Pi is Evil - Numberphile
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- เผยแพร่เมื่อ 4 ต.ค. 2023
- Featuring James Munro. See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor)
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Dragons, Dice and Evil Numbers.
James Munro is Admissions and Outreach Coordinator for Maths at Oxford University. More about outreach at Oxford can be found at www.maths.ox.ac.uk/outreach
James Munro: people.maths.ox.ac.uk/munro/
Correction: the optimal number for the original 20-sided die problem is actually 35 (this is slightly better than 34). James has been fed to the dragon as punishment for this error.
More Pi videos: bit.ly/PiNumberphile
The Most Evil Number on Numberphile: • The Most Evil Number (...
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Correction: the optimal number for the original 20-sided die problem is actually 35 (this is slightly better than 34). James has been fed to the dragon as punishment for this error.
Thanks! I've been searching for an off-by-one error in my code :D
Haha, hope the Dragon liked fried James. 🙂
It says James still works at Oxford in outreach. Is he doing this from within the dragon's belly?
@@Corwin256 It is dark in here, but on the plus side I've got more time to think about maths now! ^James
No graphic on screen for that?
The best number you can pick is Graham's Number. That way you can stall long enough for a rescue party.
it's actually kinda interesting, because with numbers this large you end up with the problem that you cannot inch towards such a number in this universe by adding small enough numbers. There's just not enough information storage space in the universe to represent all the intermediate steps.
edit. so, peculiarly, you can throw the die enough times to reach graham's number, but you'd never in this universe know whether you actually did or not
>don't want to be in a dungeon forever
>choose a number that keeps you there forever
@@Sopel997 Would it be achievable in some different number base?
Can't see the dragon for the TREE(3)?
Too bad the dragon has a time machine which defeats any time problem
What surprises me is that 21 is so unlikely even though it is exactly double the average, and it is the most likely to come up if you just have 2 throws.
If you have exactly 2 throws, that is. The extra chance for numbers 1 to 20 comes precisely from the option of stopping after the first throw.
You are correct to be surprised, as he seems to be calculating only the odds of hitting the number in a single throw after you get to within 20 of that number. He is not considering the chances of reaching the number in 2 or more additional throws once you get to within 20. Unless I missed something. So he is leaving out the possibility, for example of 1+1+1+1......+1+1 = 21.
@@GreylanderTV No, any way of reaching a number is included in the calculation, even 1+1+1+1+...+1+1 = 21.
@@GreylanderTV The recurrence relation for 21 includes the probability that you were previously at 20 and rolled one, which already includes the probability that you were at 19 and rolled one, etc.
@@GreylanderTV When he talks about "recurrence relations" he means that the computer will take his 1-roll equation and keep iterating it for an arbitrarily large number of rolls. This is why the other ways of reaching 21(or anything else) are included. If you didn't know this, then I get why it looked like he wasn't considering large numbers of rolls.
Brady you are such an amazing interviewer. Every single time I am in awe of your abilities.
Yes I didn't get his intuition about it being most likely to be in between 20 and 40, but it turned out to be correct!
Is a number even more evil if it hits 666 twice because there is a zero in that spot?
I like your thinking.
@@numberphile Then sqrt(90) would be triple evil, only counting decimal places
Obviously 666 zeroes at that spot is the most evil.
664 Neighbors of the beast.
I thought along the same lines - How many numbers are "double evil" where summing the whole as well as decimals ends up evil where the Nth decimal position number arriving at 666 is the same number as the whole?
I bet the editor had real fun with that wilhelm scream lol
1:43
I can't believe I missed that! It was quiet in my defence though 😂
This problem actually is a fantastic application of dynamic programming. If you do it without it, the time complexity of your algorithm is O(20^n). With DP it is O(n)
Even better, this is a (homogenous) linear recurrence (with constant coefficients) so has a closed form that you can compute in O(1)
@@tehdarkneswithin I guess that requires finding the 20 exact roots of a 20th order polynomial first?
Please show us the way... 😉
(And it seems to assume we can do exponentiation in O(1), which I kind of doubt when we want to be exact...)
@@tehdarkneswithinwell. i was thinking of a different problem, the finding of the max for any kind of die. which was O(s^3). Yes finding the answer to your problem however is probably like O(s*log(n))
Based comment I've been grinding dp problems, when I close my eyes all I see is `if dp[i] >-1 return dp[i]`
I've never seen this James Munro guy but please use him more! He has such a good, conversable tone, you can hear the excitement in his voice for maths as he speaks and he has the ability - seemingly - to speak to a range of ages and abilities and foster an enjoyment in maths. In fact this guy's probably what Rishi wants from all his teachers lol.
