The Royal Riddle: Can You Spot the Cheating Heir?

Imagine you're the chief advisor to a quirky king tasked with choosing his successor. The king values arithmetic skills, luck, and, most importantly, honesty. To find the worthiest heir, he's devised a competition, and your judgment will determine the future of the kingdom. Can you identify the successor?

The Royal Competition

The competition involves each potential heir receiving two six-sided dice. The red die contains the numbers 2, 7, 7, 12, 12, and 17, while the blue die has 3, 8, 8, 13, 13, and 18. Each die is fair, meaning every side has an equal chance of landing face up.

The contestants enter a secure Royal Rolling Room and roll both dice 20 times. Their score starts at zero, and with each roll, they must add the total of the two numbers to their score. After 20 turns, they report their final score. No one observes the rolls, leaving room for miscalculations or outright dishonesty.

The king instructs you to disqualify any contestant if you're at least 90% sure they've miscalculated or cheated. The remaining contestant with the highest score will be the new heir to the throne.

Unmasking the Cheaters

After explaining the rules, the children eagerly participate. When they return, their scores are as follows:

• Alexa: 385
• Bertram: 840
• Cassandra: 700
• Draco: 423

Let's analyze each score to determine who's being truthful and who's trying to deceive the king.

Bertram: The Impossible Score

Bertram's score of 840 immediately raises suspicion. Is it even possible to achieve such a high total? The highest numbers on the dice are 17 and 18, which add up to 35. Over 20 rolls, the maximum possible score would be 20 * 35 = 700. Bertram's score exceeds this limit, making it impossible. Therefore, Bertram is disqualified.

Cassandra: The Unlikely Outcome

Cassandra's score of 700 is theoretically possible, but incredibly improbable. To achieve this, she would have to roll the highest number on both dice (17 and 18) on all 40 separate occasions. The probability of this happening is 1 in 6^40, or approximately 1 in 13 nonillion (13 followed by 30 zeros).

To put this into perspective, consider that there are about 7.5 billion people on Earth. The odds of Cassandra rolling the highest number every time are far less likely than randomly picking actor Paul Rudd from the entire world's population, and then randomly picking Paul Rudd again! While it's not impossible, you can be more than 90% sure that Cassandra's score didn't happen by chance, leading to her disqualification.

Draco: The Number Theory Trick

Draco's score of 423 seems plausible at first glance, but a closer look reveals a hidden impossibility. Notice that every number on the red die is 2 more than a multiple of 5 (e.g., 2, 7, 12, 17), and every number on the blue die is 3 more than a multiple of 5 (e.g., 3, 8, 13, 18). When you add any number from the red die to any number from the blue die, the result will always be a multiple of 5.

Since each roll results in a multiple of 5, the total score after 20 rolls must also be a multiple of 5. Draco's score of 423 is not a multiple of 5, making it impossible to achieve. This is a concept explored in number theory, a branch of mathematics that studies the relationships between integers. Thus, Draco is also disqualified.

Alexa: The Worthy Successor

This leaves Alexa, whose score of 385 is both a multiple of 5 and within the achievable range. While the most likely score is 400, making her slightly unlucky, she is the only remaining heir after disqualifying the others.

All Hail Queen Alexa!

Therefore, you proclaim Alexa as the worthiest successor! While choosing a leader based on a dice game might seem unconventional, in this case, it exposed the dishonesty of the other potential heirs, leaving Alexa as the rightful ruler.