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# References — Chapter ᚅ(N): The stadium

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-7, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch07.md)

*Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.*

### 7.1 {#r-7.1}

Charles L. Bouton of Harvard University published “Nim, A Game with a Complete Mathematical Theory” in the Annals of Mathematics, second series, volume 3 (1901–1902), pages 35–39.

*Status: confirmed*

- [Bouton, “Nim, A Game with a Complete Mathematical Theory,” The Annals of Mathematics 3 (1901) 35–39 — Internet Archive item record](https://archive.org/details/jstor-1967631) — item metadata gives Title “Nim, A Game with a Complete Mathematical Theory,” Author “Bouton, Charles L.,” Journal “The Annals of Mathematics,” Volume 3, Year 1901, Pages 35-39, identifier urn:jstor-articleid:10.2307/1967631, ISSN 0003486X.
- [Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)](https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory) — byline reads “Charles L. Bouton, Harvard University, Cambridge, Massachusetts.” This is the source for the chapter’s “of Harvard.”
- [Wikipedia, Nim](https://en.wikipedia.org/wiki/Nim) — “Charles L. Bouton of Harvard University” developed the complete theory in 1901; paper cited as Annals of Mathematics vol. 3, pp. 35–39, 1901–1902.
- [Rougetet, “The Prehistory of Nim Game” (Gathering 4 Gardner G4G11 gift exchange)](https://www.gathering4gardner.org/g4g11gift/Rougetet_Lisa-Prehistory_of_Nim.pdf) — text-extracted locally: it introduces Nim “taking as a reference the one introduced by Charles Leonard Bouton in his article, 1901,” calls Joseph Leonard Walsh “a student then a colleague of Bouton at Harvard University,” and its footnote 1 cites the paper as “The Annals of Mathematics, 2nd [series].”

### 7.2 {#r-7.2}

Bouton’s theorem: in normal-play Nim the nim-sum is the bitwise exclusive-or of the heap sizes (write the heaps in binary and add each column without carrying), and the position is a loss for the player about to move exactly when the nim-sum is 0; when it is nonzero there is always a move making it 0. Two players alternately remove any positive number of counters from a single heap and whoever takes the last counter wins.

*Status: confirmed*

- [Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)](https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory) — Bouton defines a “safe combination” as heap sizes which, written in binary and aligned by columns, have every column sum ≡ 0 (mod 2); he states “If A leaves a safe combination on the table, B cannot leave a safe combination on the table at his next move” and that “the player who can first leave a safe combination on the table should win.”
- [Wikipedia, Nim](https://en.wikipedia.org/wiki/Nim) — winning strategy is to “finish every move with a nim-sum of 0”; the nim-sum is the XOR of the heap sizes; the player facing a nim-sum of zero loses against perfect play.

### 7.3 {#r-7.3}

3 ⊕ 5 ⊕ 6 = 0. In binary 3 = 011, 5 = 101, 6 = 110, and each of the three columns contains exactly two ones, so every column cancels. Among Nim positions with three nonempty heaps, 1, 2, 3 is the smallest position with nim-sum 0; smaller zero positions exist with fewer heaps (for example two equal heaps 1, 1).

*Status: confirmed*

- **Own computation** (the author's own work) — Python: 3^5^6 = 0; bin(3)='0b11', bin(5)='0b101', bin(6)='0b110', so the columns are (0, 1, 1), (1, 0, 1), (1, 1, 0) reading 4s, 2s, 1s — two ones in each. Exhaustive search over 1 <= a < b < c <= 10 with a^b^c == 0 returns (1, 2, 3) first, then (1, 4, 5), (1, 6, 7), (1, 8, 9). For three nonempty heaps a <= b <= c with a^b^c = 0, a = b would force c = 0, so all three must be distinct and (1, 2, 3) is minimal by total. Note that the two-heap position (1, 1) also has nim-sum 0 and a smaller total, so the minimality of 1, 2, 3 holds only among positions with three heaps.

### 7.4 {#r-7.4}

1 ⊕ 2 ⊕ 4 = 7, so the position is a win for the mover; reducing the heap of 4 to 3 gives 1 ⊕ 2 ⊕ 3 = 0.

*Status: confirmed*

- **Own computation** (the author's own work) — Python: 1^2^4 = 7 (001 | 010 | 100 -> 111); 1^2^3 = 0 (001 | 010 | 011). Since 7 has its 4-bit set and only the heap of 4 has that bit, the unique winning move is in that heap, taking it to 4 XOR 7 = 3.

