In a bustling kingdom, a wealthy merchant has been caught in some shady business deals. Most of his wealth is tied up in a collection of 30 beautiful Burmese rubies. The people in the town square are demanding that these rubies be taken away to pay back those he cheated. However, the merchant and his friends in the royal court argue that some of his wealth was earned fairly and through good service to the king.
The king thinks for a moment and comes up with a clever plan. Since it’s impossible to tell which rubies were obtained dishonestly, he decides to settle the matter with a game of strategy between the merchant and the king’s smartest advisor – you! You learn the rules of the game beforehand. The merchant will secretly divide his rubies into three boxes, which will then be placed in front of you. You will have three cards and must write a number between 1 and 30 on each card. Then, you place one card in front of each box. The boxes will be opened, and for each box, you will get the number of rubies written on the card, but only if the box contains at least that many rubies. If your number is higher than the rubies in the box, the merchant keeps all the rubies in that box.
The king sets two rules for how the merchant can distribute his rubies. First, each box must have at least two rubies. Second, one of the boxes must have exactly six more rubies than another box, but you won’t know which boxes those are. After a few minutes, the merchant hides the rubies, and the boxes are placed in front of you.
What numbers should you choose to ensure you get the most rubies possible from the merchant? If you want to figure it out yourself, pause here and think about it.
You don’t want to be too greedy and overshoot. However, there is a way to make sure you get more than half of the merchant’s rubies. This situation is like a tricky game of chess where you can’t see your opponent’s pieces. To figure out the minimum number of rubies you’re guaranteed to win, you need to think about the worst-case scenario, as if the merchant already knew your move and arranged the rubies to give you the least amount possible.
Since you have no way of knowing which boxes have more or fewer rubies, you should choose the same number for each box. Imagine you write three 9’s. The merchant might have divided the rubies as 8, 14, and 8. In this case, you’d get 9 rubies from the middle box and none from the others. But you can be sure that at least two boxes have at least 8 rubies.
Here’s why: Suppose two boxes have 7 or fewer rubies. These couldn’t be the two that differ by 6 because every box must have at least 2 rubies. If that were true, the third box could have at most 13 rubies (7 plus 6). Adding up all three boxes, the total would be 27, which is less than 30. So, this scenario isn’t possible.
Now you know, through a logical contradiction, that two of the boxes have 8 or more rubies. If you ask for 8 from all three boxes, you’ll get at least 16 rubies. This is the best you can guarantee, as shown by the 8, 14, 8 scenario. You’ve managed to recover more than half of the merchant’s fortune for the people. Although the merchant keeps some of his rubies, his wealth has certainly taken a hit.
Imagine you are the king’s advisor. Create a short skit with your classmates where you act out the scenario of the stolen rubies. Discuss the strategy you would use to ensure you win the most rubies. This will help you understand the importance of strategic thinking and decision-making.
Using three boxes and 30 small objects (like marbles or coins), simulate the game described in the article. Divide the objects into the boxes following the king’s rules. Then, choose numbers for your cards and see how many objects you can win. This hands-on activity will reinforce the concept of strategic allocation.
Work in pairs to solve a series of math puzzles that involve similar logical reasoning and strategy as the stolen rubies riddle. This will help you practice thinking critically and applying mathematical concepts to real-world scenarios.
Write a short story from the perspective of the merchant, the king, or the advisor. Describe the events leading up to the riddle and how you felt during the game. This activity will allow you to explore different viewpoints and enhance your creative writing skills.
Engage in a class discussion or debate about the fairness of the king’s decision and the strategy used to solve the riddle. Consider ethical implications and alternative solutions. This will develop your critical thinking and public speaking skills.
Here’s a sanitized version of the transcript:
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One of the kingdom’s most prosperous merchants has been exposed for corrupt dealings. Nearly all of his riches are invested in a collection of 30 exquisite Burmese rubies, and the crowd in the square is clamoring for their confiscation to reimburse his victims. However, the merchant and his allies at court have made a convincing case that at least some of his wealth was obtained legitimately and through good service to the crown.
