I often like to come up with games of chance. There have been times in my life when this has been profitable, but mostly I’m just interested in questions of statistics and probability.

I had considered the math behind putting together my own Deal or No Deal style game, but with greatly reduced suitcase amounts and with a cost to play. Determining a fair cost (one which I would agree to if I were the player or the banker) at first seems like a hopelessly difficult problem, but the math is actually quite simple. The player has the option of keeping the initial suitcase until the end, and the banker has the option of offering whatever small amount he wants. At any given time the chosen suitcase is worth the average of all unopened cases. The banker certainly isn’t going to offer more, and if the player accepts less it’s just because he’s hedging his bets. The cost to play should be the average of all of the cases, whatever they may be.

A couple of months ago, while discussing the Two Envelopes problem, we briefly discussed what’s known as the Monty Hall problem, after the host of Let’s Make A Deal. Thinking of that problem has inspired another gambling proposition which is this week’s Conundrum.

Let’s continue to call our two gamblers the banker and the player. The banker has three boxes and hides a $10 bill in one of the boxes and a $1 bill in each of the other two. The player pays a set amount to the banker and chooses one of the three boxes. The banker must then open one of the other two boxes and show the player a $1 bill. Then the player can decide whether to keep the contents of the box he chose or switch to the other unopened box.

What would be the fair amount for the player to pay the banker to play this game?

UPDATE: Question solved by David. See comments for the answer.

In a Venn Diagram puzzle, there are three overlapping circles, marked A, B, and C. Each circle has a different rule about who or what can go inside. The challenge is to guess the rule for each circle. You can find a more detailed explanation of Venn Diagram puzzles, along with an example, here.

Since The Tudors wasn’t on this week, I offer you this Tudor-related puzzle to hold you over until Sunday. Each of the eight people below was a member of the court of King Henry VIII.

Have you figured out one of the rules? Two? All three? Feel free to post whatever you’ve got in the comments below. Just tell us which circle you’re solving, and what the rule is.

Enjoy!

UPDATE: Circles A and B solved by Annalisa. See comments for all answers.

How do they calculate the digits of pi?

I mean, they’ve calculated the number out to billions of places. When they get a billion digits out, how do they know they’re right? Just think about how incredibly precise that is. A quark’s diameter can be described in 18 decimal places, so surely a billion places is far beyond the realm of any practical scientific purpose or authentic human experience.

From a purely mathematical standpoint, pi is defined as the ratio between a circle’s circumference and its diameter. But the only way we have of measuring such things mathematically is by using pi.

Wikipedia has this article on the subject, but I doubt you’ll be surprised when I tell you it is not helpful to me. We could ask Daniel Tammet but he’d probably just tell us what the algorithm tastes like.

Anyway, if all this math stuff is boring to you, check out this discussion thread putting a more philosophical spin on the digits of pi:

“Somewhere inside the digits of pi is a representation for all of us — the atomic coordinates of all our atoms, our genetic code, all our thoughts, all our memories. Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal.” – Cliff Pickover

This means that we exist in pi, as if in a Matrix. This means that romance is never dead. Somewhere you are running through fields of wheat, holding hands with someone you love, as the sun sets — all in the digits of pi. You are happy. You will live forever.

Silly, perhaps, but technically true. And somewhere in the digits of pi, there’s a version of the Shakespeare Teacher who understands how they calculate the digits of pi.

Most crossword puzzles are two-dimensional. They have across and down clues.

This puzzle is one-dimensional. It has forward and backward clues. And all of the answers have to do with Shakespeare.

There’s not much space here, but imagine a horizontal row of 39 squares.

There are no black squares. All answers should be run together one after another with no spaces.

Post whatever you come up with. Feel free to use the comments section of this post to collaborate. The final answer will be a string of 39 letters that can be read in both directions.

Enjoy!

Forward (Left to Right)

1 – 8: Hamlet’s home

9 – 12: Briefly betrothed to Edward IV

13 – 16: The smallest fairy?

17 – 20: “A Lover’s Complaint”

21 – 26: Speaker of “If music be the food of love, play on”

27 – 32: Does Macbeth see one before him?

33 – 39: Twelfth Night‘s Antonio once wore one (2 words)

Backward (Right to Left)

39 – 38: Scotland setting in Macbeth-like film

37 – 32: He is as constant as the northern star

31 – 29: Lear’s Fool will give you two crowns for one of these

28 – 23: The love of Venus

22 – 18: He loved Rosaline first

17 – 14: Companion to Hal and Falstaff at the Boar’s Head

13 – 11: What a piece of work it is!

10 – 5: He knows a bank where the wild thyme blows

4 – 1: Tempest setting

UPDATE: See comments for a big hint by Duane.

