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.

@WilcoVerhoef

7 ай бұрын

Thanks! I've been searching for an off-by-one error in my code :D

@hassanalihusseini1717

7 ай бұрын

Haha, hope the Dragon liked fried James. 🙂

@Corwin256

7 ай бұрын

It says James still works at Oxford in outreach. Is he doing this from within the dragon's belly?

@OxfordMathematicsPlus

7 ай бұрын

@@Corwin256 It is dark in here, but on the plus side I've got more time to think about maths now! ^James

@Bill_Woo

7 ай бұрын

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.

@Sopel997

7 ай бұрын

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

@avaraportti1873

7 ай бұрын

>don't want to be in a dungeon forever >choose a number that keeps you there forever

@Eagle3302PL

7 ай бұрын

@@Sopel997 Would it be achievable in some different number base?

@MindstabThrull

7 ай бұрын

Can't see the dragon for the TREE(3)?

@johnacetable7201

7 ай бұрын

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.

@vojtechstrnad1

7 ай бұрын

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.

@GreylanderTV

7 ай бұрын

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.

@vojtechstrnad1

7 ай бұрын

@@GreylanderTV No, any way of reaching a number is included in the calculation, even 1+1+1+1+...+1+1 = 21.

@vojtechstrnad1

7 ай бұрын

@@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.

@hakesho

7 ай бұрын

@@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.

@jschoete3430

7 ай бұрын

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?

@numberphile

7 ай бұрын

I like your thinking.

@stechuskaktus8318

7 ай бұрын

@@numberphile Then sqrt(90) would be triple evil, only counting decimal places

@mehill00

7 ай бұрын

Obviously 666 zeroes at that spot is the most evil.

@NeilABliss

7 ай бұрын

664 Neighbors of the beast.

@Omensan

7 ай бұрын

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

@nixfriarr

7 ай бұрын

1:43

@CombustibleL3mon

6 ай бұрын

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)

@tehdarkneswithin

7 ай бұрын

Even better, this is a (homogenous) linear recurrence (with constant coefficients) so has a closed form that you can compute in O(1)

@landsgevaer

6 ай бұрын

@@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...)

@narwhalergames

6 ай бұрын

@@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))

@XxZeldaxXXxLinkxX

6 ай бұрын

Based comment I've been grinding dp problems, when I close my eyes all I see is `if dp[i] >-1 return dp[i]`

If you can choose any number, just choose 0 and don't roll at all. Instant win.

@drenz1523

7 ай бұрын

choose -1 so that you would roll forever knowing that you will never reach it. bore the dragon until it succumbs and frees you

@alexhunter-meier

7 ай бұрын

The problem raises when it would demand at least one roll from you.

@claudetheclaudeqc6600

7 ай бұрын

@@drenz1523 that would be a instant lose, as it is exact. If above, you die.

@claudetheclaudeqc6600

7 ай бұрын

@@alexhunter-meier so it's result as a instant lost as well. Best would use X boom, guarantied win here!

@drenz1523

7 ай бұрын

@@claudetheclaudeqc6600 oh right... darn it!

Brady's intuition is amazing. Every time.

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.

See brilliant.org/numberphile for Brilliant and 20% off their premium service & 30-day trial (episode sponsor) James Munro is is Admissions and Outreach Coordinator for Maths at Oxford University

@godfreypigott

7 ай бұрын

You might want to pin your comment so that everyone gets to read it.

@picassodilly

7 ай бұрын

Came here to say this, so instead I’ll just bump the comment with a reply.

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!

@ashwynwadhawan7908

7 ай бұрын

Pi being evil is another reason to keep using pi. Tau is not evil and therefore boring.

@vasyan123

7 ай бұрын

Ten points to G̶r̶i̶f̶f̶i̶n̶d̶o̶r̶ Steve Mould

@dryued6874

7 ай бұрын

I'm OK with any additional reason to use Tau.

@glasswingbutterfly

7 ай бұрын

Would you rather eat pi or a tau (towel)?

@wewoweewoo

7 ай бұрын

But I have no umbra formas why would I go tau

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.

