What happens when you ask people to pick the lowest unique prime number.(robinsloan.com)
robinsloan.com
What happens when you ask people to pick the lowest unique prime number.
http://www.robinsloan.com/note/penumbra-primes/
16 comments
This reminds me of the game "clomp" (which I may or may not have invented+) but I used to play with my brother on car rides. Here are the rules:
1) Pick any integer. Your partner picks an integer without knowing yours. The first person to think of a number says "clomp". They are locked in. The other person can say their number any time after "clomp". Honor system. No cheating.
2) If your number is less then your opponent, but not one less, then add your number to your score.
3) If your number is one more than your opponent, then add your number and your opponents number together and add it to your score.
4) Otherwise no score.
5) First to 21 or more wins.
+I may have stolen this from Martin Gardner, I honestly don't remember if I came up with it or I read it. My brother came up with the name "clomp".
1) Pick any integer. Your partner picks an integer without knowing yours. The first person to think of a number says "clomp". They are locked in. The other person can say their number any time after "clomp". Honor system. No cheating.
2) If your number is less then your opponent, but not one less, then add your number to your score.
3) If your number is one more than your opponent, then add your number and your opponents number together and add it to your score.
4) Otherwise no score.
5) First to 21 or more wins.
+I may have stolen this from Martin Gardner, I honestly don't remember if I came up with it or I read it. My brother came up with the name "clomp".
Alternatively, here is a great drinking game called Fives (based on Spoof, but no coins required http://en.wikipedia.org/wiki/Spoof_(game)). Prob best for small groups.
You sit in a group, elbows on the table, fists in the air.
You take in in turns to go, clockwise.
When it is your go, you say a number that is a multiple of 5, from 0 to number of people x 10
As you say the number, everyone can either keep one or both of their fists closed, or spread their five fingers out (i.e each hand either counts as zero or five).
If the number you say is equal to the number of fingers shown, next time you use one hand not two. And then the next go, if you get it right again, you are out. Play continues until there is one person left, who then drinks the forfeit.
You sit in a group, elbows on the table, fists in the air.
You take in in turns to go, clockwise.
When it is your go, you say a number that is a multiple of 5, from 0 to number of people x 10
As you say the number, everyone can either keep one or both of their fists closed, or spread their five fingers out (i.e each hand either counts as zero or five).
If the number you say is equal to the number of fingers shown, next time you use one hand not two. And then the next go, if you get it right again, you are out. Play continues until there is one person left, who then drinks the forfeit.
I like it! Wonder if you could deal with the potential cheating by using tokens that people hide in their hand and reveal on a count?
It reminds me of a drinking game played in China (more or less the same as Liar's Dice from Red Dead Redemption) involving dice and cups. There's a street of bars ("Bar Street" I think it might be called) in Lijiang where almost every table is full of people buying beers half-a-dozen at a time and playing this game loudly.
It reminds me of a drinking game played in China (more or less the same as Liar's Dice from Red Dead Redemption) involving dice and cups. There's a street of bars ("Bar Street" I think it might be called) in Lijiang where almost every table is full of people buying beers half-a-dozen at a time and playing this game loudly.
A similar game: both players pick positive integers. The one who picked the larger one wins but only if it was larger by no more than 50.
so...is there an incentive to deviate from "clomper picks 1, opponent picks 0, first clomper wins?"
The way I read the rules, this would be the strategy:
If you pick 1, and opponent picks 1, then no score.
But if opponent picks 1, and you pick 2, then you will win 3 points.
If your opponent thinks you are going to pick 2, he can pick 3 for 5 points. But if you go 1 instead, then you win 1.
If you pick 1, and opponent picks 1, then no score.
But if opponent picks 1, and you pick 2, then you will win 3 points.
If your opponent thinks you are going to pick 2, he can pick 3 for 5 points. But if you go 1 instead, then you win 1.
Yes, this is how it's played. It really is a fun little game, especially if you and your opponent are the type of people who over-think things. Try it sometime!
I think I will - it sounds fun!
