Back in the 90s as a kid, one year I got a handheld 20 Questions electronic game as a stocking stuffer for Christmas. I remember being astonished that this dumb little plastic pod with rubber Yes/No buttons and LCD screen "beat me" by guessing the first thing I tried (the Mona Lisa) in 8 or so guesses.
The author quips that "twenty questions isn’t enough to guess almost anything", but I wonder if most people taking their first crack at the game (usually as children) pick something squarely in the 2^20 most popular things.
(My friend's dad was really good at picking words. I remember he stumped us for an entire restaurant visit with "the cub in the Chicago Cubs logo".)
My stepson has one- and I was astonished how good it is. I can’t say I’ve thought about it at length but at face value it seems a marvel. Is the logic for such a thing trivial and widely available?
I have no idea how they actually work, but if I were asked to design one:
• Choose some reasonable number of questions of the form "Is it ___"?. Let's say 256
• Come up with a list of objects, and for each one give it a 256-long bitvector encoding its answers to the questions
• Maintain a set (implemented as another bitvector) of the potential items. Figure out which question would divide the set in two most closely; ask that question.
I am the opposite of a hardware hacker or systems programmer, but it seems like this is algorithmically straightforward to implement with bit-twiddling.
It's been a longstanding question of mine how these things were programmed. How did they construct the database of answers and questions, and what the answer to each question would be for each possible answer?
This reminds me of jan Misali's analysis of Hangman (in fact the title is so similar that I'm inclined to believe that it's definitely inspired despite not mentioning the video or Misali by name) [^1]. In any case, I'm obviously inclined to agree that they're both weird games, and that the weirdness comes from their fundamental asymmetry. It definitely reminds me of a sort of ritualized play, where an asymmetry is "artificially" constructed for the purpose of being broken down. There's a fictionalized power dynamic that gets produced, yet the erosion of this dynamic isn't just the point, it's often aided by the one in power (how many times have I given my friends hints or extra chances in both games).
> It’s cooperative because everyone is ultimately working toward a common goal: deducing the answer
I think this point gets pretty close to why these games are fun or interesting to begin with. I don't think it's necessarily because that they teach you about a person based on the object they choose or the guesses they make, but the fact that the game operates at all. In some sense it's like a puzzle, but unlike a puzzle doesn't just operate on its prima facie rules - because if you can't solve a puzzle then that's just too bad, but if 20 questions doesn't end then something is off. In seems that 20 questions is interesting because there are social conventions within which it operates and playing by those conventions seems to demonstrate a social and emotional understanding that power dynamics within the relationship will be cooperatively undermined.
I haven't looked too hard for earlier examples in other sources. I see Sorel had a game called the Game of Questions in the 1600s, but it's pretty different: https://wobbupalooza.neocities.org/
I had a tip calculator back then but it was a dashboard widget for OS X rather than an iPhone app which I just made for personal use. It never occurred to me to try to make an iPhone app out of it, and I was actually looking for an idea just to have an excuse to try iPhone programming.
I know there were already many tip calculators even back then, and they were maybe fine for tips in general like when you are at a restaurant, but I never saw one except for mine that worked well for the most important case: figuring out how much to tip the pizza delivery guy when you are paying in cash.
Say your pizza comes to $16.23. If you usually tip 15% a normal tip calculator tells you that is $2.43 and the total would be $18.66. But who the heck is going to try to pay exactly $18.66 when their are standing in the open door with their home's precious heat leaking into the cold winter night? Same with handing him $20 and waiting for him to count out $1.34 in change.
What most sane people are going to do is hand him $19 and say "keep the change", or hand him $20 and say "keep the change", or hand him $20 and say "give me $1 back".
With my calculator you would enter the $16.23 price, and the calculator would give you a table something like this:
You can then easily combine that with your tipping level goal, tipping limit, and the mix of bills in your wallet to figure out what to do to leave an acceptable tip while still having a quick transaction.
Could be your app still has legs, slotted into wherever a widget like this might make sense in the Apple ecosystem. IIRC, Amazon has a program allowing plugins for Alexa that your solution might be good for.
I searched around for a screenshot and found one with an images and a description of our tipping app, Tiptotaler.