@therealax6

5 ай бұрын

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.

@YashvardhanMemoryTricks

5 ай бұрын

You are wrong.

Pi might be evil, but Numberphile is a great good.

Pi is evil?! Welp, looks like you gotta change your profile picture, Numberphile!

@physicsstudent3176

7 ай бұрын

😂

@imveryangryitsnotbutter

7 ай бұрын

Long live Tau

@fatcerberus

7 ай бұрын

@@imveryangryitsnotbutterTau is 2*pi so does that mean it’s twice as evil?

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.

@slo3337

7 ай бұрын

Ya but there is no 0 on a die

@phiefer3

7 ай бұрын

@@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).

@BinaryBolias

6 ай бұрын

@@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.

@Muhahahahaz

6 ай бұрын

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)

@therealax6

5 ай бұрын

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.

I love that you can screw around with numbers and get something meaningful out of sensless goofing around.

When the "many more sides die" was suggested, my mind immediately went to 600 - one of the 4D platonic solids.

Actually, 35 is the most likely for a 20-sided die, not 34.

@sncxyz

7 ай бұрын

Thanks for the sanity check. Just wrote code to solve this myself and it was giving me 35.

@Spoon_builds

7 ай бұрын

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.

@jamesmunro

7 ай бұрын

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

@asheep7797

7 ай бұрын

Thought I messed up the Desmos graph when I saw 35. 34 - 0.09751 35 - 0.09767

@ianstopher9111

7 ай бұрын

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.

Tried telling my maths teacher that maths was evil. She was unimpressed.

I like that under the digit definition of evil 666 itself is not an evil number

@ska4dragons

7 ай бұрын

It's a lucky number.

@JavierSalcedoC

7 ай бұрын

That's the best trick of evil

@BobStein

7 ай бұрын

The worst evildoer thinks they are righteous.

@alexhunter-meier

7 ай бұрын

​@@BobStein It's like humanity in general, it mostly does evil on this planet and thinks it's all right.

@claycon

7 ай бұрын

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.

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 😂

@gwynethjones3503

6 ай бұрын

I enjoyed differential equations in college… But had to use my fingers or scratch paper to add and subtract. 😂

More videos from Mr. Munro please! I like this guy!

@FedeDragon_

7 ай бұрын

yes he's great

@OxfordMathematicsPlus

6 ай бұрын

(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

Since (e-1)N leads to an approximation, just use √3N

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.

@TheGreatAtario

7 ай бұрын

*an N-sided die

@jamesmunro

7 ай бұрын

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

@hb1338

7 ай бұрын

@@TheGreatAtario Yes - the singular noun is die, and its' plural is dice. "The dice are loaded", not "the dice is loaded".

@tehdarkneswithin

7 ай бұрын

This is actually a fantastic observation, and could merit an entire video just exploring this topic itself.

@felipevasconcelos6736

6 ай бұрын

For the values for 1 to 20, you can think of P(n) = 0 for -19 = 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.

Fun fact: 666 is a mistranslation; the actual number is 616. Is pi still evil by the more accurate texts?

@PJandBethany

6 ай бұрын

Neat. Do you have a citation for that?

I heard the Wilhelm scream. You didn't slip it past us.

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!

I really like this James, he reminds me of classic numberphile

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

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,

@therealax6

5 ай бұрын

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.

oh crazy, James was my guide at a cambridge summer school back in 2014!

That dragon dice problem feels like convolution is involved, but I cant really put my finger on it

Putting this challenge in my D&D game!

@ianstopher9111

7 ай бұрын

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.

You summarize every signals intelligence intuition I've felt throughout my life very well

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

Thank you for using an accurate dragon cartoon

Really fun little problem and approach!

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.

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.

@jamesmunro

7 ай бұрын

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

@1ucasvb

7 ай бұрын

@@jamesmunroAh, so this is something you came up with yourself? That's cool. How did you approach the continuous case?

@yeoman588

6 ай бұрын

​@@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)))).

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.

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

Right on. Thanks for sharing.