This is not quite asking people to pick the lowest unique prime number, because it was clear in advance that the prize would go to the individuals who picked the five lowest unique prime numbers, rather than the single individual who picked the lowest. This means that there is an incentive to err on the side of picking a prime number that's too high, rather than a prime number that's too low, which probably skews the distribution.
The moderators have struck again. The title used to be "What happens when you ask people to pick the lowest prime number no one else will pick."
That's just as bad. The problem is in the word "lowest," because it is understood to mean "the one lowest".
It's not just as bad; it gives a closer to accurate picture, and it's pretty obnoxious of the mods to change it to be less illuminating. The way it's written now, I thought it was going to be about asking people a nonsensical mathematical question (which still has a correct answer), and seeing how the nonsensical curve-ball affects their answer.
Agreed. When I read the title, I thought, "WTH, what's so hard about picking the number 2? Is this article about how few people know this fact or something?"
The fact that there are 5 is important
Your strategy isn't to pick the lowest prime. It's to pick a prime that less than 5 other people would pick lower than - so you will tend to pick a higher number. The larger the group of pickers the higher you should pick.
Your strategy isn't to pick the lowest prime. It's to pick a prime that less than 5 other people would pick lower than - so you will tend to pick a higher number. The larger the group of pickers the higher you should pick.
There's no reason to require prime numbers, the game works the same with picking the lowest unique positive integer. We actually played this game at our department's research symposium two years ago with about a hundred people. Nobody picked 5 and the winning entry was 11.
Actually, there is a reason. It forces people (most people, at any rate), to actually look at a list of numbers, rather than simply thinking of one.
Like, imagine you asked a group of people to pick a number between 1-100. Some numbers are going to occur out of proportion to a random distribution. If it's a geeky crowd, you'll probably get at least 10% picking 42.
Whereas if you give them a list of the first 100 primes, and ask them to pick one, that sort of cultural bias should be mostly gone.
I realize that's not exactly the scenario involved here, but the concept still applies imo.
Like, imagine you asked a group of people to pick a number between 1-100. Some numbers are going to occur out of proportion to a random distribution. If it's a geeky crowd, you'll probably get at least 10% picking 42.
Whereas if you give them a list of the first 100 primes, and ask them to pick one, that sort of cultural bias should be mostly gone.
I realize that's not exactly the scenario involved here, but the concept still applies imo.
"Some numbers are going to occur out of proportion to a random distribution. If it's a geeky crowd, you'll probably get at least 10% picking 42."
This is so true that, for many years, magicians would use it as a "mind reading" parlor trick. The number 7 comes up with outsized frequency when people are asked to choose a number between 1 and 10 (the further out you expand the set, the lower the probability of their picking 7 -- but not by as much as you might expect). So much so, that you could fairly reliably ask someone to pick a number, guess that they'd picked 7, and be correct.
This is because most people don't actively think about the answer. They just select the number that comes most readily to mind. In Western culture, 7 is ingrained fairly heavily as a lucky/special number, and we encounter it with enough frequency that it'll be a likely choice for the brain's equivalent of the auto-fill feature.
You're much more likely to get a truly random sampling if your question requires thought or calculation, or if the bounds are unusual enough to warrant active thought. For instance, "Pick a number between 2 and 99" isn't drastically different from "Pick a number between 1 and 100." But it's different enough that it cues the brain to stop for a millisecond and actually think about the question. As a result, it's slightly more likely to generate a unique/uncommon number. (An even stranger set, such as "Pick a number between 6 and 73," will force the person to think even longer).
This is so true that, for many years, magicians would use it as a "mind reading" parlor trick. The number 7 comes up with outsized frequency when people are asked to choose a number between 1 and 10 (the further out you expand the set, the lower the probability of their picking 7 -- but not by as much as you might expect). So much so, that you could fairly reliably ask someone to pick a number, guess that they'd picked 7, and be correct.
This is because most people don't actively think about the answer. They just select the number that comes most readily to mind. In Western culture, 7 is ingrained fairly heavily as a lucky/special number, and we encounter it with enough frequency that it'll be a likely choice for the brain's equivalent of the auto-fill feature.