It apparently got an A- rating from ilounge at the time, which is heartening still today. :) [1]
App functionality description:
> You can separately input food, drinks, and tax into three fields at the top of the screen, then note the number of drinkers and non-drinkers, set the tip amount, and then get totals for individual diners.
I find this ironic, as the author treats it like a word game. I've always played that it has to be a "thing" by the typical sense of the word, e.g. an object. Sometimes with bounds, like something you saw today. The problem with "intangible things" is they are essentially imaginary, and therefore subject to the whimsy of the answerer, as they point out. Does an Air Guitar make noise? Is an Air Guitar a gesture, what about a form of dancing?
Objects don't have that problem but still can make for incredibly interesting games.
> As John Green (or Georg Cantor) taught us, some infinities are bigger than others—and the number of things is a really big infinity
I don't think this statement is true, at least not in the context it's given. At most, we'd only be able to think of countably many things, which is the smallest infinity.
No, because with infinities, even if a set is a subset of another, you may still be able to find one element in the first set that corresponds to every element of the other set.
For example, the natural numbers are a subset of the whole numbers, but there is a natural number that corresponds to every whole number. To see this, we can order the whole numbers like this: {0, -1, 1, -2, 2, -3, 3, ...}, and we can easily see that we can now assign one natural number to each of them (0 -> 0, -1 -> 1, 1 -> 2, -2 -> 3, ...). Since you'll never run out of naturals, you won't ever find a whole number that doesn't have a corresponding natural number.
Since assigning a natural number this way is equivalent to counting the elements of the other set (in this scheme, I could say that -2 is the 3rd whole number), this type of infinity is called "countable infinity". The natural numbers, the whole numbers, and the rational numbers are all countably infinite. In contrast, the irrational numbers and the real numbers are not. In fact, even the real interval [0, 1] is not countable, so this interval is considered to have more elements than N (the set of natural numbers).
Note that while there is only one countable infinity, there are many uncountable infinities - so not all uncountably infinite sets are considered as large. If you're curious about this area, the study of these concepts is done via "transfinite numbers" - particularly, the transfinite cardinal numbers (there are also transfinite ordinals).
Surprisingly, no. For example, even though every whole number is also a rational number, mathematicians would say the size (or more accurately, the cardinality) of the set of whole numbers is the same as that of the set of rational numbers.
I'm personally a fan of the Infinite Hotel Paradox as an introduction to the subject.
Not only is the answer "no" like the sibling comment says, but in fact one definition of an infinite set is that it can be put into one-to-one correspondence with a strict subset of itself. In other words, infinite sets are precisely those for which your concept of size doesn't work.
Another sibling comment used the even/odd example, but that's not necessary to dispel this particular misconception. Consider the set of non-negative integers and the set of positive integers. That is, {0,1,2,3,...} and {1,2,3,4,...}. The latter is a strict subset of the former. Maybe I have just done mathematics for too long, but to me these are intuitively, "obviously" the same size. What would it even mean for one of them to be smaller? Which one is the same size as {-1,-2,-3,...}, if either of them? Even doing folk mathematics, if the size of the first is "infinity" then the size of the second is "infinity minus one which is still infinity".
You could (using "is a subset" as a partial order), but you can't make a total order. Any way you try to compare size of sets where neither is subset of the other, while preserving your intuition of "size" will run into trouble.
You make "the ordinals" sort of using your idea, but that isn't really measurement of "size"; it's more like an assignment of ranks.
You can't think of any real number. You can think of some of them, but you need to have a way of thinking of them with some specification that can be written down, and those form a countable set. We also have finite life spans, so we actually can only think of finitely many numbers. When you think of the natural numbers, you think of them as a set and concept, which is different than being able to visualize and conceive of every natural number.
What about potentially think of? Even that does not make sense because every possible description you can think of must be written with finitely many symbols in some language, and all such descriptions form a countable set.
Therefore, there are real numbers that could never be written down in English, even potentially. Of course, that partially also depends on the kind of axiom system that you are using. In effect, the standard axioms of mathematics state the existence of things that cannot be effectively specified, which some people actually are against, although such people form a minority.
The surprising thing is that we can effectively reason with large infinities and such objects make intuitive sense (the set of all functions from the natural numbers to the natural numbers is uncountable but a very natural sounding set), and yet it is impossible even in principle to write every one down, even with an unlimited amount of time.
