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# Glossary cards — Chapter ᚅ(N): The stadium

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T7, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

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

*Combinatorial game theory and classical game theory: Nim, nimbers, and the value of matching pennies. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Nion — `ᚅ` {#g-og-nin}

*Ogham letter*

Ogham letter name meaning ‘a fork’ or ‘a loft’; the medieval glossators read it as the ash.

Nin is not a tree name: it means either ‘fork’ or ‘loft’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give costud síde, “establishing of peace”; bág ban, “boast of women”; and bág maise, “boast of beauty.” The arboreal tradition glosses uinnius, ash, and turns the first kenning inside out: “a check on peace is nin, that is, ash, for of it are made spear-shafts by which the peace is broken.” The name matters twice over, because nin also meant ‘forking branches’ and was one of the two words used for the ogham letters themselves, which is part of how the tree-alphabet idea arose in the first place. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### a losing Nim position — `3 ⊕ 5 ⊕ 6` {#g-nim356}

*Integer or rational*

011 ⊕ 101 ⊕ 110 = 000; whoever must move from here loses against correct play.

Heaps of 3, 5, and 6 are 011, 101, and 110 in binary; each column holds two ones, so the nim-sum is 000. By Bouton’s theorem of 1901 the player who must move from this position loses against correct play, whichever move they choose. It is not the smallest losing position with three heaps, which is 1, 2, 3; it was chosen because all three binary columns cancel visibly. The nim-sum is an integer, so the numerator hands 0 home directly.

Cites: [7.1](ch07-refs.md#r-7.1), [7.2](ch07-refs.md#r-7.2), [7.3](ch07-refs.md#r-7.3)

See also: [nim-sum](ch07-cards.md#g-nimsum)

### a winning Nim position — `1 ⊕ 2 ⊕ 4` {#g-nim124}

*Integer or rational*

Equal to 7, so the mover can win: reduce the heap of 4 to 3 and the nim-sum becomes 0.

Heaps of 1, 2, and 4 are 001, 010, and 100, and their nim-sum is 111, which is 7. A nonzero nim-sum means the mover can win: the winning move is to reduce the heap of 4 to 3, because 1 ⊕ 2 ⊕ 3 = 0, and from then on the mover answers every move by restoring a zero nim-sum. The value 7 is a nonzero integer, which is what a denominator needs.

Cites: [7.2](ch07-refs.md#r-7.2), [7.4](ch07-refs.md#r-7.4)

See also: [nim-sum](ch07-cards.md#g-nimsum)

### nimber multiplication — `2 ⊗ 2` {#g-nimprod}

*Integer or rational*

In Conway’s nimber field 2 ⊗ 2 = 3; the denominator is 7 + 3 = 10, because each term comes home as an integer and the + is ordinary addition.

John Conway, in On Numbers and Games (1976), defined a multiplication ⊗ that together with the nim-sum makes the nonnegative integers a field of characteristic 2, in which every number is its own negative: the nimbers. The rule is that a ⊗ b is the smallest number not of the form (a′ ⊗ b) ⊕ (a ⊗ b′) ⊕ (a′ ⊗ b′) for a′ < a and b′ < b, and it gives 2 ⊗ 2 = 3. The nimbers below 2^(2^n) form finite fields, so a product of nonzero nimbers is never zero. The value 3 is an integer, and it joins the nim-sum 7 as an ordinary sum: the denominator of this brace is 7 + 3 = 10, not a further nim-sum.

Cites: [7.18](ch07-refs.md#r-7.18), [7.7](ch07-refs.md#r-7.7), [7.8](ch07-refs.md#r-7.8)

See also: [nim-sum](ch07-cards.md#g-nimsum)

### the value of matching pennies — `v(matching pennies)` {#g-mpvalue}

*Integer or rational*

0: in the zero-sum game where one player wins a penny if the coins match and loses one if they differ, the best either player can guarantee, by flipping fairly, is to break even.

Matching pennies is a smallest zero-sum game with no pure-strategy solution: each player shows heads or tails, one wins the penny on a match, the other on a mismatch. Von Neumann’s minimax theorem of 1928 says every finite two-player zero-sum game has a value, the payoff a player can guarantee with a mixed strategy, and here it is 0, reached when both players flip fairly. Nash’s 1950 theorem extended the existence of equilibrium to every finite game. An integer, and the carrier for classical game theory.

Cites: [7.10](ch07-refs.md#r-7.10), [7.11](ch07-refs.md#r-7.11), [7.12](ch07-refs.md#r-7.12), [7.20](ch07-refs.md#r-7.20)

See also: [the minimax theorem](ch07-cards.md#g-minimax), [Nash’s theorem](ch07-cards.md#g-nasheq)

### the matching-pennies mixed strategy — `½` {#g-mixhalf}

*Integer or rational*

One half: show heads with probability ½, and the opponent can no longer exploit you. When both players do this, that is the only Nash equilibrium of the game.

A mixed strategy is a probability distribution over the moves. In matching pennies any bias toward heads or tails can be punished, so the unique equilibrium, in the sense Nash defined in 1950, is for both players to mix fifty-fifty; then each move earns the same expected payoff and neither player can gain by changing. A rational number.