You're much more likely to get a truly random sampling if your question requires thought or calculation, or if the bounds are unusual enough to warrant active thought. For instance, "Pick a number between 2 and 99" isn't drastically different from "Pick a number between 1 and 100." But it's different enough that it cues the brain to stop for a millisecond and actually think about the question. As a result, it's slightly more likely to generate a unique/uncommon number. (An even stranger set, such as "Pick a number between 6 and 73," will force the person to think even longer).
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He mentioned that someone picked 333,667 and notes that it's a Sexy Prime, but 333,667 is also a Unique Prime (http://en.wikipedia.org/wiki/Unique_prime), which makes me think someone was either being cute, or they suspected that it was a trick question.
Based on the end of the post, they might well have 'won', given the special mention in the article.
"P.S. There might, in fact, be more than five primes out there. There might, after all, have been special shipments to rogue recipients. You never know…"
"P.S. There might, in fact, be more than five primes out there. There might, after all, have been special shipments to rogue recipients. You never know…"
Wow. TFA was a fine article, but I followed a link or two, and ended up reading the posted fulltext of My. Penumbra's 24-Hour Bookstore, and that was some Good Reading.
doesn't it sound familiar? apart from the gooey ending and the google twist, i could swear i had read that - a story, with a bookstore, with that meaning and process - before.
[maybe this sounds catty, but if you had to write a pastiche of borges you'd probably come up with this, so perhaps two people simply hit the same idea....]
[maybe this sounds catty, but if you had to write a pastiche of borges you'd probably come up with this, so perhaps two people simply hit the same idea....]
It felt like Borges wrote the first half and Neal Stephenson wrote the second.
I liked the first half better -- the mystery of the situation was so fabulous that any possible resolution would be a letdown.
I liked the first half better -- the mystery of the situation was so fabulous that any possible resolution would be a letdown.
There's an audio version on Escape Pod, if you're unwilling to read this very cool story: http://escapepod.org/2009/09/10/ep215-mr-penumbras-twenty-fo...
(Rated G. May contain creepy imagery and disturbing data visualizations. Free, CC-licensed. Escape Pod is awesome.)
(Rated G. May contain creepy imagery and disturbing data visualizations. Free, CC-licensed. Escape Pod is awesome.)
I did the same, but I am confused. There appears to be a novel coming out with the same name as that short story..
I guess they must be related--I'm certainly interested in reading the longer version when it is released.
I guess they must be related--I'm certainly interested in reading the longer version when it is released.
Unsurprisingly, there's a big spike at 17 since its well known to be the most random number.
http://blogs.discovermagazine.com/cosmicvariance/2007/01/30/...
This reminds me of a question asked in a Game Theory class server years ago:
Pick a number Whoever's number is closest to one half the average of all submitted numbers wins!
I just whipped together a tiny wsgi app for this:
http://sente.cc/wsgi/number_poll/
Pick a number Whoever's number is closest to one half the average of all submitted numbers wins!
I just whipped together a tiny wsgi app for this:
http://sente.cc/wsgi/number_poll/
How could the strategy here be anything other than "pick 0"?
Even the other degenerate strategy of "pick something which is like negative infinity" is a guaranteed loser, and it's only interesting if you assume everyone else will pick positive numbers (which your web app specifies, though your problem statement doesn't).
Even the other degenerate strategy of "pick something which is like negative infinity" is a guaranteed loser, and it's only interesting if you assume everyone else will pick positive numbers (which your web app specifies, though your problem statement doesn't).
I had 10 minutes to create the poll before the library closed...I definitely could have done a better job describing the problem, you're right.
Each person was told to pick a number between 0 and 100
The number I would pick would be dependent on who else is taking the poll, how much credit do you give them, etc
I'm in my phone now with a low battery, otherwise I'd type
Each person was told to pick a number between 0 and 100
The number I would pick would be dependent on who else is taking the poll, how much credit do you give them, etc
I'm in my phone now with a low battery, otherwise I'd type
The Wikipedia entry mentions this: http://en.wikipedia.org/wiki/Guess_2/3_of_the_average
But one of the problems with selecting a strategy is assuming that everyone will act perfectly rationally. This is hardly ever the case.
But one of the problems with selecting a strategy is assuming that everyone will act perfectly rationally. This is hardly ever the case.
