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• 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?

• Pause now to figure it out for yourself.

• Upon inspection, most of these scores are concerning.

• Bertram scored 840.

• That's impressivebut 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 possiblebut 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 nonillionthat'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...

You're the chief advisor to an eccentric king

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# Can you solve the cheating royal riddle? - Dan Katz

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林宜悉 posted on 2020/10/24
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