[gwnn]

Fluffy, on 2016-April-15, 02:22, said:

So I have a button that rolls 3 dices into a transparent box, and I want to pick a single result. This has so many solutions, I could pick the result closer to me, but that would end up being subjective, so I would just roll them enough times untill there is a double/triple result, then I would pick that one.

The mod solution is better? If you think so you haven't played board games with people who consistently failed maths. And on top of that, it works for every kind of dice, not only 6-sided ones with numbers (as long as each sides are not duplciated).

EDIT: Sorry the mod solution works on any kind of numbered dice (*dumb*)

EDIT2: I saw the video and the guy says that here are 216 posibilities when rolling 3 dice... this is technically wrong as you can't distiguish 2,2,5 from 2,5,2 not 5,3,1 from 3,1,5. The right total is 56 I think, but each will have different probability.

Yea, he was a bit sloppy with his language. He's a professional mathematician so he probably knows that it's 56. Two dice have 21 combinations.

216 is "correct" in the sense that the 56 possibilities have either probability 1/216 (the triples), 3/216 (the ones of the form abb), or 6/216 (abc). You need to find a mapping of these 56 possibilities to the 21 possibilities (with likelihoods of 6/216 for doubles and 12/216 for the non-doubles) in a way that correctly reproduces the exact probabilities. I found a semi-convoluted one that I posted above but was wondering if there are other people who can find a prettier solution (the fact that there are lots of correct mappings but no obvious simple one is what makes the problem cool in my eyes). In the comment section I found another guy who has a solution that has a similar complexity and I prefer mine (for purely subjective reasons - they are probably equally good/bad).

[barmar]

Zelandakh, on 2016-April-14, 12:24, said:

You have 2 people and nothing else of any special note. I once watched a group of children playing diceless D&D on the train using a form of this method but it is also useful for picking a random number while avoiding the usual biases that the basic method involves.

It's kind of like playing rock-paper-scissors -- you can use the result to simulate a 3-sided die (player 1 wins = 1, player 2 wins = 2, tie = 3).

But this probably doesn't simulate an unbiased die very well, because anything involving humans picking things "randomly" runs into the problem that we're not very good at randomizing. So if the player who picks a finger to be 1 is more likely to select the index finger, and the player who selects a finger is more likely to select the ring finger, then 3 will be the result more often.

There's an online RPS game where the computer has an incredibly good track record, because the programmers know about our unconscious biases and the program exploits them. Actually, they don't know what our biases actually are, they just look at statistics of previous plays and assume that we're consistent, which we tend to be. But what we aren't is random!

[blackshoe]

gwnn, on 2016-April-14, 14:06, said:

Or you could easily consult Merriam-Webster:
http://www.merriam-w...dictionary/dice
Or Oxford Dictionaries:
http://www.oxforddic...on/english/dice
Before you say what is simply wrong. You could also find a bunch of articles written on the topic without any clear conclusion. In other sources die is indeed given as the singular, I know.

Using "dice" in the singular is unfortunately not wrong, but if you look closely at your links you'll find that your MW link lists "die" as a synonym for "dice (singular)" and Oxford provides a direct link to the noun "die" where that word is defined among other meanings as singular of "dice". So it's hardly a matter of "other sources".

Nero Wolfe might have equally well objected to "dice" as singular as he did to "contact" as a verb.

[barmar]

blackshoe, on 2016-April-15, 10:40, said:

Nero Wolfe might have equally well objected to "dice" as singular as he did to "contact" as a verb.

If people didn't use words in different ways from "standard", and have those changes catch on and become accepted, we'd still be speaking something like proto-Indo-European.

I expect that a very large portion of language that's currently considered "proper" was someone's pet peeve decades or centuries ago.

[kuhchung]

Thanks for the interesting problem gwnn. I have no idea since I'm terrible at math, but I am able to torture other people with the problem now as well.

[gwnn]

blackshoe, on 2016-April-15, 10:40, said:

Using "dice" in the singular is unfortunately not wrong, but if you look closely at your links you'll find that your MW link lists "die" as a synonym for "dice (singular)" and Oxford provides a direct link to the noun "die" where that word is defined among other meanings as singular of "dice". So it's hardly a matter of "other sources".

Nero Wolfe might have equally well objected to "dice" as singular as he did to "contact" as a verb.