Results are at http://sente.cc/wsgi/number_poll/results
actually -- 'better' results are here -- http://sente.cc/wsgi/number_poll/results_unique
(the other link doesn't restrict each IP to only 1 vote)
(the other link doesn't restrict each IP to only 1 vote)
This is an example of a Keynesian beauty contest [0].
0: http://en.wikipedia.org/wiki/Keynesian_beauty_contest
0: http://en.wikipedia.org/wiki/Keynesian_beauty_contest
What about negative primes? According to Wolfram Alpha:
http://www.wolframalpha.com/input/?i=-2+prime
http://www.wolframalpha.com/input/?i=-2+prime
I've always heard "prime number" defined to be greater than 1. WolframAlpha is probably just being crazy.
Wikipedia [1], and, more amusingly, Wolfram MathWorld [2] both agree with this definition.
[1]: http://en.wikipedia.org/wiki/Prime_number
[2]: http://mathworld.wolfram.com/PrimeNumber.html
Wikipedia [1], and, more amusingly, Wolfram MathWorld [2] both agree with this definition.
[1]: http://en.wikipedia.org/wiki/Prime_number
[2]: http://mathworld.wolfram.com/PrimeNumber.html
Strictly speaking, the moment you ask "Is -2 prime?" you are asking a fundamentally different question from "is 2 prime?" because -2 is not in N = {0?, 1, 2, 3, ...}, the set over which the notion "prime" is defined.
There is a more general notion of primes which can be applied to any ring (a set with "addition-like" and "multiplication-like" operations). That is the notion of "prime ideals": https://en.wikipedia.org/wiki/Prime_ideal
On this account, there are "primes" for Z, and -2 is the "same prime" as 2.
There is a more general notion of primes which can be applied to any ring (a set with "addition-like" and "multiplication-like" operations). That is the notion of "prime ideals": https://en.wikipedia.org/wiki/Prime_ideal
On this account, there are "primes" for Z, and -2 is the "same prime" as 2.
Another definition[1] of prime in an arbitrary ring is
As explanation: the only number 0 divides is 0, and Z is an integral domain[2], i.e. ab = 0 implies a = 0 or b = 0, thus 0 divides at least one of a and b if 0 divides ab.
[1]: https://en.wikipedia.org/wiki/Prime_element [2]: https://en.wikipedia.org/wiki/Integral_domain
p is prime if and only if p|ab implies p|a or p|b
As explanation: the only number 0 divides is 0, and Z is an integral domain[2], i.e. ab = 0 implies a = 0 or b = 0, thus 0 divides at least one of a and b if 0 divides ab.
[1]: https://en.wikipedia.org/wiki/Prime_element [2]: https://en.wikipedia.org/wiki/Integral_domain
That's actually really cool and reminds me that I should learn some basic abstract algebra one of these days.
There are quite a few of those scammy online auction sites that do "Unique bid auctions" -- same concept, just not restricted to prime numbers.
Since there is (possibly) a bit of skill involved, it makes it interesting from a legal perspective.
http://en.wikipedia.org/wiki/Unique_bid_auction
Since there is (possibly) a bit of skill involved, it makes it interesting from a legal perspective.
http://en.wikipedia.org/wiki/Unique_bid_auction
The challenge to win a book was really interesting, as was the results. And nice and smart way to advertise his book.
I really wonder if he will be able to keep tracks of these 5 books, since some could consider them "collector" of "deserved" and would tend to keep them instead of passing them over. I'll certainly stay tuned !
I really wonder if he will be able to keep tracks of these 5 books, since some could consider them "collector" of "deserved" and would tend to keep them instead of passing them over. I'll certainly stay tuned !
Why primes?
To make the entrants feel smarter. The whole thing is, of course, completely isomorphic to a guess-an-integer competition, but the rather minimal effort you need to go to to figure out the Nth prime makes it all worthwhile.
Perhaps it also indicates something about which primes sound primest, though. 29 is an outlying underachiever, suggesting that "29" doesn't spring to mind when you ask someone to name a prime... certainly I had to think for a moment to make sure it really was a prime. 23, 17, 37 are overachievers -- these are really prime-sounding primes, no doubt about it.