Pi and E can be thought of because there are mathematical formulas and algorithms relating them and computing them. There are only countably many such formulas & algorithms.
You can't think of an arbitrary real number, in the sense of distinguishing it every other real number(including nearby reals an ε away from it), and with no other constraints besides being a real number.
I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in some way. But there are a _lot_ of real numbers.
We could think about it this way: we can only describe (and therefore think of) numbers using a finite number of symbols out of a finite alphabet. That makes it only countable.
I played this game with physicist friends and the favorite word was 'shadow'. It was the only everyday object not composed of something in the standard model. It stumped everyone.
My kids used to have this 20 questions toy. It was always able to figure out anything I thought of in less than 20 questions. Apparently, I’m terrible at coming up with things to guess. Now I want to find that toy, put in new batteries, and try it against the list of things mentioned at the bottom of this article.
Readit News
The author quips that "twenty questions isn’t enough to guess almost anything", but I wonder if most people taking their first crack at the game (usually as children) pick something squarely in the 2^20 most popular things.
(My friend's dad was really good at picking words. I remember he stumped us for an entire restaurant visit with "the cub in the Chicago Cubs logo".)
• Choose some reasonable number of questions of the form "Is it ___"?. Let's say 256
• Come up with a list of objects, and for each one give it a 256-long bitvector encoding its answers to the questions
• Maintain a set (implemented as another bitvector) of the potential items. Figure out which question would divide the set in two most closely; ask that question.
I am the opposite of a hardware hacker or systems programmer, but it seems like this is algorithmically straightforward to implement with bit-twiddling.
> It’s cooperative because everyone is ultimately working toward a common goal: deducing the answer
I think this point gets pretty close to why these games are fun or interesting to begin with. I don't think it's necessarily because that they teach you about a person based on the object they choose or the guesses they make, but the fact that the game operates at all. In some sense it's like a puzzle, but unlike a puzzle doesn't just operate on its prima facie rules - because if you can't solve a puzzle then that's just too bad, but if 20 questions doesn't end then something is off. In seems that 20 questions is interesting because there are social conventions within which it operates and playing by those conventions seems to demonstrate a social and emotional understanding that power dynamics within the relationship will be cooperatively undermined.
[^1]: https://youtu.be/le5uGqHKll8
I recently started to learn toki pona after seeing it discussed here, his "12 days of toki pona" series is really great.
Here it is in French in 1788, as "The Twelve Questions" but still beginning with the question animal, vegetable, or mineral: https://www.google.com/books/edition/Les_soir%C3%A9es_amusan...
Here it is in English in 1796, as "Game of Twenty": https://www.google.com/books/edition/The_Juvenile_Olio_Or_Me...
I haven't looked too hard for earlier examples in other sources. I see Sorel had a game called the Game of Questions in the 1600s, but it's pretty different: https://wobbupalooza.neocities.org/
The second was 20 Questions. The app provided prompts and a paddle to keep track of the guess count.
It was a nice little app because it made iPhone social.
We called the app iQ because the iProduct pattern was still in force and it was cool to camp the name space.
We sold a bunch of copies of this game, but not nearly as many as we later made on Baby Names, the first baby name app in the App Store.
It was the gold rush era, you could still come up with simple ideas and be the first to put it up on the App Store.
https://web.archive.org/web/20090101213438/http://neutrinosl...
I know there were already many tip calculators even back then, and they were maybe fine for tips in general like when you are at a restaurant, but I never saw one except for mine that worked well for the most important case: figuring out how much to tip the pizza delivery guy when you are paying in cash.
Say your pizza comes to $16.23. If you usually tip 15% a normal tip calculator tells you that is $2.43 and the total would be $18.66. But who the heck is going to try to pay exactly $18.66 when their are standing in the open door with their home's precious heat leaking into the cold winter night? Same with handing him $20 and waiting for him to count out $1.34 in change.
What most sane people are going to do is hand him $19 and say "keep the change", or hand him $20 and say "keep the change", or hand him $20 and say "give me $1 back".
With my calculator you would enter the $16.23 price, and the calculator would give you a table something like this:
You can then easily combine that with your tipping level goal, tipping limit, and the mix of bills in your wallet to figure out what to do to leave an acceptable tip while still having a quick transaction.I searched around for a screenshot and found one with an images and a description of our tipping app, Tiptotaler.