UPDATE II: Puzzle solved by Neel Mehta. See comments for answer.

When I first started this blog, one of my very first posts suggested that almost all of the current natives of Mongolia and China were probably descendants of Genghis Khan. I literally had no readers at the time – I hadn’t yet told anyone about the blog – and so there was nobody to challenge my sweeping statement. I didn’t even make an argument. I’d like to give my argument now, and reopen the question as a Conundrum.

The idea was based on a National Geographic article about the biological legacy of Genghis Khan:

An international group of geneticists studying Y-chromosome data have found that nearly 8 percent of the men living in the region of the former Mongol empire carry y-chromosomes that are nearly identical. That translates to 0.5 percent of the male population in the world, or roughly 16 million descendants living today.

I went on to note:

16 million descendants. And that’s only men descended from Khan directly through the male line, father to son, for the past 800 years. The total number of Khan’s descendants living today is truly incalculable.

If you figure an average of four generations per century, that’s 32 generations between Genghis and his living descendants. Each person living today should have around 2 to the power of 32, or roughly 4.3 billion, living ancestors that are contemporary with Khan. Obviously, many individuals will have to be counted more than once, so let’s take a different tack.

Let’s pick a year somewhere between 1200 and 2000, say 1500. The total population of mainland Asia in 1500 was 268,400,000. Each living person today would have approximately 2 to the power of 20, or about a million, ancestors who were around in 1500 (and that’s if we don’t count anyone with a living parent).

So how many of the 268,400,000 around in 1500 were Khan’s descendants? Well, there are 16 million men living today that share the Y chomosome. If Khan and his direct male heirs had an average of 1.68 sons over 32 generations, that would give us our 16 million. That would only account for 505 men carrying that Y chromosome in 1500. But that calcuation leaves out two factors.

First, by 1500, Khan’s seed had been pretty well spread. The factors that account for his prevalence today came mostly into play during Khan’s life and the few generations following (see the article for details). So the distribution was a lot more top-heavy than the calculation above would suggest.

Second, we’re only counting direct male-line heirs. Passing a Y chromosome down from father to son over 32 generations is only one of 4.3 billion different permutations of inheritance. Each of those 16 million Y chromosome carriers alive today probably has an average of at least one sister or daughter. That doubles the known descendants right there. Extend that back over 32 generations, then consider all of their descendants, and you get the idea. If we change “average of 1.68 sons over 32 generations” (which we know is true) to “average of 2 children of either sex over 32 generations” (which doesn’t seem like too great of a leap from there), then 16 million becomes 4.3 billion, greater than the population of mainland Asia today.

It seems to me that today’s ethnic Mongolians and Chinese would almost all have to be descended from Khan, some many times over.

Now I am no math expert. I’m a Shakespeare Teacher. It’s very possible I could be wrong about this. I’d be interested to hear what other people think, particularly people with more professional experience with statistical analysis.

And I should also point out that I pin no political, moral, or judgmental significance to being a descendant of Genghis Khan. This is simply a math, history, and logistical Conundrum. I truly hope no offense is taken (though if you read my original post and the Economist article it is based on, it actually seems to be a point of pride for both Mongolia and China to be the descendants of Khan). And my family comes from Belarus, so this would mean I’m probably a descendant of Khan as well. So don’t screw with me.

Now, with all that in mind, for this week’s Conundrum, I hereby submit my original conclusion up for public scrutiny:

So, China and Mongolia should probably stop arguing over which of their people are the true heirs of Genghis Khan. My guess is, almost all of them are.

In a Venn Diagram puzzle, there are three overlapping circles, marked A, B, and C. Each circle has a different rule about who or what can go inside. The challenge is to guess the rule for each circle. You can find a more detailed explanation of Venn Diagram puzzles, along with an example, here.

You’ve told me that there’s not enough math on this site, and I have listened. Each of the eight items below is a number.

Have you figured out one of the rules? Two? All three? Feel free to post whatever you’ve got in the comments below. Just tell us which circle you’re solving, and what the rule is.

Enjoy!

UPDATE: Circles B and C solved by Kenneth W. Davis. See comments for all answers.

We all know words that end with -ly are adverbs. Except when they aren’t.

Can you name a noun, verb, adjective, conjunction, and interjection that end in -ly?

(I wasn’t able to think of a pronoun or preposition, but if you can, post that too!)

How about a holiday, an insect, a country, and three characters from Shakespeare that end in -ly?

Can you name two former U.S. presidents with “ly” somewhere in their first names?

Post whatever you have in the comments below, and I’ll try to respond promptly!

UPDATE: Correct responses submitted by Erin (5), Kenneth W. Davis (4), and DeLisa (4). See comments for answers.

