playing exercise will enhance your understanding and make the learning process more engaging.
Create your own version of the counterfeit coin puzzle with a different number of coins or a different twist. Share your puzzle with classmates and challenge them to solve it. This activity will encourage creativity and deepen your understanding of logical reasoning.
Write a journal entry reflecting on the experience of solving the counterfeit coin riddle. Describe the challenges you faced, the strategies you used, and what you learned from the activity. This will help you consolidate your learning and improve your problem-solving skills.
You’re the realm’s greatest mathematician, but ever since you criticized the emperor’s tax laws, you’ve been locked in the dungeon with only a marker to count the days. One day, you’re suddenly brought before the emperor, who looks even angrier than usual. One of his 12 governors has been convicted of paying his taxes with a counterfeit coin, which has already made its way into the Treasury. As the kingdom’s greatest mathematician, you’ve been granted a chance to earn your freedom by identifying the fake.
Before you are the 12 identical-looking coins and a balance scale. You know that the false coin will be very slightly lighter or heavier than the rest. However, the emperor’s not a patient man. You may only use the scale three times before you’ll be thrown back into the dungeon. You look around for anything else you can use, but there’s nothing in the room—just the coins, the scale, and your trusty marker.
How do you identify the counterfeit? Pause now to figure it out yourself!
To solve the problem, you can’t weigh each coin against all of the others, so you’ll have to weigh several coins at the same time by splitting the stack into multiple piles and narrowing down where the false coin is. Start by dividing the 12 coins into three equal piles of four. Placing two of these on the scale gives us two possible outcomes. If the two sides balance, all eight coins on the scale are real, and the fake must be among the remaining four.
To keep track of these results, use the marker to mark the eight authentic coins with a zero. Now take three of them and weigh them against three unmarked coins. If they balance, the remaining unmarked coin must be the fake. If they don’t, draw a plus on the three unmarked coins if they’re heavier, or a minus if they’re lighter.
Next, take two of the newly marked coins and weigh them against each other. If they balance, the third coin is fake. Otherwise, look at their marks. If they are plus coins, the heavier one is the impostor. If they are marked with minus, it’s the lighter one.
But what if the first two piles you weigh don’t balance? Mark the coins on the heavier side with a plus and those on the lighter side with a minus. You can also mark the remaining four coins with zeros, since you know the fake one is already somewhere on the scale.
Now you’ll need to think strategically to remove all remaining ambiguity in just two more weighings. To do this, you’ll need to reassemble the piles. One method is to replace three of the plus coins with three of the minus coins and replace those with three of the zero coins.
From here, you have three possibilities. If the previously heavier side of the scale is still heavier, that means either the remaining plus coin on that side is actually the heavier one, or the remaining minus coin on the lighter side is actually the lighter one. Choose either one of them and weigh it against one of the regular coins to see which is true.
If the previously heavier side became lighter, that means one of the three minus coins you moved is actually the lighter one. Weigh two of them against each other. If they balance, the third is counterfeit. If not, the lighter one is. Similarly, if the two sides balanced after your substitution, then one of the three plus coins you removed must be the heavier one. Weigh two of them against each other. If they balance, the third one is fake. If not, then it’s the heavier one.
The emperor nods approvingly at your finding, and the counterfeiting lord takes your place in the dungeon.
Coin – A small, flat, round piece of metal used as money, often used in math problems to teach counting and probability. – Example sentence: “If you flip a coin, what is the probability of it landing on heads?”
Scale – A tool used to measure weight or mass, often used in math to understand proportions and ratios. – Example sentence: “We used a scale to weigh the objects and compare their masses in our math class.”
Weigh – To measure the weight or mass of an object, often used in problems involving units of measurement. – Example sentence: “In the math lab, we had to weigh different fruits to find out which was the heaviest.”
Group – A set of elements or numbers that are considered together because of a shared property or characteristic. – Example sentence: “We learned how to group numbers to make addition easier by using the associative property.”
Balance – A state where different elements are equal or in the correct proportions, often used in equations and algebra. – Example sentence: “To solve the equation, you need to balance both sides by adding the same number to each.”
Fake – Something that is not genuine, often used in math problems to identify counterfeit items or errors in data. – Example sentence: “In the math puzzle, we had to find the fake coin that weighed less than the others.”
Track – To follow or monitor the progress of something, often used in math to keep a record of data or changes over time. – Example sentence: “We used a graph to track the growth of the plant over several weeks.”
Mark – A symbol or notation used to indicate a position, value, or characteristic in math problems and graphs. – Example sentence: “Place a mark on the number line to show the solution to the inequality.”
Strategy – A plan or method used to solve a problem or achieve a goal, often used in math to find solutions efficiently. – Example sentence: “Using a strategy like breaking the problem into smaller parts can help solve complex math problems.”
Puzzle – A problem designed to test ingenuity or knowledge, often used in math to develop critical thinking skills. – Example sentence: “The math puzzle required us to use logic and reasoning to find the missing number.”
