It's common to use the term "non-euclidean geometry" in video games where the level geometry does not follow logical physics. For example you are seemingly inside a close room with a column at its center, but if you walk around the column you end up in a completely different place (typically there's some kind of "portal" surface at one side of the column that connects the level to another completely different level.) For example the game Antichamber is based on this.
Is "non-euclidean geometry" a misnomer in this case? It doesn't sound like a case of hyperbolic or elliptic geometry.
It's common to use the term "non-euclidean geometry" in video games where the level geometry does not follow logical physics. For example you are seemingly inside a close room with a column at its center, but if you walk around the column you end up in a completely different place (typically there's some kind of "portal" surface at one side of the column that connects the level to another completely different level.) For example the game Antichamber is based on this.
Is "non-euclidean geometry" a misnomer in this case? It doesn't sound like a case of hyperbolic or elliptic geometry.
Who wins at Memory?
While playing Memory today, I thought about 1) the best strategy to win and 2) who wins statistically, the player to go first or the player to go second?
Let's assume that both players have perfect memory, so that all that matters is statistics. Let's also assume that you must reveal at least one new card on your turn, so we don't get stuck with both players only choosing known cards because revealing a new card decreases your chances of winning.
1) I have a good idea about the optimal strategy. When there are lots of cards left, it's is better to only reveal one new card, and then pick a card which is already known.
Reasoning: if you reveal a card which you don't know the match for, you should not reveal another card, because that increases the risk that your opponent can match that card with an already known card.
If there are only a few cards left, it can be better to switch to reveal two new cards, if the first one isn't a match to a known card. For example: 6 cards left (AABBCC), 2 cards known (AB). You reveal a card which is C. If you guess on one of the three remaining cards, you have 1/3 chance of winning. If you instead pick one of the known cards, you have 0 chance of winning, because all of the new cards which the opponent turns up will match a known card.
2) For a small number of pairs, calculating player 1's chance of winning is easy. For example, for 2 pairs is 1/3. Reveal one card, A. You now have 1/3 chance of guessing the matching card. Whoever gets the first pair also gets the second.
For 3 pairs, it is 1/15 (win directly) + 8/15 (win on second turn), or 3/5.
How about x number of pairs? Is there a way to generalize this?
Who wins at Memory?
While playing Memory today, I thought about 1) the best strategy to win and 2) who wins statistically, the player to go first or the player to go second?
Let's assume that both players have perfect memory, so that all that matters is statistics. Let's also assume that you must reveal at least one new card on your turn, so we don't get stuck with both players only choosing known cards because revealing a new card decreases your chances of winning.
1) I have a good idea about the optimal strategy. When there are lots of cards left, it's is better to only reveal one new card, and then pick a card which is already known.
Reasoning: if you reveal a card which you don't know the match for, you should not reveal another card, because that increases the risk that your opponent can match that card with an already known card.
If there are only a few cards left, it can be better to switch to reveal two new cards, if the first one isn't a match to a known card. For example: 6 cards left (AABBCC), 2 cards known (AB). You reveal a card which is C. If you guess on one of the three remaining cards, you have 1/3 chance of winning. If you instead pick one of the known cards, you have 0 chance of winning, because all of the new cards which the opponent turns up will match a known card.
2) For a small number of pairs, calculating player 1's chance of winning is easy. For example, for 2 pairs is 1/3. Reveal one card, A. You now have 1/3 chance of guessing the matching card. Whoever gets the first pair also gets the second.
For 3 pairs, it is 1/15 (win directly) + 8/15 (win on second turn), or 3/5.
How about x number of pairs? Is there a way to generalize this?
There may be some deep theory here, but for starters, let's consider one pair and two pairs.
If there's just one pair, the first player has a trivial advantage. They'll turn over the pair and automatically win every time.