Great video Brady. Interesting subject topic. I've literally just got home from a tutoring session with a student regarding probability so this was a nice continuation from that for me.
Brady's intuition is amazing. Every time.
If you can choose any number, just choose 0 and don't roll at all. Instant win.
choose -1 so that you would roll forever knowing that you will never reach it. bore the dragon until it succumbs and frees you
The problem raises when it would demand at least one roll from you.
@@drenz1523 that would be a instant lose, as it is exact. If above, you die.
@@alexhunter-meier so it's result as a instant lost as well.
Best would use X
boom, guarantied win here!
@@claudetheclaudeqc6600 oh right... darn it!
Pi might be evil, but Numberphile is a great good.
I love that you can screw around with numbers and get something meaningful out of sensless goofing around.
Pi is evil?! Welp, looks like you gotta change your profile picture, Numberphile!
😂
Long live Tau
@@imveryangryitsnotbutterTau is 2*pi so does that mean it’s twice as evil?
Update: I have verified the first 3000 values for N. Was able to compute each result in O(N^2) cause of knowing what place to start looking and an improved algorithm that does not compute things based on number of rolls. The answers to the question of what the best guess > N is seems to follow f(N) = round((e - 1)N + 4 - pi) for N=2...3000 EXCEPT N=232 and N=1233 both in which case the actual answer is f(N)+1 but thats not something im completely sure about since the difference in probability for N=232 is only 6.182648143449043e-12 and for N=1233 is a meager 5.937702586555904e-13.
Edit: I am sure now. have tried with 112 bit percision floating point numbers.
The issue with picking the closest value to the peak is that doing that only works for specific kinds of functions. (They at least require that, around a peak x, f(x + k) = f(x - k).) It doesn't work for generalised functions, but if the function can be approximated by one that fulfills this property, it will usually get you the right answer. Any quadratic function meets this rule, so if the function has a very small third derivative, you can get away with it.
You are wrong.
The calculation for the optimal dragon dungeon number can use pi too. Instead of (e-1)N, you can do the average of the die * pi to get close to the optimal guess. 6 sided die average is the sum of both sides divided by two, so 3.5, and 3.5 * pi = 11. Same for 20 sided die, but 10.5 * pi = 33, not 34
Interesting that the 0 is ignored in the average of the digits in pi. If you include it to get an average digit of 4.5, then the value 1/4.5 represents the expected number of times that a specific very high total will be hit, since it can be more than once if one or more zeros occur immediately after hitting the total.
Ya but there is no 0 on a die
@@slo3337 Any number can appear on any die.
In fact, most 10-sided die are numberd from 0-9 (as well as some numbered from 00 to 90, that when paired with a 0-9 allow you to roll 0-99 for percentages). This would also perfectly simulate rolling a 9-sided die in this application (since we only care about the sum, not the number or rolls).
@@phiefer3 Well, in a literal sense, the only numbers which can appear on a die... are the numbers which are marked on the faces of said die.
A way to have a guaranteed victory against the video's dragon is by putting your chosen number (or a factor of it) on all the faces of the dragon's die.
If you want the probability that a given number will be hit, then zero must be ignored
But if you want the expected number of digits, then zero should be included
(He was only doing the former, not the latter)
Remember the recurrence relation? p(k) = 1/n p(k - 1) + 1/n p(k - 2) + ... + 1/n p(k - n).
(In the digits example, n = 9.)
If we allow zero, the probability becomes 1/(n + 1) (because there's now an additional possible outcome), and you get the extra addend for zero:
p(k) = 1/(n + 1) p(k - 0) + 1/(n + 1) p(k - 1) + ... + 1/(n + 1) p(k - n).
But p(k - 0) is obviously p(k), so you can move that term to the other side, and you get p(k) - 1/(n + 1) p(k) = n/(n + 1) p(k) on the left-hand side, meaning you have to divide all terms by n/(n + 1) - which gets you back to the original equation without zero.
Pi being evil is another reason to use Tau instead - It reaches 663 on the 139th digit and 668 on the 140th, nicely skipping over 666 as any well behaved number should do!
Pi being evil is another reason to keep using pi. Tau is not evil and therefore boring.
