Can you solve the cheating royal riddle Dan Katz
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 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…