But if there are two pairs, the tables turn in favor of the second player. The first player will fail to make a match two-thirds of the time. After that, the second player will pick one of the two remaining cards, which is assured to match with one of the two revealed ones. From there, they win. (I'm assuming that when a player makes a match, their turn continues.)
I'm guessing that the pattern may continue such that an odd number of pairs favors the first player and an even number of pairs favors the second player. I'll give this a little more thought.
Edit: I took a few minutes to write out what I believe are all the possibilities of two- and three-pair games. I've assumed (without loss of generality) that players turn over cards from left to right. If they can make a match, they choose the appropriate card. If they can't, they pick the next card to the right. Also without loss of generality (I think-- someone should work out the details here...), we'll label the cards in ascending order from left to right. Note that this is different from your strategy, but I'll give that some thought as well.
1122 - P1 win
1212 - P2 win
1221 - P2 win
112233 - P1 win
112323 - P2 win
112332 - P2 win
121233 - P2 win
121323 - P1 win
121332 - P2 win
122133 - P2 win
122313 - P1 win
122331 - P2 win
123123 - P1 win
123132 - P1 win
123213 - P1 win
123231 - P1 win
123312 - P2 win
123321 - P2 win
Using this strategy, player 1 wins seven out of fifteen times, so my intuition was wrong. Player 2 seems to have the advantage here.
Okay, now let's consider your strategy, in which player 2 "wastes" a move by turning over just one new card and also turns over a card he knows is not a match. As far as I can tell, that strategy affects the following games:
121323
121332
122313
122331
123123
123132
123213
123231
123312
123321
Or more succinctly:
1213XX
1223XX
123XXX
In the first two games, player 2 finds themselves ahead by a match. Because the 3 is uncovered, they must take a chance on one of the Xs. Otherwise, player 1 will uncover one of the Xs and be assured a victory.
The third game is marginally more interesting. Player 2 has just uncovered a 3 and can uncover one of the Xs, hoping it's a 3 as well. Two-thirds of the time, however, it's not a 3 and player 1 is assured a victory. Their other option is to uncover one of the first two cards, kicking it back to player 1. Player 1 would then uncover one of the Xs and be positively guaranteed a victory. In this case, the optimal strategy is to uncover one of the Xs.
So assuming I didn't make any mistakes, the original analysis stands. A perfectly played game of memory with three pairs favors player 2 just slightly, as they'll win 8 out of 15 times.
>(I'm assuming that when a player makes a match, their turn continues.)
Correct.
We seem to interpret the problem the same, but still your answer for a 3-pair game is different than mine. We agree this far at least:
1 pair:
p1: 1/1
draw: 0/1
p2: 0/1
2 pairs:
p1: 1/3
draw: 0/3
p2: 2/3
Let's see where one of us goes wrong...
3 pairs:
p1: 1/5 for finding one pair (AA known, BBCC left). After this, the situation is identical to a 2 pair game:
-p1: 1/3 for finding two pairs to win 3-0
-p2: 2/3 for finding two pairs to win 2-1
total chance to win in this scenario:
p1 = 1/5*1/3 = 1/15
p2 = 1/5*2/3 = 2/15
p1: 4/5 for no pair (AB known, ABCC left)
-p2: 2/4 for 1 pair directly (B known, BCC left)
--p2: 1/3 for B, 2 more pairs to win 3-0
--p2: 2/3 for C
---p2: 1/2 to win 3-0
---p1: 1/2 to win 3-0
-p2: 2/4 for turning up C
--p2 1/3 to find the other C, 3 pairs to win 3-0
--p2 2/3 to find A or B, which means p1 wins 3-0
p2 chance to win is then
4/5*2/4*1/3 = 8/60 = 2/15
4/5*2/4*2/3*1/2 = 16/120 = 2/15
4/5*2/4*1/3 = 8/60 = 2/15
p1 chance to win is
4/5*2/4*2/3*1/2 = 16/120 = 2/15
4/5*2/4*2/3 = 16/60 = 4/15
In total
p2:
2/15 to win 2-1
6/15 to win 3-0
p1:
7/15 to win 3-0
So my original calculation of 9/15 for p1 was wrong, and 7/15 like you had it was right.