Ten points to G̶r̶i̶f̶f̶i̶n̶d̶o̶r̶ Steve Mould
I'm OK with any additional reason to use Tau.
Would you rather eat pi or a tau (towel)?
But I have no umbra formas why would I go tau
When the "many more sides die" was suggested, my mind immediately went to 600 - one of the 4D platonic solids.
I like that under the digit definition of evil 666 itself is not an evil number
It's a lucky number.
That's the best trick of evil
The worst evildoer thinks they are righteous.
@@BobStein It's like humanity in general, it mostly does evil on this planet and thinks it's all right.
The Biblical reference relates 666 to “the number of the beast (or antichrist).” The Antichrist mimics the true Christ with stolen power & knowledge. So naturally the original 666 from the true Christ is not evil.
More videos from Mr. Munro please! I like this guy!
yes he's great
(waves) Hi! I've made about 70 episodes of an online maths club over on this channel, and I'm doing a bunch of livestreams on maths admissions test questions. ^James
Watching hardcore mathmeticians trying to do simple mental arithmatic is always amusing, a friend is doing a phd is maths and we always find it funny when he stuggles to work out his portion of the bill 😂
I enjoyed differential equations in college… But had to use my fingers or scratch paper to add and subtract. 😂
The recursive rule can be seen as a Infinite Response Filter that just average the previous N values for a N sided dice. So it's a low-pass filter hence the smoothing effect.
*an N-sided die
I'm so glad that someone picked up on the smoothing effect! This was not the right video to talk about how there's a jump discontinuity then a jump in the first derivative (in the continuous version), but it's the sort of thing I see when I look at the graph :) ^James
@@TheGreatAtario Yes - the singular noun is die, and its' plural is dice. "The dice are loaded", not "the dice is loaded".
This is actually a fantastic observation, and could merit an entire video just exploring this topic itself.
For the values for 1 to 20, you can think of P(n) = 0 for -19 =< x =< -1 and P(0) = 1, which makes sense conceptually, since you always start with 0, and you never have negative numbers.
It doesn’t make sense if you keep extending it backwards, though: you get that |P(n)| = 20 for n in [-39, 20], alternating in sign. And then you get P(-40) = 800.
I really like this James, he reminds me of classic numberphile
oh crazy, James was my guide at a cambridge summer school back in 2014!
Putting this challenge in my D&D game!
That is a bit meta. The dragon in the game is asking the characters to choose numbers for dice rolls which the players then perform on behalf of the characters.
I heard the Wilhelm scream. You didn't slip it past us.
Tried telling my maths teacher that maths was evil. She was unimpressed.
You summarize every signals intelligence intuition I've felt throughout my life very well
Great puzzle! Tried to solve it in my head before firing up python, and the incorrect heuristic I got was roughly (2 - 1/e)*N. I figured out the probabilities for 1 through 20 to be P(x) = p*(1+p)^(x-1), where p = 1/20, which are exponentially increasing. Then P(21) would be the average of P(1) through P(20), and so on. The approximation I then made was that the maximum probability among P(21) through P(40) would be the average starting from the first number between 1 and 20 that had an above-average probability (i.e. more than P(21)), and if you work through the math on that, this would give an optimal value of N + log(e-1)/log(1+p), in the limit as N becomes large; and using series expansions this is roughly (2 - 1/e)*N. This is an approximation because the probabilities P(21) through P(40) are also (initially) increasing, so instead of finding the first x such that P(x) > P(21), we should find the first x such that P(x) > P(x+20). But that seemed too complicated to think about.
Pleasantly surprised to see the actual correct heuristic also involves e, and to be more specific, a simpler expression of e that lies between 1 and 2!
That dragon dice problem feels like convolution is involved, but I cant really put my finger on it
Actually, 35 is the most likely for a 20-sided die, not 34.
Thanks for the sanity check. Just wrote code to solve this myself and it was giving me 35.
Seeing the same by just rolling a lot in python:
number of success for 32: 96891
number of success for 33: 97156
number of success for 34: 97212
number of success for 35: 97605
number of success for 36: 97127
1 million rolls each time.
Ah I wondered if anyone would spot this. Sorry - I have no idea why I said 34 on the day. In my defense, 34 and 35 have very very similar probabilities. ^James
Thought I messed up the Desmos graph when I saw 35.
34 - 0.09751
35 - 0.09767
Perhaps because (e-1)*20 is closer to 34 than 35 the intuition would be that 34 is slightly better than 35. However, a bit of experimentation shows that 35 is favoured.