It apparently got an A- rating from ilounge at the time, which is heartening still today. :) [1]
App functionality description:
> You can separately input food, drinks, and tax into three fields at the top of the screen, then note the number of drinkers and non-drinkers, set the tip amount, and then get totals for individual diners.
[1] https://www.ilounge.com/index.php/articles/comments/iphone-g...
I find this ironic, as the author treats it like a word game. I've always played that it has to be a "thing" by the typical sense of the word, e.g. an object. Sometimes with bounds, like something you saw today. The problem with "intangible things" is they are essentially imaginary, and therefore subject to the whimsy of the answerer, as they point out. Does an Air Guitar make noise? Is an Air Guitar a gesture, what about a form of dancing?
Objects don't have that problem but still can make for incredibly interesting games.
> Does an Air Guitar make noise?
"Maybe, but only in your mind" or "not directly by itself"
> Is an Air Guitar a gesture,
Yes.
> what about a form of dancing?
"Yeah, I guess it is'
> As John Green (or Georg Cantor) taught us, some infinities are bigger than others—and the number of things is a really big infinity
I don't think this statement is true, at least not in the context it's given. At most, we'd only be able to think of countably many things, which is the smallest infinity.
So if a set is infinite but a provably strict subset of another - would we not say that set/infinity is smaller?
For example, the natural numbers are a subset of the whole numbers, but there is a natural number that corresponds to every whole number. To see this, we can order the whole numbers like this: {0, -1, 1, -2, 2, -3, 3, ...}, and we can easily see that we can now assign one natural number to each of them (0 -> 0, -1 -> 1, 1 -> 2, -2 -> 3, ...). Since you'll never run out of naturals, you won't ever find a whole number that doesn't have a corresponding natural number.
Since assigning a natural number this way is equivalent to counting the elements of the other set (in this scheme, I could say that -2 is the 3rd whole number), this type of infinity is called "countable infinity". The natural numbers, the whole numbers, and the rational numbers are all countably infinite. In contrast, the irrational numbers and the real numbers are not. In fact, even the real interval [0, 1] is not countable, so this interval is considered to have more elements than N (the set of natural numbers).
Note that while there is only one countable infinity, there are many uncountable infinities - so not all uncountably infinite sets are considered as large. If you're curious about this area, the study of these concepts is done via "transfinite numbers" - particularly, the transfinite cardinal numbers (there are also transfinite ordinals).
I'm personally a fan of the Infinite Hotel Paradox as an introduction to the subject.
Another sibling comment used the even/odd example, but that's not necessary to dispel this particular misconception. Consider the set of non-negative integers and the set of positive integers. That is, {0,1,2,3,...} and {1,2,3,4,...}. The latter is a strict subset of the former. Maybe I have just done mathematics for too long, but to me these are intuitively, "obviously" the same size. What would it even mean for one of them to be smaller? Which one is the same size as {-1,-2,-3,...}, if either of them? Even doing folk mathematics, if the size of the first is "infinity" then the size of the second is "infinity minus one which is still infinity".
You make "the ordinals" sort of using your idea, but that isn't really measurement of "size"; it's more like an assignment of ranks.
What about potentially think of? Even that does not make sense because every possible description you can think of must be written with finitely many symbols in some language, and all such descriptions form a countable set.
Therefore, there are real numbers that could never be written down in English, even potentially. Of course, that partially also depends on the kind of axiom system that you are using. In effect, the standard axioms of mathematics state the existence of things that cannot be effectively specified, which some people actually are against, although such people form a minority.
The surprising thing is that we can effectively reason with large infinities and such objects make intuitive sense (the set of all functions from the natural numbers to the natural numbers is uncountable but a very natural sounding set), and yet it is impossible even in principle to write every one down, even with an unlimited amount of time.
You can't think of an arbitrary real number, in the sense of distinguishing it every other real number(including nearby reals an ε away from it), and with no other constraints besides being a real number.
I don't think that's true. We can certainly think of certain numbers which we gave a name to and have defined it in some way. But there are a _lot_ of real numbers.
We could think about it this way: we can only describe (and therefore think of) numbers using a finite number of symbols out of a finite alphabet. That makes it only countable.
Is a ghost?
A finish line?