The Cheating Royal Riddle

You’re the chief advisor to an eccentric king who needs to declare his successor. He wants his heir to be good at arithmetic, lucky, and above all else, honest. So he’s devised a competition to test his children, and ordered you to choose the winner. The future of the kingdom is in your hands. Can you find the worthiest successor? Dan Katz shows how.

Transcript:

You’re the chief advisor to an eccentric king who needs to declare his successor. He wants his heir to be good at arithmetic, lucky, and above all else, honest. So he’s devised a competition to test his children, 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 one has 3, 8, 8, 13, 13, and 18. The dice are fair, so each side is equally likely to come up.

Each contestant will be sent into a Royal Rolling Room, where they’ll 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. That 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 mis-added 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?

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 1 over 6 to the 40th power, or 1 in about 13 nonillion— that’s 13 followed by 30 zeros.

To put that in perspective, there are about 7.5 billion people in the world, and 7.5 billion squared is a lot less than 13 nonillion. Rolling the highest number all 40 times is much less likely than if you picked a completely random person on Earth, and it turned out to be actor Paul Rudd… and then you randomly picked again, and got Paul Rudd 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. That’s because every red number is 2 more than a multiple of 5, and every blue number is 3 more than a multiple of 5. This means that when you add them together, you’ll always get an exact multiple of 5.

And when you add rolls that are multiples of 5, the result will also be a multiple of 5. These sorts of relationships between integers are studied in a branch of math called number theory.

Here number theory shows us that 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! At least if you agree that the best way to organize your government is a roll of the dice…

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