Excellent video. I don't believe I've seen a video with James before, but that was really an excellent topic, and made entertaining! Would live to see more. My first intuition was that this had to do with partitions, but it doesn't seem needed (or maybe you can use them but it's overkill for this particular topic as you can use more traditional methods?).
Cheers,
I'm pretty sure that expanding that recurrence to a non-recurring version requires solving partitions along the way, but that's just my intuition.
Fun fact: 666 is a mistranslation; the actual number is 616. Is pi still evil by the more accurate texts?
Neat. Do you have a citation for that?
Really fun little problem and approach!
What a great video for Spooktober! Thanks Brady
It got cute there at the end. Your intuition at the start was so good! Exactly the right reasoning that led you to the right conclusion and you got there quickly while I was still pondering. Now I'll never know if my intuition was gonna be that good! haha cheers
Missed a great chance for a Matt Parker cameo with whatever highest number fair die he created a while ago.
Could you please include some reference links in the description so people who are interested in these problems can look more into them? I'm interested in the continuous generalization he mentioned, but I don't know how to search for this problem or the related work on it.
Let me know if you find anything - I'm the person in the video and I don't have a reference for this! Just a small bit of maths I did on the back of an envelope. Might have to write it up now it's on numberphile... ^James
@@jamesmunroAh, so this is something you came up with yourself? That's cool. How did you approach the continuous case?
@@jamesmunro I figured out continuous functions for
(1) the segment for S between 0 and n: P_1(S) = (1 / n + 1)^S / (n + 1), with a maximum probability at the point (n, (n + 1)^(n - 1) / n^n), and
(2) the segment for S between n and 2 n: P_2(S) = n^(-S) (n + 1)^(S - 1) - S n^(n - S) (n + 1)^(S - n - 2), with a maximum probability at the point ((n + 1)^(n + 1) / n^n + 1 / ln(n / (n + 1)), -(n / (n + 1))^(n - n^(-n) (n + 1)^(n + 1)) / (e (n + 1)^2 ln(n / (n + 1)))).
Thank you for using an accurate dragon cartoon
Not discussed in the video, but a nice result anyway: the probability of hitting N with an N-sided die approaches e/(N+3/2) for large N.
I appreciate the Wilhelm scream.
Re: Evil I tried doing my own and wasn't getting 666, but then saw that you stuck a note in the animation about ignoring the leading number.
Once I started from the decimal, we get matching results,. I went position by position for the first 200 are here are the counts of different end points (where I stopped adding up digits when adding the next one puts the total over 666):
666: 45
665: 42
663: 25
664: 24
662: 20
660: 17
661: 15
659: 10
658: 2
Nice problem for a computer simulation. You have to be careful when you simulate a dice because what you get may not be evenly divisible by the number of sodes. The rng I use fills an integer with random bits which doesn’t work well if you want to simulate a dice with, say, 17 sides. So you have to discard some of the values close to the max value.
If the dragon let you choose 2 target numbers upfront, what would the best choices be for a 20-sided die? Will they maybe be 34 & 35, or will they be spread out?
That's a really good question. I thought it would obviously be 34 and 35 but the more I think about it the more I'm unsure.
My intuition tells me that since the average roll of a d20 is 10.5, choosing 33 and 34 is the best option. To explain why spread out numbers wouldn't work, imagine the dragon gave you 10 numbers to choose. You could choose any 10 random numbers, or you could choose 10 numbers around 34. Since we know from this video that 34 is the best choice, this gives us 4/5 numbers to select as a sort of "early exit", and the other 5/4 numbers as a "late exit", essentially allowing our number to "fall through" a "bigger hole"
That’s an interesting question because you want to pick two that are both likely but cover different sequences of rolls.
Right. P1 + (P2 | !P1). Interesting.
This is an excellent follow-up question. ^James
Combinatorics comes to mind. Partitions in particular. With the added complexity of a limited number of faces in the dice. The discontinuities are borne in the denominator I believe…
(There’s a scene about these partitions in the Ramanujen movie)
There is a fragrance called Pi, by Givenchy (and created by one of the most prolific and successful perfumers: Alberto Morillas.) Pi smells really great, warm sweet, suitable for cold weather season, so this the best time to wear it in the northern hemisphere . I'd encourage all math/Pi enthusiasts to check it out.
