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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.

  • Let's start with the highest.

  • 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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B1 score cassandra contestant highest heir draco

Can you solve the cheating royal riddle? - Dan Katz

  • 7 1
    林宜悉 posted on 2020/10/24
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