Missed a great chance for a Matt Parker cameo with whatever highest number fair die he created a while ago.

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?

@byzatic8507

7 ай бұрын

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.

@thefidgetspinnerofdoom

7 ай бұрын

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"

@jakobr_

7 ай бұрын

That’s an interesting question because you want to pick two that are both likely but cover different sequences of rolls.

@mrjava66

7 ай бұрын

Right. P1 + (P2 | !P1). Interesting.

@jamesmunro

7 ай бұрын

This is an excellent follow-up question. ^James

okay but bold of you to assume I wouldn't want to stay with the dragon forever

1:43 obligatory Wilhelm scream

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.

I appreciate the Wilhelm scream.

A Parker Number is where the digits do not add up to ANY literary sum. Not 42, not 666, not 451...

What a great video for Spooktober! Thanks Brady

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

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.

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)

11:04 I find it interesting that the corresponding decimal place is 12 squared

@MuhammadFrazAslam

6 ай бұрын

Yes, In Analog Clock, Hour Needle passes 2 times from 12... And there are 1440 minutes in a Day..

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?

@jpdemer5

6 ай бұрын

Matt obviously would have gone with 34, and gotten eaten before he could answer this.

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.

"Zeros dont really get you closer to your rolling total, they just delay the inevitable." 🤯

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.

Good idea Sir

James: "I don't know how to make a nine-sided die" Most recent Curiosity Box: "hold my light-year of water"

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

Brady choosing 42 then there’s the Wilhelm’s scream during animation, welcome to nerdphile ;)

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.

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

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.

Amazing video! Btw, I'm in uni and just started learning matlab, how would we graph something like that in matlab? (if this is a stupid question pardon me👀)

@TheFrewah

5 ай бұрын

Two loops, an inner and an outer. The inner loop stops when your sum has reached a threshold value. The outer should be large enough so that matlab runs for a few hours. Keep track of the sums in arrays. Then divide by what the outer loop gives so you get something between 0 and 1. Now you can create a graph

Legend is still Continuing

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.

@leefisher6366

7 ай бұрын

Oh, if you're allowing an interval now, I choose [1,20] inclusive.

And that's why we should be using tau instead. Periodicity of the circle for the win!

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.

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?

@Tangwuji67

7 ай бұрын

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.

@kindlin

7 ай бұрын

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.

@NateSchoonoversAdventures

6 ай бұрын

That makes sense! I just missed that part. Thanks!@@Tangwuji67

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.

I wonder. On a smooth graph, if we're using N-sided dice, what's happening between N and N+1? Does it tend to infinity? Or it's smoothness just breaks somewhere between these numbers? (where exactly?) Or it just doesn't exist in that range?

@Sugarush-bc4wh

7 ай бұрын

The graph jumps down a bit (jump discontinuity), it does not tend to infinity. Here are the exact formulas if you want to graph it yourself, where T is the target number and N is the number of sides: P(T,N) = ((N^(-1)+1)^(T))/(N+1) for 1=

@Sugarush-bc4wh

7 ай бұрын

The break occurs exactly right after T=N

Short bit of (python) code to calculate and print the probabilities given an N=20 sided die: N, P = 20, [ ] for i in range(43): P.append((i == 0) + sum(P[i-k-1] for k in range(min(N, i))) / N) print(f"{i:2}:{P[-1] * 100:7.3f} %") Results in: 0:100.000 % 1: 5.000 % 2: 5.250 % 3: 5.513 % 4: 5.788 % 5: 6.078 % 6: 6.381 % 7: 6.700 % 8: 7.036 % 9: 7.387 % 10: 7.757 % 11: 8.144 % 12: 8.552 % 13: 8.979 % 14: 9.428 % 15: 9.900 % 16: 10.395 % 17: 10.914 % 18: 11.460 % 19: 12.033 % 20: 12.635 % 21: 8.266 % 22: 8.430 % 23: 8.589 % 24: 8.743 % 25: 8.890 % 26: 9.031 % 27: 9.163 % 28: 9.287 % 29: 9.399 % 30: 9.500 % 31: 9.587 % 32: 9.659 % 33: 9.714 % 34: 9.751 % 35: 9.767 % 36: 9.761 % 37: 9.729 % 38: 9.670 % 39: 9.580 % 40: 9.458 % 41: 9.299 % 42: 9.350 %

@eatpant1412

7 ай бұрын

Nice code.

