Talk:Nimber

message: Pleas for the love of pete one of you super smart peole make a youtube video... some of us cant even read and there is literally nothing relevant that come up on yt when searching nimber. I'm sorry that my life has turned out the way it has but please somebody....
68.13.124.209 (talk)  —Preceding undated comment added 04:30, 2 August 2017 (UTC)
thread_title: I just can't

message: I think it would be good to include addition and multiplication tables for the range 0-15.  I've done the computation but I'm not sure how to format it.  (I'm also not sure whether decimal, hex, or binary would be most illustrative.  Thoughts?)  75.36.182.157 00:56, 15 April 2007 (UTC)
thread_title:  Tables? 

message: The sources cited and those I have read are set up so that nimbers are not a countable set, but rather are a proper class, larger than any set.  Various subclasses of nimbers are interesting, including both a very small subclass which happens to be a set and happens not to be algebraically closed, and also another subclass which happens to be a set and an algebraically closed field.  The former was presumably intended by 195.69.84.154, but they also deleted the latter.
It is important to cite sources for wikipedia.  Note that the material challenged by 195.69.84.154 is taken fairly directly from the sources, but the edits by 195.69.84.154 are unsourced and contradict the given sources.
It would not be a problem to include a new source that defines the nimbers to be one of these tiny sets that 195.69.84.154 is probably talking about, and discuss its properties, but this should not interfere with talking about the larger sets and classes of nimbers already discussed in the article and the current sources. JackSchmidt (talk) 17:24, 5 December 2007 (UTC)
replies: text: Rereading the online source, it appears that perhaps only a sentence or two is dedicated to the nimbers.  Almost everything else is background or explanation of the surreal numbers.  Someone might want to check if the reference should be included there, but there are already nice references.  At any rate, despite nimbers only getting a quick mention in the online reference, it definitely indicates that they form a proper class. JackSchmidt (talk) 17:53, 7 December 2007 (UTC)
thread_title: Size of the nimbers and revert

message: I have some doubts about algebraically closedness
of the nimbers. Of course, they have all roots of irreducible polynomials of degree 2^k... But, by the way, what is the root of x^3+x+1??? 195.69.84.154 (talk) 13:18, 26 February 2008 (UTC)
replies: text: Well, it's more than ω and less than \omega^{\omega^\omega}.  There's no reason the nim-product of two infinite ordinals can't be finite.  But if you want to understand how to solve equations like this in general I'm afraid you have to read On Numbers And Games.  18.21.0.86 (talk) 22:53, 3 May 2008 (UTC)

text: I once calculated that \omega^2 + 3 \omega is a root of the polynomial x^3+x+1.  I just checked, and it seems right.  However, I  don't have a source to justify addition to the article.  Perhaps, it's insignificant enough not to require a source.  As hint to the verification, use the fact that \omega^3 = 2. DRLB (talk) 16:01, 28 October 2008 (UTC)
thread_title:  On algebraically closedness 

message: 131.215.167.232 changed "nim-addition" (resp. "nim-multiplication") to "nim-dition" (resp. "nim-tiplication").  I've reverted that, as the former spellings have some Google hits including the original books by Conway et al, while the latter spellings seem to have no Web presence other than this page itself.  Joule36e5 (talk) 04:45, 17 November 2008 (UTC)
thread_title:  Terminology 

message: Does anyone know if there exists a game-theoretic interpretation of nimber multiplication? I've been scouring the net for some time now, and there are a few pages that mention that some such interpretation exists, but none of them actually demonstrate it.  —Preceding unsigned comment added by 83.10.99.117 (talk) 23:59, 13 March 2009 (UTC)
replies: text: I was wondering that myself.  There must be some reason that nimber multiplication is defined as it is, whether related to nim or to game theory more generally.  I know the significance of addition to nim – under "normal play", a position is winning for the player who has just moved iff the nim-sum of heap sizes is zero.  The article ought to state this, but I'm not sure how best to fit it into the section as it stands.  But in any case, what about multiplication? — Smjg (talk) 00:11, 5 January 2014 (UTC)
thread_title:  Nimber multiplication vs Game Thoery 

message: To quote "This subset is closed under both operations, since 16 is of the form 2^2^n."
If addition of two nimbers is indeed just XOR, then 16 being of the form 2^n is entirely sufficient.  This can also be seen in the table for n=0,1,2,3,4 ie. 2^n = 1,2,4,8,16. Of course, that is not true for multiplication. MaZe Pallas (talk) 05:46, 7 March 2010 (UTC)
replies: text: I think you answered your own question.  The quote says both operations, and nimber multiplication is only closed within ordinals taking the form 2^2^n (as mentioned in article). TricksterWolf (talk) 19:15, 7 July 2013 (UTC)
thread_title:  Why 2^2^n? 