### 7.5 {#r-7.5}

Nim’s origin is not known. Variants have been played since ancient times, a Chinese origin has been proposed and disputed, and Bouton himself said he had “not been able to discover much concerning its history”; the name is usually derived from the German imperative nimm, “take.”

*Status: confirmed*

- [Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)](https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory) — “The writer has not been able to discover much concerning its history, although certain forms of it seem to be played at a number of American colleges, and at some of the American fairs. It has been called Fan-Tan, but as it is not the Chinese game of that name, the name in the title is proposed for it.”
- [Wikipedia, Nim](https://en.wikipedia.org/wiki/Nim) — “Variants of nim have been played since ancient times”; a Chinese origin (jiǎn-shízǐ) is suggested but uncertain; earliest European references from the beginning of the 16th century; the OED derives the name from German nimm, ‘take’.
- [Rougetet, “The Prehistory of Nim Game”](https://www.gathering4gardner.org/g4g11gift/Rougetet_Lisa-Prehistory_of_Nim.pdf) — text extracted locally: the Fan-Tan link “was wrong, as chance is part of the Fan-Tan”; Ross (Mathematical Gazette, May 1953) denied a Chinese origin and proposed German nimm; Walsh, Bouton’s Harvard colleague, confirmed it in December 1953; “Games as simple as Nim ... are handed over verbally; therefore this orality is difficult to track.”

### 7.6 {#r-7.6}

Complete winning rules for the one-heap ancestor of Nim were in print four centuries before Bouton. Luca Pacioli’s De Viribus Quantitatis, written between 1496 and 1508, sets two players to reach 30 by adding a number from 1 to 6 in turn, names the safety steps 2, 9, 16 and 23, and gives a rule for finding them in any such game. Problem XIX of Claude-Gaspard Bachet’s Problèmes plaisans et delectables (Lyon, first edition 1612) does the same for reaching 100 by adding a number from 1 to 10, with the steps 1, 12, 23 and so on to 89. What was new in 1901 is generality: Bouton’s article is where the solution of a combinatorial game was first formalized and studied as a general case with any number of piles and any number of chips in each pile.

*Status: confirmed*

- [Rougetet, “The Prehistory of Nim Game” (Gathering 4 Gardner G4G11 gift exchange)](https://www.gathering4gardner.org/g4g11gift/Rougetet_Lisa-Prehistory_of_Nim.pdf) — and text-extracted locally. Pacioli’s Effect XXXIIII asks two players to reach 30 by adding, in turn, a number from 1 to 6: “it is a simplified version of Bouton’s Nim game ... the solution remains based on the same principle as Bouton’s Nim: there exist ‘safe combinations’ that secure the win, provided that one plays correctly.” For the target 30 Pacioli “quickly gives the winning strategy to be applied: the four safety steps, 2, 9, 16 and 23 have to be reached,” and states a general rule — “always divide the number that you wish to arrive at by one more than has been taken and the remainder of the said division will always be the first [step of the] progression” — with a worked example for the target 35: “he takes whatever he wants up to 6, and you will take, or actually you will make 7 the first degree, and then 14, 21, 28, and 35.” The paper then says exactly what is new in 1901: “it was not before 1901, with Bouton’s article, that the solution of a combinatorial game was to be formalized and studied as a general case with any number of piles and any number of chips in each pile.” Problem XIX of Bachet’s Problemes plaisans et delectables, qui se font par les nombres (Lyon, 1st edition, 1612) proposes “a version of Nim that is similar to the one formulated by Pacioli”: two players must reach 100 by adding numbers from 1 to 10, and Bachet’s rule is to “add 1 to the number that cannot be exceeded, here 10; you obtain 11 and then, always remove 11 of the number to be reached, 100; you will obtain these numbers 89, 78, 67, 56, 45, 34, 23, 12, 1.”
- [Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)](https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory) — Bouton makes no priority claim of any kind. He says only that the game “has interested the writer on account of its seeming complexity, and its extremely simple and complete mathematical theory.”

Editorial note: Bouton makes no priority claim himself, and the unqualified form of one — that in solving Nim he did something never done for any game — fails on Pacioli and Bachet, whose one-heap game was solved completely and in print. So the chapter and the ⊕ card claim generality rather than priority: not the first game solved, but the first solved for any number of heaps of any size, which is the distinction Rougetet draws.