Cites: [7.11](ch07-refs.md#r-7.11), [7.12](ch07-refs.md#r-7.12)

See also: [Nash’s theorem](ch07-cards.md#g-nasheq)

### the prisoner’s dilemma equilibrium payoff — `u(D, D)` {#g-pdpayoff}

*Integer or rational*

1: with the usual payoffs 5, 3, 1, 0, both prisoners defect and each gets 1, though both would have got 3 by cooperating.

Merrill Flood and Melvin Dresher devised the game at RAND in 1950 and Albert Tucker gave it the prisoners, and with them the name. With temptation 5, reward 3, punishment 1, and sucker’s payoff 0, defecting is better whatever the other player does, so mutual defection is the only Nash equilibrium and pays each player 1; mutual cooperation would pay 3. It is the standard example of an equilibrium that is not the best outcome for either player. An integer.

Cites: [7.13](ch07-refs.md#r-7.13), [7.14](ch07-refs.md#r-7.14)

See also: [Nash’s theorem](ch07-cards.md#g-nasheq)

### nim-sum — `⊕` {#g-nimsum}

*Function or operator*

Bitwise exclusive-or of heap sizes; a position is lost for the mover exactly when it is 0, under the last-move-wins rule (Bouton 1901).

The nim-sum ⊕ is bitwise exclusive-or: write the heap sizes in binary and add each column without carrying, which is exactly what a binary adder computes for its sum bit: A ⊕ B in a half adder, A ⊕ B ⊕ C_in in a full one. Charles Bouton proved in 1901 that a Nim position in normal play is a loss for the player about to move exactly when the nim-sum of the heaps is 0, the first complete solution of a combinatorial game in general form, with any number of heaps of any size. The game’s own origin is unknown: 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 traced to the German imperative nimm, take. The Sprague–Grundy theorem (1935, 1939) extends it to every finite impartial game under the same convention. It is an operation rather than a value, hence the operator tag.

Cites: [7.1](ch07-refs.md#r-7.1), [7.17](ch07-refs.md#r-7.17), [7.2](ch07-refs.md#r-7.2), [7.5](ch07-refs.md#r-7.5), [7.6](ch07-refs.md#r-7.6), [7.9](ch07-refs.md#r-7.9)

See also: [the Sprague–Grundy value](ch07-cards.md#g-sg)

### the Sprague–Grundy value — `𝒢(G) = mex` {#g-sg}

*Function or operator*

Every finite impartial game under normal play is worth exactly one Nim heap, of size given by the mex, the least number not among the values of the positions one move away.

Roland Sprague (1935) and Patrick Grundy (1939) independently proved that any finite impartial game under normal play is equivalent to a single Nim heap, whose size 𝒢(G) is computed recursively as the mex, the minimum excluded value, of the Grundy values of the positions reachable in one move. A position is a loss for the mover exactly when 𝒢 = 0, and the value of a sum of games is the nim-sum of the values, which is why ⊕ and Bouton’s theorem decide every such game and not only Nim. A function on games, hence the tool tag.

Cites: [7.2](ch07-refs.md#r-7.2), [7.9](ch07-refs.md#r-7.9)

See also: [nim-sum](ch07-cards.md#g-nimsum)

### the minimax theorem — `max min = min max` {#g-minimax}

*Function or operator*

Every finite two-player zero-sum game has a value: a payoff each player can guarantee by mixing moves at random in the right proportions. Von Neumann, 1928.

John von Neumann proved it in Zur Theorie der Gesellschaftsspiele (Mathematische Annalen 100, 1928, pages 295–320): in a finite two-player zero-sum game the most the first player can guarantee against the worst reply equals the least the second player can be held to, max_x min_y f(x, y) = min_y max_x f(x, y), once both are allowed mixed strategies; the two players agree on one number, and that number is the value of the game. Every hypothesis earns its place: two players, finitely many strategies each, strictly opposed payoffs, and mixing. Without mixing there may be no solution at all: matching pennies has no pure-strategy equilibrium, and under mixing its value is 0. Nash’s theorem of 1950 drops the zero-sum condition, but what survives the generalization is the existence of an equilibrium rather than a single guaranteed value.

Cites: [7.10](ch07-refs.md#r-7.10), [7.11](ch07-refs.md#r-7.11), [7.12](ch07-refs.md#r-7.12)

See also: [Nash’s theorem](ch07-cards.md#g-nasheq)

### Nash’s theorem — `Nash` {#g-nasheq}

*Function or operator*

Every finite game—finitely many players, finitely many strategies each—has at least one equilibrium in mixed strategies. John Nash, 1950.

In Equilibrium points in n-person games (Proceedings of the National Academy of Sciences 36, 1950, pages 48–49) John Nash proved that a game with finitely many players, each having finitely many strategies, always has at least one profile of mixed strategies from which no player can gain by changing their own strategy alone. Finiteness is the hypothesis popular statements drop. The theorem is existence only: it does not say how many equilibria a game has, which one gets played, or that an equilibrium is good for anybody. Matching pennies has exactly one, and it is a mixture rather than a move. The prisoner’s dilemma’s only equilibrium pays both players less than cooperating would. Von Neumann’s minimax theorem of 1928 is the two-player zero-sum case, where existence comes with a single number both players can guarantee.

Cites: [7.10](ch07-refs.md#r-7.10), [7.11](ch07-refs.md#r-7.11), [7.12](ch07-refs.md#r-7.12), [7.14](ch07-refs.md#r-7.14)

See also: [the minimax theorem](ch07-cards.md#g-minimax)