Since (e-1)N leads to an approximation, just use √3N
a formula would really be best for comprehension, extension, and application.this is A start: with r rolls of an s sided dice the probability that the rolls' result sum above total t is given by ... this would be very helpful on a type on functor category scheme i'm looking into.
LOL the wilhelm scream at 1:43
okay but bold of you to assume I wouldn't want to stay with the dragon forever
James: "I don't know how to make a nine-sided die"
Most recent Curiosity Box: "hold my light-year of water"
12:35 You can make an "any-sided" die by taking a pointy prism and create as many faces on the side as you need. Within reason, from 3 to 10 works fine.
12:34 The easiest way to make a 9-sided die is with a 9-sided prism and doming the ends so it can't land on the nonagonal faces.
Right on. Thanks for sharing.
1:43 obligatory Wilhelm scream
I'd pick something like 9239328574395749 and leave once the dragon gets bored.
Another level of irrelevance is to consider whether the number of the beast was actually 616.
its interesting to hear that Brady's intuition was something that had to be small because my mind immediately went that extremely large numbers could be better because there are more and more ways to reach that number. My intuition was telling me that it should be a very large multiple of 21 since it would be a multiple of the average roll (10.5) of a 20 sided die
This is a cool bit of maths, I will try to remember it the next time I am trapped in a dungeon with an angry dragon and a dice.
If you had a full set of dice (D6 D8 D10 D12 D20) and you can choose a different one for each roll, what number is best to pick then? Is there a strategy? What if to have to plan the sequence of rolls before you start?
p=0.30013020833333337 when guess is 8 and your plan is to use: 8, 6, 10, 12, 20. (this is the optimum)
I'd like to hear Matt Parker's position on whether the leading digit should be included in the evil check. I'd also be curious how the check is done with 10 and numbers larger than 10. Is everything before the decimal place tossed out, or just the first digit?
Matt obviously would have gone with 34, and gotten eaten before he could answer this.
You’re not trying to get to 21, you’re trying to beat the deal while not going over 21.
Brady choosing 42 then there’s the Wilhelm’s scream during animation, welcome to nerdphile ;)
A Parker Number is where the digits do not add up to ANY literary sum. Not 42, not 666, not 451...
What if you allow 1-20, but disallow “wins” on the first roll? (So it always has to be a proper “sum” of 2 or more numbers)
Player 1 has a standard 6 sided die, sides labelled 1-6. Player 2 has a unique 6 sided die, with 5 sides labelled "0" and 1 side labelled "21" (Sum:1-6). The goal of the game is to reach a number, "x", or exceed that number, in the least rolls. What dice should you pick? Does it matter? Does the value of "x" have an impact on the dice you should choose?
Does anyone know the name of that problem? I would like to look at written math about that.
It should be possible to envisage or even draw, and therefore construct, a fair n-sided dice (not die!), with equally sized faces, even though there's no classically named corresponding solid. Or am I wrong?
"Zeros dont really get you closer to your rolling total, they just delay the inevitable." 🤯
What's even better than doing the math is just refusing the dragon's game. If he then tries to keep you there, it's wrongful imprisonment.
Really cool! I love it! I am going to work this into a game for sure! I don't understand that initial probability spike. How is it more likely to roll a 20 on a d20 than a 1?
That's because you can hit 20 with 1 throw (20), but if you miss it you can still get 20 with 2 throws (1 + 19, 2 + 18, etc), but if you miss it and if you are still below 20 you can get 20 with 3 throws, etc. Whereas to get 1 you only have one throw.
If you notice the size of the jump in that graph, it's exactly 0.5, the initial value for rolling 1, which is equal for all numbers 1 to 20. So, essentially there is zero ways to add numbers to get 1, more ways to get each subsequent number, so the probably increases and increases, and at 20 the extra 0.5 suddenly DROPS OFF, and now you have a curve with no discontinuity, and just an extra 0.5 bonus for the initial 1 to 20.
That makes sense! I just missed that part. Thanks!@@Tangwuji67
Great Video! I stumbled over two things:
5:50 What does he mean by "You have to make 21 in two rolls"? That the highest probability is achieved with two rolls? Because of course there is also 5+5+11, 5+5+5+6, 21*1 etc.