### 7.7 {#r-7.7}

In On Numbers and Games (1976) John Horton Conway defined a nimber multiplication ⊗ which, with the nim-sum ⊕, makes the non-negative integers a field of characteristic 2 in which every element is its own additive inverse; the multiplication rule is a ⊗ b = mex{(a′ ⊗ b) ⊕ (a ⊗ b′) ⊕ (a′ ⊗ b′) : a′ < a, b′ < b}, giving 2 ⊗ 2 = 3; and for every n the nimbers below 2^(2^n) form the finite field GF(2^(2^n)), so a product of nonzero nimbers is never zero.

*Status: confirmed*

- [Wikipedia, Nimber](https://en.wikipedia.org/wiki/Nimber) — nim-addition α ⊕ β = mex({α′ ⊕ β} ∪ {α ⊕ β′}), equal to bitwise XOR for finite ordinals; nim-multiplication α ⊗ β = mex{(α′ ⊗ β) ⊕ (α ⊗ β′) ⊕ (α′ ⊗ β′) : α′ < α, β′ < β}; “2² = 3” in nim-multiplication; “an algebraically closed field of characteristic 2” in which each element is “its own additive inverse”; “For all natural numbers n, the set of nimbers less than 2^(2^n) form the Galois field GF(2^(2^n))”; the field comes “with the nimber multiplicative inverse of a nonzero ordinal α,” so every nimber but 0 can be divided by; Conway, On Numbers and Games, 1976 (2nd edition 2000).

### 7.8 {#r-7.8}

2 ⊗ 2 = 3 follows directly from the mex rule.

*Status: confirmed*

- **Own computation** (the author's own work) — Recursive implementation of a⊗b = mex{(a′⊗b) xor (a⊗b′) xor (a′⊗b′) : a′<a, b′<b} in Python. For a = b = 2 the excluded set from (a′,b′) in {0,1}² is {0, 2, 1}, whose mex is 3. Full table for 0..3: row 0 = 0,0,0,0; row 1 = 0,1,2,3; row 2 = 0,2,3,1; row 3 = 0,3,1,2 — the multiplication table of GF(4), confirming both 2 ⊗ 2 = 3 and the absence of zero divisors.

### 7.9 {#r-7.9}

The Sprague–Grundy theorem: every impartial game under the normal play convention is equivalent to a single Nim heap, whose size is the Grundy value computed recursively as the mex (minimum excluded value) of the Grundy values of the options; the Grundy value of a sum of games is the nim-sum of the summands’ values, and a position is a loss for the player to move exactly when its Grundy value is 0. Roland Sprague, “Über mathematische Kampfspiele,” Tôhoku Mathematical Journal 41 (1935) 438–444; Patrick Michael Grundy, “Mathematics and games,” Eureka 2 (1939) 6–8.

*Status: confirmed*

- [Wikipedia, Sprague–Grundy theorem](https://en.wikipedia.org/wiki/Sprague%E2%80%93Grundy_theorem) — “Every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim.” Grundy value defined by mex; “The Grundy value of the sum of a finite set of positions is just the nim-sum of the Grundy values of its summands”; a position is a loss for the next player iff its Grundy value is zero. Cites Sprague, Über mathematische Kampfspiele, Tohoku Math. J. 41, 438–444, and Grundy, Mathematics and games, Eureka 2, 6–8, 1939.
- [R. Sprague, “Über mathematische Kampfspiele,” Tohoku Mathematical Journal, First Series, vol. 41 — J-STAGE record](https://www.jstage.jst.go.jp/article/tmj1911/41/0/41_0_438/_article) — from the publisher’s own record: author R. Sprague, Tohoku Mathematical Journal First Series, volume 41, pages 438–444, publication year 1935.
- [Wikipedia, Roland Sprague](https://en.wikipedia.org/wiki/Roland_Sprague) — “Über mathematische Kampfspiele, Tôhoku Mathematical Journal, vol. 41 (1935), pp. 438–444”; Sprague and Grundy discovered the function independently “in 1935 and 1939 respectively.”
- [Wikipedia, Patrick Michael Grundy](https://en.wikipedia.org/wiki/Patrick_Michael_Grundy) — “Mathematics and games,” Eureka, 2: 6–8, 1939; introduced the Sprague–Grundy function, independent of and later than Sprague’s 1935 work.

Editorial note: The volume Tôhoku Math. J. 41 spans 1935–1936 and a few secondary listings give 1936. The publisher’s own J-STAGE record and the Wikipedia biography of Sprague both give 1935, which is also the date universally used in the combinatorial game theory literature, so the chapter’s “Sprague in 1935” stands.