This branches into a lot of possibilities pretty quickly. Maybe I'll work out 4 pairs by hand too, if one player gets a pair directly (1/7 chance of this happening) the game basically becomes a 3 pair game which we already know the solution to. In that case 2-1 above becomes a draw.
There should be a better way than doing all possibilities by hand though...
The only thing that article tells me is that "yes, using the term in this context is a misnomer".
Then you read it wrong. As the article states, non-Euclidean geometry "arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one". Looking into the metric requirements, we see that the triangle inequality is a requirement. Basically, this states that the direct line between two points A and B is always the shortest possible distance, and that when traveling via another point C not on that line, the distance is always greater. However, in a game like you specified, this does not always hold. Look at the game Portal, for example: The direct line from one side of the room A to the other B may be very long, whilst going through portals placed near them (C), this distance might be shorter.
Looks extremely relevant. I noticed the same strategies that I figured out, labeled in the paper as risky, safe, and pass. I'll take a closer look when I get a chance.
I haven't had time to put pencil to paper, but I've been thinking about how to analyze the game. The game's state can be completely determined by the following variables:
1) Total number of pairs remaining. Let's call it P.
2) Number of points the current player is ahead (or behind) by. This variable effectively ranges from -P to P because in those extreme cases, the best (or worst) you could expect is a tie. For example, if you're 20 points behind and there are just 5 pairs left in the game, you've already lost and there's no point in conducting any analysis. We can call this variable S for score. (The range of S is giving me flashbacks to raising and lowering operators in quantum mechanics.)
3) Number of known cards. This ranges from 0 to P because if it exceeds P, then a pair is already known, which can't happen with optimal play. We'll call this variable N.
To each state, we assign the expectation value of the round's point value (loss=-1, tie=0, win=1) given optimal play. I'll call this variable X, so in short, we're looking for X as a function of P, S, and N. From there, I would work on a constructive argument. Start with the trivial P=1 cases and then reduce all higher states to this case based on an analysis of the probability.
This analysis should probably be carried out on a computer. For P pairs, there are P*(2*P+1)*(P+1) = 2P^3 + 3P^2 + P different states to analyze. For P=1, we already have five states to analyze and for P=2, this balloons to 29 different states and P=3 has 84 different states.
(This is not really a math question, but given that it's about math, I thought it would be appropriate here.)
The memetic phrase "you keep using that word; I do not think it means what you think it means" is a quote from the movie The Princess Bride.
The word in question being referred to is "inconceivable". The character (who is using it "incorrectly") is using it to mean "incredible" or "hard to believe". Yet, as the other character says, there may be a subtle difference.
"Inconceivable" means "difficult or impossible to be conceived" (rather than difficult or impossible to believe.) But what could be a concrete example of something that's inconceivable?
The human mind perceives the "size" of numbers mostly by comparing them to real things, or multiples of the size of real things. We compare very large numbers to things like "the number of atoms on Earth" (or even "the number of atoms in the universe"), or things like "the size of our galaxy/universe".
Even if we don't really have a good grasp of how large that amount is, it nevertheless anchors it to something real that we can compare to. Although the exact size may be more intangible to our brains, it nevertheless allows us to compare bug numbers between themselves. For example if something is "as big as Earth" and something else is "as big as the solar system", we get a concept of which one is significantly larger than the other.
We often also use multiples of these real things to get a picture of the quantity, by saying things like "a trillion times the number of atoms in the universe", or "a trillion times the mass of the Sun". Again, while the actual quantity remains pretty abstract, it nevertheless gives us a comparison point, so we can compare the large numbers between themselves, and know which one is clearly larger than the other. Something that's "a trillion times the size of our galaxy" is much larger than something that's "the size of our solar system".