Numberphile: Pi is evil Also, Numberphile (9 years ago): Pi is beautiful 😂

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?

@narwhalergames

6 ай бұрын

p=0.30013020833333337 when guess is 8 and your plan is to use: 8, 6, 10, 12, 20. (this is the optimum)

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?

This guy taught me chaos theory

It is worth noting that tau is not evil (with or without the initial 6). τ > π

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.

@ericbarr734

6 ай бұрын

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!

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.

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.

Another level of irrelevance is to consider whether the number of the beast was actually 616.

That's interesting. Intuitively I would have said, since the expected value of the dice is 10.5, you ought to go for a large value that is a multiple of that, so 84 or thereabouts. Something I noted - 10.5*3 is (rounded) 32, which is quite close to the actual ideal value 34. I wonder if that's coincidental or there's an underlying relationship.

@FightingTorque411

7 ай бұрын

Talking of rounded, 10.5 * π gets you even closer to 34...

@AdelaeR

7 ай бұрын

@@FightingTorque411 And the average of a 6-sided die is 3.5 and multiplying it by π is ... 11! Code cracked?

@kolkonut

7 ай бұрын

@@AdelaeR Fascinating. Could there be a relationship between e and pi in this context, with respect to the (e-1)N heuristic? If I could get my hands on a spreadsheet, I'd love to tabulate the difference between pi*E[N] and (e-1)N for a range of N. It may prove to be a very interesting graph.

@lowbudgetmaths

7 ай бұрын

The stated code is (e-1)N ~ 1.7N. So pi*(N+1)/2 ~ 1.6N + 1.6 which will be in the ballpark of the above for small N, but gradually deviate more and more. Similarly, finding m so that m(N+1)/2 ~ 1.7N will be hit or miss for different N.

@NoNameAtAll2

7 ай бұрын

(N+1)/2*3 = 1.5N + C so also close by coincedence

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?

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

more evidence for tau to replace pi

@PhilBoswell

7 ай бұрын

Are you sure that-for this definition of evil-tau is not also evil?

@heathrobertson2405

7 ай бұрын

@@PhilBoswell I haven’t check but I remain in hope

Gonna implement this into my next DnD game.

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.

I was hoping the right answer to the dragon's riddle was somehow Pi related :'(

If an sphere is an infinite sided dice, then would the number be the limit of (e-1)N as N approaches infinity?

"The fractional part of Pi is evil, maybe the 3 saves it." There's a theology joke in there somewhere.

With any number greater than 20 there will reach a point the difference between the running total and the target number is 20 or less. Since prior rolls don’t effect the next roll wouldn’t all of this go out the window when you hit this point?

@SgtSupaman

7 ай бұрын

Yes, your odds eventually come down to 1/20 on your last roll (though, as long as you need more than a 1, it's perhaps more accurate to say your odds match the starting slope on the graph he showed), but that doesn't discount the original calculation of what is the most likely to occur from 0. But yeah, once you are stepping through it, every roll is basically a repetition of the same game with a lower number that you didn't get to pick.

That e-1 is not an integer is the house (or dungeon) advantage.

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)

I'm so here for British Peter Parker teaching me about dice roll probabilities

can there be a dice whose sides are equally likely, but not the same shape?

@ianstopher9111

7 ай бұрын

An interesting question. I wonder if the probability then becomes about whether the die has to energy to roll further, i.e. overcome friction to tip over to the next side, so it would be perhaps about the angle between adjacent sides that plays a role in whether the die can rotate. In addition if the die has different sides then assuming equal density throughout then there will also be a lopsidedness as well. Hmm, I will stick with spherical cows for now.

Graham's number - Because you (or the dragon) would die of old age before hitting it.