### 7.10 {#r-7.10}

John von Neumann proved the minimax theorem in 1928: every finite two-player zero-sum game has a value, max_x min_y f(x, y) = min_y max_x f(x, y), when both players are allowed mixed strategies. Published as “Zur Theorie der Gesellschaftsspiele,” Mathematische Annalen 100 (1928) 295–320. The value is a payoff each player can guarantee with a suitable mixed strategy, whatever the other does.

*Status: confirmed*

- [Wikipedia, Minimax theorem](https://en.wikipedia.org/wiki/Minimax_theorem) — “Von Neumann published this in 1928 in the paper ‘Zur Theorie der Gesellschaftsspiele’ (Mathematische Annalen, volume 100, pages 295–320)”; states max_x min_y f(x, y) = min_y max_x f(x, y); applies to finite two-player zero-sum games with mixed strategies, where f(x, y) is the expected payoff to the first player.
- [Wikipedia, Game theory](https://en.wikipedia.org/wiki/Game_theory) — von Neumann published “On the Theory of Games of Strategy” in 1928; “Von Neumann’s original proof used Brouwer’s fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.”
- [Minimax, Wikipedia](https://en.wikipedia.org/wiki/Minimax) — “For every two-person zero-sum game with finitely many strategies, there exists a value V and a mixed strategy for each player”; “Equivalently, Player 1’s strategy guarantees them a payoff of V regardless of Player 2’s strategy, and similarly Player 2 can guarantee themselves a payoff of −V.”

### 7.11 {#r-7.11}

Matching pennies is a two-player zero-sum game with no pure-strategy Nash equilibrium; its unique Nash equilibrium has each player choosing heads or tails with probability one half, and the value of the game (the expected payoff at equilibrium) is 0.

*Status: confirmed*

- [Wikipedia, Matching pennies](https://en.wikipedia.org/wiki/Matching_pennies) — Even and Odd each choose heads or tails; Even wins on a match, Odd on a mismatch; the game is zero-sum; “There is no pure strategy (heads or tails) that is a best response to a best response”; there is a unique mixed-strategy equilibrium in which “each player chooses heads or tails with equal probability”; “When either player plays the equilibrium, everyone’s expected payoff is zero.”
- **Own computation** (the author's own work) — With payoff +1 to the matcher on a match and −1 on a mismatch, and mixing probabilities p and q on heads, the matcher’s expected payoff is pq − p(1−q) − (1−p)q + (1−p)(1−q) = (2p−1)(2q−1). At p = 1/2 this is 0 for every q, so the matcher can guarantee 0; by the symmetry of the game the opponent can too, so the value is 0. If p ≠ 1/2 the opponent’s best reply is pure, against which the matcher would want to switch to the opposite pure strategy, so p = q = 1/2 is the only equilibrium.

### 7.12 {#r-7.12}

John Nash proved in 1950 that every finite game with finitely many players has an equilibrium in mixed strategies; “Equilibrium points in n-person games,” Proceedings of the National Academy of Sciences 36 (1950) 48–49.

*Status: confirmed*

- [Wikipedia, Nash equilibrium](https://en.wikipedia.org/wiki/Nash_equilibrium) — “Equilibrium points in n-person games,” PNAS volume 36, pages 48–49, 1950; “there is a Nash equilibrium, possibly in mixed strategies, for every finite game”; equilibrium defined as a profile where “no player could gain more by changing their own strategy (holding all other players’ strategies fixed)”; “Nash equilibria need not exist if the set of choices is infinite and non-compact”: “A game where two players simultaneously name a number and the player naming the larger number wins does not have a NE, as the set of choices is not compact because it is unbounded.”
- [Wikipedia, Game theory](https://en.wikipedia.org/wiki/Game_theory) — “Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies,” in 1950.

### 7.13 {#r-7.13}

The prisoner’s dilemma was devised by Merrill Flood and Melvin Dresher at the RAND Corporation in 1950, and Albert W. Tucker supplied the prison-sentence story that gave the game its name.

