One of the kingdom’s most prosperous merchants
has been exposed for his corrupt dealings.
Nearly all of his riches are invested in a collection of 30
exquisite Burmese rubies,
and the crowd in the square is clamoring for their confiscation
to reimburse his victims.
But the scoundrel and his allies at court
have made a convincing case that at least
some of his wealth was obtained legitimately,
and through good service to the crown.
The king ponders for a minute and announces his judgment.
Because there’s no way to know which portion of the rubies
were bought with ill-gotten wealth,
the fine will be determined through a game of wits
between the merchant and the king’s most clever advisor – you.
You’re both told the rules in advance.
The merchant will be allowed to discreetly divide his rubies
among three boxes, which will then be placed in front of you.
You will be given three cards,
and must write a number between 1 and 30 on each,
before putting a card infront of each of the boxes.
The boxes will then all be opened.
For each box, you will receive exactly as many rubies as the number
written on the corresponding card, if the box has at least that many,
but if your number is greater than the number of rubies actually there,
the scoundrel gets to keep the entire box.
The king puts just two constraints on how the scoundrel distributes his rubies.
Each box must contain at least two rubies
and one of the boxes must contain exactly six more rubies than another,
but you won’t know which boxes those are.
After a few minutes of deliberation,
the merchant hides the gems,
and the boxes are brought in front of you.
Which numbers should you choose
in order to guarantee the largest possible fine for the scoundrel
and the greatest compensationfor his victims?
Pause the video now if you want to figure it out for yourself.
3 2 1
Answer in 3 Answer in 2 Answer in 1
You don’t want to overshoot by being too greedy.
But there is a way you can guarantee
to get more than half of the scoundrel’s stash.
The situation resembles anadversarial game like chess –
only here you can’t see theopponent’s position.
To figure out the minimum number
of rubies you’re guaranteed to win,
you need to look for the worst casescenario,
as if the merchant already knew your move
and could arrange the rubies to minimize your winnings.
Because you have no way of knowing which boxes will have more or fewer rubies,
you should pick the same number for each.
Suppose you write three 9’s.
商人可能将宝石分成8 14 8
The scoundrel might have allocated the rubies as 8, 14 and 8.
In that case, you’d receive 9
from the middle box and no others.
On the other hand, you can be sure that at least two boxes
have a minimum of 8 rubies.
Here’s why. We’ll start by assuming the opposite,
that two boxes have 7 or fewer.
Those could not be the two thatdiffer by 6,
because every box must have at least 2 rubies.
In that case, the third box would have
at most 13 rubies—that’s 7 plus 6.
Add up all three of those boxes,
and the most that could equal is 27.
Since that’s less than 30, this scenario isn’t possible.
You now know, by what’s calleda proof by contradiction,
that two of the boxes have8 or more rubies.
If you ask for 8 from all three boxes
you’ll receive at least 16—
and that’s the best you can guarantee,
再次考虑 8 14 8这种情况就会明白
as you can see by thinking again about the 8, 14, 8 scenario.
You’ve recovered more than half thescoundrel’s fortune
as restitution for the public.
And though he’s managed to hold on to some of his rubies,
his fortune has definitelylost some of its shine.