Perhaps it also indicates something about which primes sound primest, though. 29 is an outlying underachiever, suggesting that "29" doesn't spring to mind when you ask someone to name a prime... certainly I had to think for a moment to make sure it really was a prime. 23, 17, 37 are overachievers -- these are really prime-sounding primes, no doubt about it.
Author here: I am also fascinated by this sense of certain numbers sounding prime (or more prime than others). For instance -- maybe this is just me -- I think 109 (the first winner) doesn't sound prime at all.
But 17! Yes! It can't shut up about how prime it is!
But 17! Yes! It can't shut up about how prime it is!
A number theory class I once took provided us with the following "proof", claiming, in fact, that it was the most useful thing we would learn, and would make a great party trick.
Theorem: Anything less than 91 which looks prime is prime.
Proof: To do this, we will calculate the smallest number that looks prime but isn't. The numbers 2, 3, 4, 5, and 6 all have easy and quick divisibility rules, so our number cannot be divisible by any of them. 7, however, is hard to check for - so clearly the first number that looks prime but isn't will be divisible by 7. Sadly, everyone realizes 49 isn't prime, because it's a square, so we need to find the next difficult prime factor. 8, 9, 10, 11, and 12 are all also subject to quick tests and divisibility rules, so discount those. 13, however, is ugly - thus, the first number that looks prime but isn't is 7 times 13, which equals 91. Q.E.D. (My professor insisted this stood for "Quite Easily Done".)
Theorem: Anything less than 91 which looks prime is prime.
Proof: To do this, we will calculate the smallest number that looks prime but isn't. The numbers 2, 3, 4, 5, and 6 all have easy and quick divisibility rules, so our number cannot be divisible by any of them. 7, however, is hard to check for - so clearly the first number that looks prime but isn't will be divisible by 7. Sadly, everyone realizes 49 isn't prime, because it's a square, so we need to find the next difficult prime factor. 8, 9, 10, 11, and 12 are all also subject to quick tests and divisibility rules, so discount those. 13, however, is ugly - thus, the first number that looks prime but isn't is 7 times 13, which equals 91. Q.E.D. (My professor insisted this stood for "Quite Easily Done".)
Isn't this just the inverse of memorizing multiplication tables?
Did you receive any emails choosing composite numbers? To me, 51 sounds prime.
Yes, many! On the submission form, I even referred folks to Wikipedia's list of prime numbers but, alas.
Right. And numbers like 117 and 119 secretly trick (at least) me into thinking they are prime.
Hm, I wonder what Douglas Hofstadter's superrational (http://www.gwern.net/docs/1985-hofstadter) strategy would be.
From what I understand the "super-" only makes a difference in games where cooperation/defection make sense. The superrational player chooses cooperation because they know that all the other players are also superrational and will also do so. This is not the case in the game described, so I think that the superrational player would play like a plain old rational one.
2 people on 61 is thinking, damn it, almost!
I would assume that the right strategy here varies depending on how many other people are going to enter the competition.
Actually, a "guess how many people will enter this competition" competition would be fun.
What would be interesting now is to repeat the competition after publishing these results; in fact, make sure these results are given on the entry page for the new competition. What would happen this time? Well, a bunch of people would start picking numbers around 109 this time. Except everybody expects everybody to do that, so maybe they'll start picking lower. Meanwhile smartasses will still be picking "2", and smartasses who think they can outsmart those smartasses will be picking "3".
Actually, a "guess how many people will enter this competition" competition would be fun.
What would be interesting now is to repeat the competition after publishing these results; in fact, make sure these results are given on the entry page for the new competition. What would happen this time? Well, a bunch of people would start picking numbers around 109 this time. Except everybody expects everybody to do that, so maybe they'll start picking lower. Meanwhile smartasses will still be picking "2", and smartasses who think they can outsmart those smartasses will be picking "3".
http://swopec.hhs.se/hastef/abs/hastef0671.htm
Calculating the equillibrium strategy for rational actors is difficult because each player doesn't know how many other players there are. In the paper above, game theorists calculate it and show that the distributions seen in the lottery match up fairly well to a rational strategy.