*Status: judgement*

- [Wikipedia, Prisoner’s dilemma](https://en.wikipedia.org/wiki/Prisoner%27s_dilemma) — “designed by Merrill Flood and Melvin Dresher in 1950 during their work at the RAND Corporation”; “Albert W. Tucker later named the game the ‘prisoner’s dilemma’ by framing the rewards in terms of prison sentences”; mutual defection is “the only strong Nash equilibrium in the game.”
- [Stanford Encyclopedia of Philosophy, Prisoner’s Dilemma](https://plato.stanford.edu/entries/prisoner-dilemma/) — substantive revision 24 October 2025, fetched: “The ‘discovery’ of the prisoner’s dilemma is usually attributed to Merrill Flood and Melvin Dresher,” recruited by RAND after the war; “In early 1950, they conducted an experiment.” On the name: “The illustrative story with prison sentences as payoffs is due to Albert Tucker, who wanted to make the ideas behind Flood and Dresher’s game more accessible to an audience of Stanford psychologists,” but “The title prisoner’s dilemma is often ... credited to Tucker, but Guerra-Pujol and Martinez-Garcia note that the earliest use of this label in print is Luce and Raiffa 1957.”

Editorial note: The prison-sentence story is securely Tucker’s; the label is less securely his. Wikipedia credits him with both. The Stanford Encyclopedia credited him with both too until its revision of 24 October 2025, which withdraws the naming — “often (for example in Poundstone p117, Peterson (2015a) p1 and in earlier versions of this entry) credited to Tucker” — on Guerra-Pujol and Martinez-Garcia’s finding that the label’s earliest appearance in print is Luce and Raiffa (1957). A correction by the same reference work outranks a restatement, so the page gives Tucker the prisoners and lets the name follow from them rather than crediting him with printing it.

### 7.14 {#r-7.14}

With the usual payoffs — temptation 5, reward 3, punishment 1 and sucker’s payoff 0 — defecting strictly dominates cooperating, mutual defection is the unique Nash equilibrium and pays each player 1, and mutual cooperation would have paid each 3. These are the values Axelrod used and the values standard in the literature.

*Status: confirmed*

- [Rapoport, Seale and Colman, “Is Tit-for-Tat the Answer? On the Conclusions Drawn from Axelrod’s Tournaments,” PLOS ONE 10(7): e0134128](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0134128) — the payoff matrix of Axelrod’s original tournament is given as “(T, R, P, S) = (5, 3, 1, 0).”
- [Wikipedia, Prisoner’s dilemma](https://en.wikipedia.org/wiki/Prisoner%27s_dilemma) — the payoff ordering is T > R > P > S (temptation, reward, punishment, sucker); mutual defection is the only strong Nash equilibrium, and “the collectively ideal result of mutual cooperation is irrational from a self-interested standpoint.”
- **Own computation** (the author's own work) — With T = 5, R = 3, P = 1, S = 0: against a cooperator, defect pays 5 > 3; against a defector, defect pays 1 > 0. Defection strictly dominates, so (D, D) is the unique Nash equilibrium and u(D, D) = 1, while u(C, C) = 3. Also R > (T + S)/2 = 2.5, so these values satisfy the usual repeated-game condition as well.

### 7.15 {#r-7.15}

The chapter’s epigraph is Conway’s own wording, as recorded by the MacTutor History of Mathematics archive.

*Status: confirmed*

- [MacTutor History of Mathematics, Quotations by John Horton Conway](https://mathshistory.st-andrews.ac.uk/Biographies/Conway/quotations/) — verbatim: “You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.” The chapter quotes the second and third sentences exactly.

### 7.16 {#r-7.16}

Conway arrived at the surreal numbers by analyzing Go endgames — that is, by asking what a game position is worth — and every surreal number is a game but not every game is a surreal number; the nimbers, star among them, are games that are not numbers.

*Status: confirmed*

- [Wikipedia, Surreal number](https://en.wikipedia.org/wiki/Surreal_number) — “Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers”; “He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames”; “Every surreal number is a game, but not all games are surreal numbers, e.g. the game {0 | 0} is not a surreal number” — {0 | 0} being star. Conway used the surreals for game analysis in On Numbers and Games (1976).
- [Wikipedia, Nimber](https://en.wikipedia.org/wiki/Nimber) — the nimbers form a field of characteristic 2 in which each element is its own additive inverse — the chapter’s “every one being its own negative.” Read again on 23 September 2026, its lead also says the nimbers “form a subclass of surreal numbers”; the book does not follow that phrase (the Star line below: ∗ = {0 | 0} is a value but not a number) and keeps the link for the field facts.
- [Star (game theory), Wikipedia](https://en.wikipedia.org/wiki/Star_(game_theory)) — read as raw wikitext: star, “written as ∗ or ∗1,” may be denoted as “{0|0}” and “is a value, but not a number in the traditional sense”; it is “confused with” 0.