Graham's Number, however, defies all visualizations and comparisons to real things, even if using multipliers. There exists no way to adequately compare it to anything. You can't say something like "Graham's number is as big as a trillion trillion trillion times the number of subatomic particles in our universe", because that would still be way, way too small of a number. Graham's number is so immensely large, that there is no simple sentence, comparing it to the size of anything real, that can be used to describe its size. It is basically impossible for our brains to have even a vague concept of how large it is.
In other words, the size of Graham's number is truly inconceivable.
(This is not really a math question, but given that it's about math, I thought it would be appropriate here.)
The memetic phrase "you keep using that word; I do not think it means what you think it means" is a quote from the movie The Princess Bride.
The word in question being referred to is "inconceivable". The character (who is using it "incorrectly") is using it to mean "incredible" or "hard to believe". Yet, as the other character says, there may be a subtle difference.
"Inconceivable" means "difficult or impossible to be conceived" (rather than difficult or impossible to believe.) But what could be a concrete example of something that's inconceivable?
The human mind perceives the "size" of numbers mostly by comparing them to real things, or multiples of the size of real things. We compare very large numbers to things like "the number of atoms on Earth" (or even "the number of atoms in the universe"), or things like "the size of our galaxy/universe".
Even if we don't really have a good grasp of how large that amount is, it nevertheless anchors it to something real that we can compare to. Although the exact size may be more intangible to our brains, it nevertheless allows us to compare bug numbers between themselves. For example if something is "as big as Earth" and something else is "as big as the solar system", we get a concept of which one is significantly larger than the other.
We often also use multiples of these real things to get a picture of the quantity, by saying things like "a trillion times the number of atoms in the universe", or "a trillion times the mass of the Sun". Again, while the actual quantity remains pretty abstract, it nevertheless gives us a comparison point, so we can compare the large numbers between themselves, and know which one is clearly larger than the other. Something that's "a trillion times the size of our galaxy" is much larger than something that's "the size of our solar system".
Graham's Number, however, defies all visualizations and comparisons to real things, even if using multipliers. There exists no way to adequately compare it to anything. You can't say something like "Graham's number is as big as a trillion trillion trillion times the number of subatomic particles in our universe", because that would still be way, way too small of a number. Graham's number is so immensely large, that there is no simple sentence, comparing it to the size of anything real, that can be used to describe its size. It is basically impossible for our brains to have even a vague concept of how large it is.
In other words, the size of Graham's number is truly inconceivable.
First, as a simple example, the vast majority of numbers are literally inconceivable because they are not computable or definable. Put one way, we might attempt to list all possible programs a computer might run (say, through assembly language) and make a list of every possible output. We can do this because the list of all algorithms is at most countably infinite (which requires infinite disk space) but the cardinality of the continuum is strictly higher than that. Therefore, the number line is densely populated by what I like to think of as "phantom numbers"-- numbers that have never been considered, never will be considered, and are outright impossible to be considered. I think it's fascinating that the list of numbers everyone (or every machine) has ever written down or imagined is surprisingly small-ish (maybe in the trillions) while all those numbers are buoyed by a ghostly aether of numbers that lie beyond our imagination.
Second, I mostly agree with you on Graham's number. However, it can be defined rather succinctly. If you accept some notion of chained arrow notation, then it's not all that hard to "conceive" of a number greater than Graham's number.
Still, I've toyed with the idea of "conceivability". Somewhere in our psychology is an approximate quantity that is the largest we can wrap our brains around. Perhaps it's about a thousand and if I showed you a thousand beads, you would look at them and say, "that's about a thousand", demonstrating that you have a notion of what you're looking at. (Of course, the largest conceivable number may be much smaller or much greater depending on what you mean by "wrapping your brain around it".) A number greater than this amount is inconceivable, but most simple examples don't really stand out from one another. For example, is 10^23 (roughly Avogadro's number) less conceivable than 10^200 (roughly the number of Planck volumes in the universe)? I'd argue that they equally defy human imagination, despite the fact that they are 177 orders of magnitude apart!