message: The non prose starts of with the recursive definition of nim addition, but I as a reader have no clue what a nimber is, or how to represent it. Also the recursive definition assumes the reader knows what the primes (') mean applied to nimbers...  — Preceding unsigned comment added by 83.134.138.194 (talk) 21:53, 3 May 2012 (UTC)
replies: text: A nimber is an ordinal number with two new operations defined on it (nimber addition and nimber multiplication).  Nimber addition on two nimbers is the bitwise XOR of the nimbers; multiplication is more complex to define.  Finite nimbers are simply the natural numbers (0, 1, 2, 3, etc.).  See ordinal number for more detail on transfinite ordinals.  TricksterWolf (talk) 18:56, 7 July 2013 (UTC)
thread_title:  Formal (non-prose) definition of nimber please? 

message: Nimbers are a really neat mathematical concept and there's a lot of fun things you can do with them from a recreational-math standpoint. But the article doesn't convey any of the wonder of nimbers and the useful bits are hidden under deep mathematical symbiology which most people might not even be equipped to unpack, which is a shame, because nimbers are really easy to understand if you explain them in plain English. For crying out loud, the article doesn't even link to the article on Nim!
I wrote this to explain nimbers to friends and think it might be useful to integrate into this article, perhaps rewritten in a more encyclopedic fashion:
There's a game called Nim. In Nim, you have a certain number of piles of objects, and a turn consists of removing some number of objects from a pile. You win if you remove the last object. So you might have a Nim game where you have a pile of 2 objects and a pile of 3 objects. Player A takes an object from the second pile; now you have two piles of 2 objects. Player B takes two objects from the first pile, and then player A wins the game by taking both objects from the second pile.
Nim is of mathematical interest because a wide variety of simple games with specific traits are mathematically equivalent to games of Nim. These games are called impartial games because each player's available moves are identical, and the only difference is whose turn it is to play.
What makes Nim particularly interesting from an analytical standpoint is that you can "add" games of Nim. For instance, let's take the game described above with a pile of 2 and a pile of 3. This is, in itself, two games of Nim - it's a game of Nim with a single pile of 2 added to a game of Nim with a single pile of 3. If we add the "nimbers" for 2 and 3, we should get the nimber for the combined game.
Let's say we're looking at piles of 2, 5, and 11 and want to find the nimber for the combined game. Here's how we do that.
First, we turn the numbers into binary numbers:
0010
0101
1011
Then, we count the number of 1s in each column. If it's odd, that column's value is a 1 in the resulting nimber. If it's even, that column's value is a 0. There's one 1 in the leftmost column, one 1 in the next columnm two 1s in the next column, and two in the last column. So the nimber we get is 1100 (the binary number), which is equivalent to 12. Note that if you add two games with the same nimber, you get a value of 0, because each column will either have two 1s or zero 1s.
You can use nimbers to identify whether the first or second player can force a win in a game. If the nimber of a game is 0, then the first player loses given optimal play; if it's any other value, then the first player wins, and their optimal play is to change the game state to make the nimber 0. 24.136.1.234 (talk) 14:37, 8 December 2015 (UTC)
thread_title:  Explaining nimbers with simpler English? 

message: why is it that the text uses the same symbol for both XOR and nim-sum? don't your think that's confusing?
:Because, as the article explains, they are more or less the same operation. —David Eppstein (talk) 16:53, 11 May 2019 (UTC)
thread_title:  same notation for both XOR and nim-sum? 

message: What is meant by the subtraction operation in the expression (1+(<alpha>'-<alpha>))<beta>-1?  Why is this not rendered as an addition since nimbers are additive self-inverse? Olthe3rd1 (talk) 23:14, 28 June 2019 (UTC)
thread_title:  Calculation of reciprocal 

message: I came to this article from one on game theory and surreals, where the motivating construction is {?|?} where we fill in the blanks. This article doesn't seem to have any of this context. It isn't clear from this article exactly what inductively defined games correspond to the nimbers. It seems to jump directly to assuming all nimbers can be talked about by proxy using a corresponding ordinal. Matthew Pocock (talk) 12:38, 27 August 2023 (UTC)
replies: text: Nimbers are not surreal numbers. They both come from the inductive construction of an arithmetic on infinite games, but they are a different subset of the games. The only value they have in common is zero. —David Eppstein (talk) 15:06, 27 August 2023 (UTC)
thread_title:  Nimbers as surreals? 

message: Over ten years ago based on what my physics teacher showed us between lessons, we created "Pawn Duel" as an online game as simple illustration for counting moves in chess. It turned out to be a human-comprehensible version of Northcott's game, practical to set up if you have a chess board. Over the course of these years somebody else even got inspired and created a small flash game called "Spartans vs Goblins", which is a graphic version for kids.
I have tried creating a page describing this example as comprehensible, even for kids. It was deleted as too specific. I am not an active wiki contributor and don't know all the practices. Hence, could somebody support me, or point to a place where this practical example of Northcott's game (https://nim.ai1.co/pawnduel/) is appreciated by community?  — Preceding unsigned comment added by Okkay (talk • contribs) 23:12, 9 September 2023 (UTC)
thread_title:  Online version of Northcott's game