### Brain Teasers

# The Tyrant King and the Marbles

Probability
Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

You are currently being held prisoner by King Fischer, the tyrannical king of the faraway land of Fakefictio, for crimes involving a banana peel, a mechanical pencil, and the king's favorite Dragonforce album. Of course, as you know, you were not responsible for said crimes; these crimes never even took place. As a matter of fact, the king doesn't even like Dragonforce and is more of a fan of Journey and Kansas, but he is so tyrannical that he's arrested you anyway. The punishment for the crimes is a very graphic death that will probably involve much bleeding, and the date of your execution is today. The king could just kill you, but that wouldn't make for a very interesting puzzle at all, and he is allergic to much bleeding, so he decides to offer you a game to give you a chance to escape with your life. Win, and you will go free. Lose, and you will not only not go free, but you'll probably die, too.

The king presents to you four urns and a bowl filled with red and blue marbles. As two guards pin you to the dungeon wall by the arms, a third guard will place 5 marbles in each urn. You get to choose the distribution of red and blue marbles in each urn. For example, you could have the guard place 3 red and 2 blue marbles in one urn, and 1 red and 4 blue in another urn. You can also place 5 red or 5 blue in a single urn. King Fischer's watchful eye will observe the process, so he will know your distribution.

After all of the urns have 5 marbles each, the guard will shake the urns, and the king will predict whether you will draw all red marbles or at least one blue marble. The guards will then cover your eyes with a blindfold and place your hand in each urn; you will be required to draw exactly one marble from each urn. If the king's prediction proves correct, you will be executed. However, if the prediction is incorrect (that is, you draw a blue marble when the king predicts you'll draw all red marbles, or vice versa), then you will go free.

Note that if you put 5 red marbles in every urn, then the king will know for certain that you'll draw all red marbles, and use this knowledge to your disadvantage to get you killed. Contrariwise, if you put 5 blue marbles in one urn, then you'll be forced to draw at least one blue marble, and the king will predict this, too. In order to have hope of survival, you'll need to distribute the marbles in a fashion that makes the outcome less certain.

How should you distribute the marbles to maximize your chances of survival, and what are your chances of survival under this strategy?

The king presents to you four urns and a bowl filled with red and blue marbles. As two guards pin you to the dungeon wall by the arms, a third guard will place 5 marbles in each urn. You get to choose the distribution of red and blue marbles in each urn. For example, you could have the guard place 3 red and 2 blue marbles in one urn, and 1 red and 4 blue in another urn. You can also place 5 red or 5 blue in a single urn. King Fischer's watchful eye will observe the process, so he will know your distribution.

After all of the urns have 5 marbles each, the guard will shake the urns, and the king will predict whether you will draw all red marbles or at least one blue marble. The guards will then cover your eyes with a blindfold and place your hand in each urn; you will be required to draw exactly one marble from each urn. If the king's prediction proves correct, you will be executed. However, if the prediction is incorrect (that is, you draw a blue marble when the king predicts you'll draw all red marbles, or vice versa), then you will go free.

Note that if you put 5 red marbles in every urn, then the king will know for certain that you'll draw all red marbles, and use this knowledge to your disadvantage to get you killed. Contrariwise, if you put 5 blue marbles in one urn, then you'll be forced to draw at least one blue marble, and the king will predict this, too. In order to have hope of survival, you'll need to distribute the marbles in a fashion that makes the outcome less certain.

How should you distribute the marbles to maximize your chances of survival, and what are your chances of survival under this strategy?