11:30 He is talking about the average one digit makes in the summation of the digits, shouldn't zero be included here? Zero occurs with the same likeliness in Pi as all other digits, so the average should be 0+1+2+3+4+5+6+7+8+9/10 (Mind the divided by 10 instead of 9!). In effect, only 1/4.5 = 22.23 % of numbers are 3v1l, on average.
is it possible to get python code that show that 35 is the best sum to chooses in dice with 20 faces .
my code show different answer
11:04 I find it interesting that the corresponding decimal place is 12 squared
Yes, In Analog Clock, Hour Needle passes 2 times from 12... And there are 1440 minutes in a Day..
So what's the oeis number for the sequence of optimal picks?
Good idea Sir
I'm gonna call this rule34 of dice 😈
I really like the animation!!
Wow! I estimated the best number would be either 22 or 33. Based on what I know from backgammon, the average roll would be approximately half the highest possible roll plus one (11). Then look at the multiples of 11 higher than 20 and less than 40.
"The fractional part of Pi is evil, maybe the 3 saves it."
There's a theology joke in there somewhere.
This guy taught me chaos theory
Not a mathematician, and no experience with advance calculation. What occurred to me was adding the integers from 1 to 20. 1+2+3+4+......20 and that got me 210. 210 then halved to get the average of 105, which when divided by 3 = 35(also noticed the 10.5 average you mention is a factor of 105{10.5x10}). So out of curiosity : Total of integers on all faces/sides of die lets call "T" divided by 2 to get T/2 then divided by 3, something like (T/2)/3? The question is does this hold up on a greater sided die and or even on a twenty sided die with random integers?
Numberphile: Pi is evil
Also, Numberphile (9 years ago): Pi is beautiful 😂
The Wilhelm scream is a nice easter egg.
Wait... i have a question. if pi doesnt repeat a number more than 3 times so why isnt the shape of a circle random?
My initial guess was 3 times the average of the faces, which would have rounded up to 32 for the 20-sided die and 11 for the 6-sided die. And now I'm wondering how well that scales up with the number of faces.
3 times the average, where the average is (N+1)/2. So 3*(N+1)/2. Which expands to 1.5N + 1.5
They gave the exact answer as (e-1)*N, which is ~1.7N. Or 1.5N + 0.2N
So your approximation works for values where 0.2N is about 1.5. When you get bigger N your approximation will deviate more and more. But it's not bad for a quick guess!
I'm curious. What if you had M number of dice that have D faces? So what is the probably of landing on a value greater than N*D?
There was only a 20% shot for pi to be evil, and yet it is.
How can the recurrence relation for p_100 (at 4:20) be possibly correct beyond p_99? Numbers between 2 and 20 all have a different number of partitionings, as we Indeed later see, so their probabilities are not equal.
If an sphere is an infinite sided dice, then would the number be the limit of (e-1)N as N approaches infinity?
What is the smallest number? As in Graham's Number or TREE() for the largest number?
The definition of evil number is corrected in the end of the video. The typical definition (by Pegg and Lomont) is that digits of the fractional part accumulate to 666 at some position (here the non-fractional digits, that is 3 for pi, was included). e, by the way is not evil by this definition.
Update: I have checked the first 10000 values of N. g(N) = round((e - 1)N + q) where 0.8591145934369706 < q < 0.8592248809409284 which is a very narrow interval of just 0.00011. There are 8 values of g(N) outside the formula round((e - 1)N + 4 - pi) which disproves my earlier thought! i wonder if this constant, q, has a name?
Surely you just pick a humongous number - so you have to roll the dice (essentially) foreveer? Or at least: Long enough for the dragon to get bored, or you to escape...
Mathematicians: here's the most statistically probable number you can get
Me: the dragon never said I had to say what the number was out loud, I'll just the roll the dice 3 times and say the sum of the 3 rolls is the number I've chosen
as long as the dragon is waiting, you get to live. the answer is bigger than the number of rolls you can make for the rest of your natural life
And that's why we should be using tau instead. Periodicity of the circle for the win!
I'm seeing the long term average strategy for huge values... sort of... however:
You will always have a last roll - which either hits your value or exceeds it. (If it does neither, it isn't your last roll). When you are thus within range, there is ALWAYS a 1 in 20 chance of getting the right number. To put it another way, there is never a situation where more than one value on the d20 will end the game in a victory... so... all numbers are equal. Please explain how this is wrong, since it clearly seems to be.
Oh, if you're allowing an interval now, I choose [1,20] inclusive.
Legend is still Continuing
"because the digits of pi are evil, I vote we cut them off and just make pi = 3"
- engineers probably