Researchers in Germany are working on a way to predict the intentions of human subjects by observing their brain activity. Damn!

For some reason it’s a little disturbing to me that something as personal and ephemeral as an intention can have a physiological manifestation that can be measured. Or maybe I’m just disturbed that they are now starting to measure it. What new “mind reading” technologies might be developed from this science? Could it become prosecutable to merely intend to commit a crime? Intent is already used as a legal concept, and attempted murder is considered a crime, even if nobody is hurt as a result. Could market researchers measure the intent of potential consumers? Will we one day have little handheld devices that can measure intent at a poker table or when our daughter’s date arrives to pick her up?

It all reminds me of a thought experiment made popular by Robert Nozick, which will be this week’s Conundrum. Before we get to it, though, it might be helpful to consider another thought experiment known as Kavka’s Toxin.

Let’s say I offer you $100,000 if you can form an intention to drink a particular toxin. This toxin will make you violently ill for about five or six hours, after which you will be perfectly fine. You’d drink it for the money, but you’re not being asked to drink it. You’re being asked to intend to drink it. After you have the money, you are free to change your mind and not drink it. The question is, can you actually form a genuine intention of doing something unpleasant that you will have no motivation to do?

Turn that one over in your mind for a few moments before moving on to this week’s Conundrum, Newcomb’s Problem.

Imagine there are two boxes, Box A and Box B. You will have the option of choosing to take both boxes, or to take Box B alone. You will keep what you find inside. Box A is transparent and contains one thousand dollars. Box B is opaque. A super-intelligent alien scientist with a proven track record of accurately predicting human behavior has analyzed you and has secretly made a prediction about which you will choose. If he believes you will choose Box B alone, he has put one million dollars inside. If he believes you will take both boxes, then he has left Box B empty. Which do you choose?

The super-intelligent scientist has run this trial with several hundred other humans, and has made a correct prediction each time. The only people who have ended up with the million are the ones who chose Box B alone. On the other hand, our alien friend has already made his prediction and left. Your choice can no longer affect the amounts that are in the boxes. You may as well take them both, right?

Fans of game theory might recognize this as a variation of the Prisoner’s Dilemma. Game theory would likely suggest that you flip a coin, so we’re going to disallow that option. You must rely on reasoning alone.

Unlike last week’s math puzzler, this one doesn’t have a right or wrong answer. It’s a thought experiment designed to test your conceptions of free will vs. determinism.

Or as Nozick put it:

To almost everyone, it is perfectly clear and obvious what should be done. The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the opposing half is just being silly.

It will be interesting to hear how people answer this.

Will you take both boxes, or Box B alone?

Feel free to answer the question, or continue the discussion of any of the topics covered above.

I overheard this once on a train and was never able to figure it out. Maybe someone here can help me.

Imagine I have two envelopes and I tell you truthfully that both contain money and that one envelope contains twice as much money as the other. I offer you your choice of envelope and you choose one of them without opening it.

Now I ask you if you would like to switch envelopes. You chose yours randomly, so it’s a 50/50 chance whether the other envelope contains half as much or twice as much. So, if the amount you now have is x, there’s a 50 percent chance that switching would get you 2x and a 50 percent chance it will get you x/2. You have twice as much to gain as you have to lose, regardless of how much is in your envelope, so it makes sense mathematically to switch envelopes.

But of course, this is ridiculous, since you have no new information about the two envelopes than you had before. Once you’ve made that switch, by the same logic, you should want to switch again. This much seems obvious. So where’s the flaw in the math above?

By the way, I consulted our good friend Wikipedia before posting this, and it was little help. It just mumbled something about Bayesian Decision Theory and said the problem would be easy if I were a mathematician. It then went on to pose a harder problem in which you can look inside one of the envelopes, and an even harder problem that was way over my head at 5:30 am. Thanks, Wikipedia.

In a Venn Diagram puzzle, there are three overlapping circles, marked A, B, and C. Each circle has a different rule about who or what can go inside. The challenge is to guess the rule for each circle. You can find a more detailed explanation of Venn Diagram puzzles, along with an example, here.

This week, we’re going to try a twist on the increasingly-classic Venn Diagram puzzle. The entry for the center section has been removed. Can you still guess the rules without the most important clue?

Have you figured out one of the rules? Two? All three? Feel free to post whatever you’ve got in the comments below. Just tell us which circle you’re solving, and what the rule is.

And, for an extra challenge, can you figure out what should replace the question mark in the center?

Enjoy!

UPDATE: Circles A, B, and C solved by Annalisa. The center clue has been solved by Kenneth W. Davis. Alternate answer for the center suggested by Brian. See comments for answers.