The king ponders for a moment and announces his judgment. Because there’s no way to know which portion of the rubies were acquired through dishonest means, the fine will be determined through a game of wits between the merchant and the king’s most clever advisor – you. You are informed of the rules in advance. The merchant will be allowed to discreetly divide his rubies among three boxes, which will then be placed in front of you. You will be given three cards and must write a number between 1 and 30 on each before placing a card in front of each box. The boxes will then be opened. For each box, you will receive exactly as many rubies as the number written on the corresponding card, if the box contains that many. However, if your number exceeds the number of rubies actually present, the merchant retains the entire box.
The king imposes two constraints on how the merchant distributes his rubies. Each box must contain at least two rubies, and one of the boxes must contain exactly six more rubies than another – but you won’t know which boxes those are. After a few minutes of deliberation, the merchant hides the gems, and the boxes are placed in front of you.
Which numbers should you choose to guarantee the largest possible fine for the merchant and the greatest compensation for his victims?
Pause the video now if you want to figure it out for yourself.
You don’t want to overshoot by being too greedy. However, there is a way you can guarantee to get more than half of the merchant’s stash. The situation resembles an adversarial game like chess – only here you can’t see the opponent’s position. To determine the minimum number of rubies you’re guaranteed to win, you need to consider the worst-case scenario, as if the merchant already knew your move and could arrange the rubies to minimize your winnings.
Since you have no way of knowing which boxes will have more or fewer rubies, you should choose the same number for each. Suppose you write three 9’s. The merchant might have allocated the rubies as 8, 14, and 8. In that case, you’d receive 9 from the middle box and none from the others. On the other hand, you can be sure that at least two boxes contain a minimum of 8 rubies.
Here’s why: Let’s assume the opposite, that two boxes have 7 or fewer. Those could not be the two that differ by 6, because every box must have at least 2 rubies. In that case, the third box would have at most 13 rubies—that’s 7 plus 6. Adding up all three boxes, the maximum total would be 27. Since that’s less than 30, this scenario isn’t possible.
You now know, through a proof by contradiction, that two of the boxes have 8 or more rubies. If you ask for 8 from all three boxes, you’ll receive at least 16—and that’s the best you can guarantee, as you can see by considering the 8, 14, 8 scenario. You’ve recovered more than half the merchant’s fortune as restitution for the public. Although he has managed to retain some of his rubies, his fortune has certainly lost some of its luster.
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This version maintains the core content while removing any potentially sensitive or inappropriate language.
Rubies – Precious stones often used in mathematical problems to represent variables or unknown quantities. – In the math puzzle, each ruby represented a different number that we needed to find.
Merchant – A person who buys and sells goods, often used in math problems to illustrate concepts of profit and loss. – The merchant calculated his profit by subtracting the cost of the goods from the selling price.
Boxes – Containers used in math problems to group items or represent sets in probability and statistics. – We used boxes to organize the different outcomes of our probability experiment.
Numbers – Mathematical objects used to count, measure, and label. – We learned how to add and subtract negative numbers in our math class today.
Strategy – A plan of action designed to achieve a specific goal, often used in problem-solving. – Our strategy for solving the equation was to isolate the variable on one side.
King – A figure often used in logic puzzles to represent a decision-maker or leader. – The king had to decide the best way to distribute resources among his subjects in the math problem.
Wealth – An abundance of valuable resources or material possessions, often used in math problems to discuss distribution and allocation. – The problem asked us to calculate the total wealth of the kingdom by adding up all the gold coins.
Logical – Characterized by clear, sound reasoning, often used in math to describe a sequence of steps that lead to a conclusion. – We used logical reasoning to determine the next number in the sequence.
Scenario – A hypothetical situation used to illustrate a concept or problem in math. – The scenario described in the math problem involved calculating the area of a garden.
Critical – Involving careful judgment or evaluation, often used in math to describe the importance of certain steps in problem-solving. – It was critical to check our calculations to ensure the solution was correct.