But there are numbers that stretch beyond inconceivability. For a simple and familiar example, consider a googolplex, which is 10^10^100. We can't even physically write down the number! I'd say that a googolplex is "inconceivably inconceivable".
All of this is a very roundabout way of saying that the inconceivableness of a number is the number of times you must successively take its logarithm before it becomes conceivable. A number like 2^3^5^7^11 is basically an inconceivably inconceivably inconceivably inconceivable number.
Furthermore, Graham's number is not just inconceivable, its inconceivableness is inconceivable and the inconceivableness of its inconceivability is likewise inconceivable. Trying to put together a sensible string of "inconceivables" to describe Graham's number (or its inconceivability...) is extremely difficult and maybe impossible.
First, as a simple example, the vast majority of numbers are literally inconceivable because they are not computable or definable.
I'm talking about magnitudes, not about exact numbers. The magnitude of something like pi is easy to describe by comparing it to a real thing, even though you could never know its exact representation. Exact representations are not the point.
Sure, there are infinitely many uncomputable and undefinable numbers between 0 and 1, but their magnitude is trivial to grasp. After all, they are between 0 and 1.
Second, I mostly agree with you on Graham's number. However, it can be defined rather succinctly. If you accept some notion of chained arrow notation, then it's not all that hard to "conceive" of a number greater than Graham's number.
To the average person the arrow notation says absolutely nothing. It's just an abstract notation. It doesn't give them a notion of the magnitude of the number because it can't be compared to anything else that's more tangible.
But there are numbers that stretch beyond inconceivability. For a simple and familiar example, consider a googolplex, which is 10^10^100.
The magnitude of googolplex can be more easily understood because it uses familiar notions, namely exponentiation, and only two levels of it. I'm sure one can come up with a visualization of the magnitude that can give even a vague idea of its size. It's not inconceivable.
Graham's number, however, uses too many levels of exponentiation to be describable by comparing it to anything. It just escalates way too quickly and defies all comparison. It defies all description.
The magnitude of googolplex can be more easily understood because it uses familiar notions, namely exponentiation, and only two levels of it. I'm sure one can come up with a visualization of the magnitude that can give even a vague idea of its size. It's not inconceivable.
Graham's number, however, uses too many levels of exponentiation to be describable by comparing it to anything. It just escalates way too quickly and defies all comparison. It defies all description.
But Graham's number is the upper bound on a solution to a problem that-- with a little bit of effort-- the layperson can easily understand. I don't really know how you can decide that a googolplex is conceivable just because the operation used is more familiar to people but at the same time discard Graham's number.
You yourself said you're talking about magnitudes and said exact representations are not the point. By your own criteria, Graham's number is the most inconceivable, a googolplex is more conceivable but still inconceivable, and pi is most conceivable.
But Graham's number is the upper bound on a solution to a problem that-- with a little bit of effort-- the layperson can easily understand. I don't really know how you can decide that a googolplex is conceivable just because the operation used is more familiar to people but at the same time discard Graham's number.
While true (although the upper bound has been drastically improved to 2↑↑↑6 last year), I don't think this says anything about its conceivability. After all, any number greater than or equal to the solution of a problem is an upper bound for it. So while the number that's the solution to this problem (does it have a name?) might be conceivable, it doesn't necessarily have a logical or intuitive relation to Graham's number.
But Graham's number is the upper bound on a solution to a problem that-- with a little bit of effort-- the layperson can easily understand. I don't really know how you can decide that a googolplex is conceivable just because the operation used is more familiar to people but at the same time discard Graham's number.
While true (although the upper bound has been drastically improved to 2↑↑↑6 last year), I don't think this says anything about its conceivability. After all, any number greater than or equal to the solution of a problem is an upper bound for it. So while the number that's the solution to this problem (does it have a name?) might be conceivable, it doesn't necessarily have a logical or intuitive relation to Graham's number.