Sorry, I was unclear about what I meant by the "other sources" part. I meant that there are sources which give both forms as possible/correct and some only give a singular "die." Still, if at least a few (reputable) sources give a form as correct, then it is not "simply wrong", which is all I set out to disprove. Of course, which one one uses is a matter of simple preference. I find "one die" grating and "one dice" slightly less grating but still awkward. So I would usually try to avoid talking about dice altogether, at least when there is a reasonable chance that I will need to talk about only one of them at some point of the conversation (and I can't get around the issue by using a construction such as the one in this sentence). Needless to say, starting a thread about rolling dice is one of the less wise things I can do in this regard.

[barmar]

"throw a dice" sounds more wrong to my ears than "throw a die".

[kenberg]

barmar, on 2016-April-15, 13:27, said:

"throw a dice" sounds more wrong to my ears than "throw a die".

The one where the correct usage still sounds wrong to me is "the data show". I usually convert it to "the set of data shows" or "the data points show".

If we look in
http://www.merriam-w...tionary/agendum
we find that agenda is the plural of agendum, but I am quite sure I have never heard anyone say "His agenda are".
So my agendum is to insist that anyone who corrects me when I say "the data shows" must forever after say "the agenda for this meeting are" or "the agendum for this meeting is".

Or we could all just agree that English is a concoction of several languages and does not really conform well to logic.

[gwnn]

Is anyone interested in the dice problem too? I'm not complaining about any hijacking since I am solely culpable for it. I just wanted to remind people that there's a reasonably cool (if useless) problem on the top of this page.

[kenberg]

gwnn, on 2016-April-15, 15:31, said:

Is anyone interested in the dice problem too? I'm not complaining about any hijacking since I am solely culpable for it. I just wanted to remind people that there's a reasonably cool (if useless) problem on the top of this page.

Yes, I fin it interesting. A couple of things:

a. Using three dice, you can in fact model two distinguishable dice. if the three dice come up either {1,1,2} or {2,2,1} then call this (r,g) =(1,2) while of the three dice come up {1,3,3} or (1,1,3} ca;; this (r,g)=(2,1). I haven't worked out any pattern, but clearly this can be done.

b. The idea of a "simple" matching is of interest, but hard to pin down. At firat I thought this could mean that we must use an algebraic expression and work modulo 6, but in fact I imagine any mathcing could be put in such a form, albeit maybe with many terms.

c. Generalize!


d. Spouse is getting impatient for dinner.

[gwnn]

You don't need to distinguish between (1,2) and (2,1), FWIW. The other solution I saw (the only other one in the video comments that I know works) has one of the rules as:

(a,b,b),(a,a,b),(6,a,b) -> (a,b), if a<>b and a,b<>6.

And then go on from there. Certainly it makes sense to have (1,2,2) correspond to (1,2), but it's a bit asymmetric.

I was thinking of defining a metric of similarity, for example the overlap between the three-dice result and the corresponding two-dice result. Then, maybe I could optimize this using a genetic algorithm? Maximum overlap is definitely not necessarily "easiest to explain", I know.

[kenberg]

gwnn, on 2016-April-15, 16:05, said:

You don't need to distinguish between (1,2) and (2,1), FWIW. The other solution I saw (the only other one in the video comments that I know works) has one of the rules as:

(a,b,b),(a,a,b),(6,a,b) -> (a,b), if a<>b and a,b<>6.

And then go on from there. Certainly it makes sense to have (1,2,2) correspond to (1,2), but it's a bit asymmetric.

I was thinking of defining a metric of similarity, for example the overlap between the three-dice result and the corresponding two-dice result. Then, maybe I could optimize this using a genetic algorithm? Maximum overlap is definitely not necessarily "easiest to explain", I know.

You do not need to distinguish between (1,2) and (2,1), but I find it interesting that you can. That is, three indistinguishable dice can be used to model the throw of two distinguishable dice. Obviously two indistinguishable dice can not be used to model two distinguishable dice. I have not thought about whether four indistinguishable dice can be used to model three distinguishable dice. I doubt it.

We all just take this where it leads us.

[nige1]

Thank you Gwnn for telling us about this intriguing problem and your ingenious solution.

It might be easier to tackle the general case i.e. using the throw of m identical dice to simulate the throw of n dice (distinguishable or not).

e.g. to simulate the throw of 1 die from m dice, you can use your mod 6 trick.

[hrothgar]

This is all very interesting, however, from a practical perspective, I would simple decide that I will roll the dice and then chose the left-most die as #1 and the second left-most die as #2...

If there is a "tie", I'll chose the bottom die before the top...

[Fluffy]

Csaba I suppose you mean the second problem. I felt that the only viable solution for me was to write down the 56 posibilities and make groups that got the same probabilities as 2 dice. In other words: brute force. Similar to the famous problem that finished with "the oldest plays piano", I see nothing mathematical or entertaining on using brute force.