### 7.17 {#r-7.17}

Exclusive-or is the arithmetic of a binary adder: the sum bit of a half adder is A ⊕ B and of a full adder A ⊕ B ⊕ C_in.

*Status: confirmed*

- [Wikipedia, Adder (electronics)](https://en.wikipedia.org/wiki/Adder_(electronics)) — “The Boolean logic for the sum (in this case S) will be A ⊕ B whereas for the carry (C) will be A ⋅ B”; for the full adder “S = A ⊕ B ⊕ C_in.”

### 7.18 {#r-7.18}

The combinatorial-game denominator 1 ⊕ 2 ⊕ 4 + 2 ⊗ 2 is nonzero on either reading, but the two readings give different numbers: under the book’s homecoming rule, where each term returns an integer and the + is ordinary addition, it is 7 + 3 = 10; if the + were read as a further nim-sum it would be 7 ⊕ 3 = 4.

*Status: judgement*

- **Own computation** (the author's own work) — 1^2^4 = 7; 2 ⊗ 2 = 3 by the mex rule (verified by recursion, see 7.8); 7 + 3 = 10 and 7 XOR 3 = 4. Both are nonzero, so the value of the chapter’s factor (0 over a nonzero denominator, times a second such fraction) is 0 either way.
- [Wikipedia, Nimber](https://en.wikipedia.org/wiki/Nimber) — nim-addition of finite nimbers is bitwise XOR, and the multiplication table gives 2 ⊗ 2 = 3 — the two facts that make both readings available.

Editorial note: Three symbols sit side by side in this denominator — ⊕, ⊗ and + — so the + could be read as a third nimber operation. The book’s homecoming rule settles it: each term returns an ordinary integer and the + is ordinary addition, so the denominator is 7 + 3 = 10. The tip on both formula copies, the chapter key and the 2 ⊗ 2 card all say so. Under the nim reading it would be 7 ⊕ 3 = 4; both values are nonzero, so the chapter’s factor is 0 either way and nothing but the reading is at stake.

### 7.19 {#r-7.19}

Dr. Nim, the marble-powered plastic computer that plays Nim, was invented by John Thomas Godfrey and manufactured by E.S.R., Inc. in the mid-1960s.

*Status: judgement*

- [Wikipedia, Dr. Nim](https://en.wikipedia.org/wiki/Dr._Nim) — “Dr. Nim is a toy invented by John Thomas Godfrey and manufactured by E.S.R., Inc. in the mid-1960s. It consists of a marble-powered plastic computer capable of playing the game of Nim.” No specific year is given.
- [Dr. Nim (Instruction Manual), E.S.R. Inc. — Internet Archive item record](https://archive.org/details/dr-nim) — the scanned manual is cataloged with a publication date of 1 January 1966.

Editorial note: No reference work pins a year. Wikipedia says only “mid-1960s”; the Internet Archive catalogues the scanned instruction manual as 1966, which is also the year collectors’ listings give, while some listings say 1965. The chapter says “mid-1960s,” the one description all of them support, and the toy is mentioned only to describe a video about it.

### 7.20 {#r-7.20}

No zero-sum game smaller than two strategies apiece can lack a pure-strategy equilibrium, and matching pennies is a 2×2 zero-sum game that lacks one, so it is a smallest game of its kind. It is also the standard textbook example of one.

*Status: judgement*

- [Wikipedia, Matching pennies](https://en.wikipedia.org/wiki/Matching_pennies) — describes matching pennies as the basic two-player zero-sum game with no pure-strategy Nash equilibrium, but does not call it the smallest such game.
- **Own computation** (the author's own work) — A zero-sum game needs at least two strategies per player for a pure equilibrium to fail (any 1×n or m×1 game has one), so 2×2 is the smallest possible shape and matching pennies is a 2×2 game with none. In that sense the claim is exact; it is not exact in the sense of a unique smallest game, since the 2×2 game obtained by relabeling or rescaling the payoffs behaves the same.

Editorial note: The literature treats matching pennies as the standard example but does not use the superlative, so the page rests “smallest” on the argument rather than on an attribution. It means smallest in shape: any 1×n or m×1 zero-sum game has a pure equilibrium, so 2×2 is the least shape in which one can fail. It does not mean unique, since relabeling or rescaling the payoffs gives another 2×2 game that behaves the same.