Yeah, but I was trying to make a point about how we're defining conceivability. It seems in this context we're saying that a conceivable number is one that is "manageably small", not necessarily one that is easy to write down in some form. By the latter definition, a googolplex might be regarded as conceivable because it can be easily written down and also approximates the solution to a problem in statistical mechanics. My point is that a googolplex, regardless of how succinctly it can be defined, is still unfathomably unfathomably huge.
Wikipedia itself gives examples of how you can describe the magnitude of googolplex:
"Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes."
"A typical book can be printed with 106 zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore it requires 1094 such books to print all zeros of googolplex."
"if a person can write two digits per second, then writing a googolplex would take around about 1.51×1092 years, which is about 1.1×1082 times the accepted age of the universe."
Graham's number, however, has no such easily understood description because of its sheer magnitude.
Wikipedia itself gives examples of how you can describe the magnitude of googolplex:
"Written out in ordinary decimal notation, it is 1 followed by 10100 zeroes."
"A typical book can be printed with 106 zeros (around 400 pages with 50 lines per page and 50 zeros per line). Therefore it requires 1094 such books to print all zeros of googolplex."
"if a person can write two digits per second, then writing a googolplex would take around about 1.51×1092 years, which is about 1.1×1082 times the accepted age of the universe."
Graham's number, however, has no such easily understood description because of its sheer magnitude.
You're talking about the conceivability of writing down a googolplex, not the conceivability of the number itself. It's the difference between saying a million is easy to conceive because it has just six zeros versus actually imagining a million objects.
Also, if you can imagine 1094 books, I'm impressed...
No, a million is easy to conceive because you can say, for example, "10x10x10 meters of water weights a million kilograms".
The thing is, even if you consider those verbal descriptions of googolplex highly abstract, my point is that there just is no such description for Graham's number. You can't just say, for example, "writing down Graham's number would take 1010000000000 books", because that would be way too little.
No, a million is easy to conceive because you can say, for example, "10x10x10 meters of water weights a million kilograms".
The thing is, even if you consider those verbal descriptions of googolplex highly abstract, my point is that there just is no such description for Graham's number. You can't just say, for example, "writing down Graham's number would take 1010000000000 books", because that would be way too little.
Graham's number is irrelevant to the conversation at this point. We both agree that the number itself is inconceivable, the number of digits in it is inconceivable, the number of digits in the number of digits is inconceivable, and the number of times I would need to continue with this string of "inconceivables" is inconceivable. There is no argument.
You still haven't said anything about how to conceive of a googolplex itself, only descriptions of how one might write down the number. It's big. It defies all comprehension and even a googol is too big to truly understand in any meaningful sense. We cannot play some trick to relate it to our day to day lives, like saying, "It's such-and-such amount of water." You need to take the logarithm of it twice before the average person can comprehend it on some level.
even a googol is too big to truly understand in any meaningful sense. We cannot play some trick to relate it to our day to day lives, like saying, "It's such-and-such amount of water." You need to take the logarithm of it twice before the average person can comprehend it on some level.
The first logarithm of googol is 100. I'd say the average person can comprehend 100 quite well. It's also perfectly easy to say how much water it is to weigh googol kilograms. Quite simply, it's a 100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x
100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100x100 meter 50-hypercube full of water.
Current TAS:
[SNES] Jelly Boy
[NES] Street Fighter 2010
The size of googol is trivial to demonstrate. After all, it's 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.
The size of googol is trivial to demonstrate. After all, it's 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.
Yes, and in base half-a-googol it's ``2''. Your capacity to write down the definition of a number in a system of choice is a metric you yourself dismiss: after all, if it were one, the possibility of defining Graham's number would make it conceivable. Unless, of course, your notion of conceiving is limited strictly to that set of mathematical operations you're comfortable with and that are written down in base 10, in which case it's pointless arguing with you.