### Answer

Suppose there are A red marbles in the first urn, B red marbles in the second urn, C red marbles in the third urn, and D red marbles in the fourth urn. Then your probability of drawing all red marbles is A*B*C*D/625, and your probability of getting at least one blue marble is (625-A*B*C*D)/625. The king will choose whichever option is more likely. To minimize the probability of a correct prediction, you want to get these values as close as possible to each other. Equivalently, you want A*B*C*D/625 to be as close as possible to 1/2, or A*B*C*D to be as close as possible to 312.5. Since A, B, C, and D can only take on the values of 0, 1, 2, 3, 4, and 5, the possibilities are limited. The closest possible value to 312.5 is 320=5*4*4*4; none of the closer values from 306 through 319 can be achieved because each one has a prime factor greater than 5. For example, 306=2*3*3*17, 307 is a prime, 308 = 2*2*7*11, and so on. Therefore, you should place 4 red marbles and 1 blue marble in three urns, and 5 red marbles in the remaining urn. King Fischer will predict that you will get all red marbles, and you'll have a 61/125=.488 probability of drawing a blue marble and surviving.In the end, the king takes some medicine to control his allergic reactions to much bleeding and executes you anyway. So much for all that mathematics and probability you just went through. What an anticlimax!

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## Comments

I can't understand why this isn't rated higher ("fun"). Great teaser, not overly difficult, but certainly challenging enough to be fun. Plus, written in a very entertaining way. These used to be more common on here and it would be great to bring back more like this. Thanks.

As for the answer, the key (as was said) is realizing the symmetrical nature of the problem - and that you have to get the probability of either outcome as close as possible to .5. And it's much easier to calculate the p of picking one red. Five urns; each with the probability choices limited to multiples of .2. As the answer suggests, .8x.8x.8x1x1 = .512 (the king will give you the .48 is best. A close second would be .8x.6x1x1x1 = .48 - the case with one urn holding 4, and one holding 3 reds; the rest hold 5. Nothing else gets even close.

Thanks again - a fun diversion.

As for the answer, the key (as was said) is realizing the symmetrical nature of the problem - and that you have to get the probability of either outcome as close as possible to .5. And it's much easier to calculate the p of picking one red. Five urns; each with the probability choices limited to multiples of .2. As the answer suggests, .8x.8x.8x1x1 = .512 (the king will give you the .48 is best. A close second would be .8x.6x1x1x1 = .48 - the case with one urn holding 4, and one holding 3 reds; the rest hold 5. Nothing else gets even close.

Thanks again - a fun diversion.

Minor correction to my previous - I had thought there were 5 urns, not 4. But the best two cases are still the same: .488 chance of survival if you have 5R, 4R, 4R, 4R. .480 chance of survival if you have 5R, 5R, 4R, 3R. (So I had to remove one urn of 5R from each of my previous, as there are only 4 urns.) No other combination is close to the desired "coin flip" situation.

Thanks for the kind words, tpg76.

I tried to write this teaser in an entertaining way, while also making the goal as clear as possible (make the outcome as uncertain as possible, as opposed to most teasers where you are trying to maximize the probability of a desired outcome), but maybe it's still too arcane for some readers. Hey, you can't please everyone; thankfully "everyone" is not my target audience.

I tried to write this teaser in an entertaining way, while also making the goal as clear as possible (make the outcome as uncertain as possible, as opposed to most teasers where you are trying to maximize the probability of a desired outcome), but maybe it's still too arcane for some readers. Hey, you can't please everyone; thankfully "everyone" is not my target audience.

You can't go by the Fun rating, obviously. It would be nice, however, if people would just skip the teasers they can't solve without rating them. There are lots of people out there who welcome a challenge.

While it was a pretty easy puzzle, it was a nice depature from the "calculate the probability" or "maximize the probability" problems to instead "neutralize the probability". More of these (but harder) would be good.

I loved the story and the explanation. Thanks for the entertainment!

Thank you! I strove to make the story as amusing as possible, and the explanation as understandable as possible.

Interestingly, the answer is the same for any number of urns >2, Put one red in each of three urns - the probability of survival (assuming a smart and observant "tyrant king") is always exactly .488. A great average in baseball; not-so-great in the executioner's cross hairs.

Only people who love or have a love of Math can appreciate the comic or fun of this teaser. Math solutions and problems can be fun to those who know and understand the formula and its results. I myself can appreciate a good math problem and its results when the situational prose is devised for fun, but this teaser was boring, the storyline was good, yet, predicatable. Nevertheless, the chance of a blue marble selected to save the guy's life was null unless one Urn contained all Blue marbles. Good thought provoking teaser, but still Boring.

TLR

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