I think your solution was good, and I don't expect the ideal solution too look much less complex

[gwnn]

hrothgar, on 2016-April-16, 05:31, said:

This is all very interesting, however, from a practical perspective, I would simple decide that I will roll the dice and then chose the left-most die as #1 and the second left-most die as #2...

If there is a "tie", I'll chose the bottom die before the top...

I think applying the first solution twice would be my solution. Of course, a sledgehammer might be even simpler. I definitely don't claim that my solution is practicable for late-night drinking/strip games (or both).

Fluffy said:

Csaba I suppose you mean the second problem. I felt that the only viable solution for me was to write down the 56 posibilities and make groups that got the same probabilities as 2 dice. In other words: brute force. Similar to the famous problem that finished with "the oldest plays piano", I see nothing mathematical or entertaining on using brute force.

I think your solution was good, and I don't expect the ideal solution too look much less complex

Of course my solution was also based on brute force. I started from realizing that ssB->ss can cover all "ss" tosses and went from there and filled up the space with as simple mappings as I could. This problem strongly reminded me of bridge system design so I thought others here might enjoy it. idk

[Vampyr]

barmar, on 2016-April-15, 13:27, said:

"throw a dice" sounds more wrong to my ears than "throw a die".

Yes, and perhaps it also depends on how often you roll dice. I play backgammon, and it is often important to distinguish between one die and two (or more) dice.

[blackshoe]

kenberg, on 2016-April-15, 15:17, said:

The one where the correct usage still sounds wrong to me is "the data show". I usually convert it to "the set of data shows" or "the data points show".

If we look in
http://www.merriam-w...tionary/agendum
we find that agenda is the plural of agendum, but I am quite sure I have never heard anyone say "His agenda are".
So my agendum is to insist that anyone who corrects me when I say "the data shows" must forever after say "the agenda for this meeting are" or "the agendum for this meeting is".

Or we could all just agree that English is a concoction of several languages and does not really conform well to logic.

Sorry, Ken, but I don't think "agenda" as a singular noun signifying a list of items (individually each an "agendum") is any more wrong that "die" is wrong as a singular for "dice" or "data" used in the singular is wrong.

"I rolled a die" and "the data are" both sound right to me. So, I confess, does "the agenda (meaning the list of items to be discussed in a meeting) is". That said, I have occasionally (mis)used "the data says". I don't recall ever saying "I rolled a dice" or "the agendum is" - the latter would refer to a single item on the agenda (list), but most people, I think, call them "agenda items", as do I.

[kenberg]

blackshoe, on 2016-April-16, 19:02, said:

Sorry, Ken, but I don't think "agenda" as a singular noun signifying a list of items (individually each an "agendum") is any more wrong that "die" is wrong as a singular for "dice" or "data" used in the singular is wrong.

"I rolled a die" and "the data are" both sound right to me. So, I confess, does "the agenda (meaning the list of items to be discussed in a meeting) is". That said, I have occasionally (mis)used "the data says". I don't recall ever saying "I rolled a dice" or "the agendum is" - the latter would refer to a single item on the agenda (list), but most people, I think, call them "agenda items", as do I.

I am fine with people using agenda as a singular, as long as they don't go ballistic if I inadvertently use data as singular. I try to remember it is plural, and I usually do, but I sometimes slip. The two cases seem equivalent. So I think people should get in a dither over both "the data shows" and "the agenda is" or they should get in a dither over neither. I prefer letting it be. I suppose it is forum and fora also, but I am not living in Ancient Rome.

And Caesar can say"The dice is cast" if he wants to, he is Caesar, but it sounds weird.

[Zelandakh]

Csaba, just in case you are still interested in solutions, here is mine. It is unfortunately not any simpler than the others.

Take the 3 numbers and order them from smallest to largest as abc.

For a triple, a=b=c -> 6:6
For a double, a=b or b=c -> a:c
For 3 different numbers a<>b and b<>c,
- if a=1 -> b:c
- if a=2 and b=3 -> 1:c-3 (or a+b+c-8)
- if a=2 and b>3 -> 1:b+c-5 (or a+b+c-7)
- if a>2 -> double a+b+c-10

It would be nice to have a simple formula for this last part, especially if it could keep the property that, roughly, a better 3-dice throw gives a better 2-dice result.

Intuitively, the general case of any number of dice ought to have a solution, since the number of possibilities is simply a factor greater and the single outcomes always sum to a whole factor. But proving that might be quite difficult - it would be a lot easier to disprove if someone can come up with a counter example.