Imagine you’re the chief advisor to a king who needs to choose the next ruler of the kingdom. The king wants his heir to be good at math, lucky, and most importantly, honest. To find out which of his children has these qualities, he creates a competition and asks you to pick the winner.
Each child will get two special six-sided dice. The red die has the numbers 2, 7, 7, 12, 12, and 17. The blue die has 3, 8, 8, 13, 13, and 18. These dice are fair, meaning each number is equally likely to appear when rolled. Each contestant will roll both dice 20 times. They start with a score of zero and add the sum of the two dice to their score each time they roll. After 20 rolls, they report their final score. The rooms are secure, so no one can see the rolls, which means a contestant could make a mistake or even cheat by reporting a false score.
Your job is to disqualify anyone if you’re at least 90% sure they miscalculated or cheated. The contestant with the highest valid score will become the new heir to the throne.
After the competition, the children report their scores:
Now, it’s up to you to decide who is the worthiest successor. Let’s analyze their scores.
Bertram’s Score: Bertram claims a score of 840. The highest possible roll with both dice is 17 + 18 = 35. If you roll 35 every time for 20 rolls, the maximum score is 20 x 35 = 700. Bertram’s score of 840 is impossible, so he must be disqualified.
Cassandra’s Score: Cassandra reports a score of 700. While this is the highest possible score, it’s extremely unlikely to roll the highest numbers every single time. The chance of this happening is so low that it’s almost impossible, so you can be 90% sure she didn’t achieve this score honestly. Cassandra should also be disqualified.
Draco’s Score: Draco’s score is 423. However, when you add any number from the red die to any number from the blue die, the result always ends in 0 or 5. Therefore, the total score must be a multiple of 5. Since 423 is not a multiple of 5, Draco’s score is impossible, and he should be disqualified.
This leaves Alexa, who scored 385. Her score is a multiple of 5 and within the range of possible scores. In fact, the most common score is around 400, so she was just a bit unlucky. With everyone else disqualified, Alexa is the last heir standing. All hail Queen Alexa, the worthiest successor!
Roll a pair of regular six-sided dice 20 times and record the results. Calculate the average score and compare it to the theoretical average. Discuss how probability affects the outcomes and relate it to the dice used in the riddle.
Using the numbers on the special dice from the article, simulate 20 rolls and calculate the total score. Compare your results with the scores reported by the contestants. Discuss why some scores were deemed impossible or unlikely.
Engage in a class debate about the importance of honesty in competitions. Discuss how probability can be used to detect dishonesty, using examples from the riddle to support your arguments.
Design a riddle involving a competition that tests honesty, luck, and skill. Use mathematical concepts to ensure the riddle has a logical solution. Share your riddle with classmates and solve each other’s challenges.
Write a short story about a kingdom where math is used to solve problems and make decisions. Include elements of probability and honesty, inspired by the riddle. Share your story with the class and discuss the role of math in decision-making.
Here’s a sanitized version of the transcript, with sensitive or potentially inappropriate content removed or modified for clarity:
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You’re the chief advisor to a king who needs to declare his successor. He wants his heir to be good at arithmetic, lucky, and above all else, honest. To test his children, he has devised a competition and ordered you to choose the winner. Each potential heir will be given the same two six-sided dice. The red die has the numbers 2, 7, 7, 12, 12, and 17. The blue die has 3, 8, 8, 13, 13, and 18. The dice are fair, so each side is equally likely to come up. Each contestant will roll both dice 20 times. A contestant’s score starts at zero, and each turn, they should add the total of the two numbers rolled to their score. After 20 turns, they should report their final score. The rooms are secure, and no one observes the rolls. This means a contestant could add incorrectly or, worse, be dishonest and make up a score they didn’t achieve. This is where you come in. The king has instructed you that if you’re at least 90% sure a contestant misadded or cheated, you should disqualify them. The highest-scoring player who remains will be the new heir to the throne.
After you explain the rules, the children run to their rooms. When they return, Alexa announces her score is 385. Bertram says 840. Cassandra reports 700, and Draco declares 423. The future of the kingdom is in your hands. Whom do you proclaim to be the worthiest successor? Pause now to figure it out for yourself.
Upon inspection, most of these scores are concerning. Let’s start with the highest. Bertram scored 840. That’s impressive, but is it even possible? The highest numbers on the two dice are 17 and 18. 17 plus 18 is 35, so in 20 rolls, the greatest possible total is 20 times 35, or 700. Even if Bertram rolled all the highest numbers, he couldn’t have scored 840. So he’s disqualified.
Cassandra, the next-highest roller, reported 700. That’s theoretically possible, but how hard is it to be that lucky? In order to get 700, Cassandra would have to roll the highest number out of six on 40 separate occasions. The probability of this is extremely low. To put that in perspective, rolling the highest number all 40 times is much less likely than picking a completely random person on Earth and getting the same person again. You can’t be 100% sure that Cassandra’s score didn’t happen by chance, but you can certainly be 90% sure, so she should be disqualified.
Next up is Draco, with 423. This score isn’t high enough to be suspicious, but it’s impossible for a different reason. Pick a number from each die and add them up. No matter which combination you choose, the result ends in a 0 or a 5. This means that when you add rolls that are multiples of 5, the result will also be a multiple of 5. Draco’s score, which is not a multiple of 5, cannot be achieved. So he should be disqualified as well.
This leaves Alexa, whose score is a multiple of 5 and is in the achievable range. In fact, the most likely score is 400, so she was a little bit unlucky. But with everyone else disqualified, she’s the last heir standing. All hail Queen Alexa, the worthiest successor!
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This version maintains the essence of the original content while ensuring clarity and appropriateness.
Math – The study of numbers, quantities, shapes, and patterns and the relationships between them. – In math class, we learned how to calculate the area of a triangle.
Score – The number of points achieved in a game or test. – After solving all the problems correctly, Maria received a perfect score on her math quiz.
Dice – Small cubes with numbers or symbols on each side, used in games to generate random numbers. – We used dice to determine the starting player in our math board game.
Contest – A competition where individuals or teams compete to demonstrate their skills or knowledge. – The math contest challenged students to solve complex problems within a limited time.
Analyze – To examine something in detail to understand it better or to draw conclusions. – We need to analyze the data from our experiment to find patterns and make predictions.
Honest – Being truthful and fair, especially in sharing information or results. – It is important to be honest about your calculations to ensure the accuracy of your math work.
Possible – Capable of being done or achieved. – It is possible to solve the equation using different methods, but each should give the same result.
Multiple – A number that can be divided by another number without leaving a remainder. – Twelve is a multiple of three because it can be divided by three evenly.
Disqualify – To declare someone ineligible for a competition due to breaking rules or not meeting requirements. – The team was disqualified from the math contest for using a calculator when it was not allowed.
Successor – A number that follows another in a sequence or progression. – In the sequence of even numbers, 8 is the successor of 6.
