{
  "index": "2014-B-5",
  "type": "ALG",
  "tag": [
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "In the 75th annual Putnam Games, participants compete at mathematical games.\nPatniss and Keeta play a game in which they take turns choosing an element \nfrom the group of invertible $n \\times n$ matrices with entries in the field\n$\\mathbb{Z}/p \\mathbb{Z}$ of integers modulo $p$, where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are:\n\\begin{enumerate}\n\\item[(1)]\nA player cannot choose an element that has been chosen by either player on any previous turn.\n\\item[(2)]\nA player can only choose an element that commutes with all previously chosen elements.\n\\item[(3)]\nA player who cannot choose an element on his/her turn loses the game.\n\\end{enumerate}\nPatniss takes the first turn. Which player has a winning strategy?\n(Your answer may depend on $n$ and $p$.)",
  "solution": "We show that Patniss wins if $p=2$ and Keeta wins if $p>2$ (for all $n$).\nWe first analyze the analogous game played using an arbitrary finite group $G$.\nRecall that for any subset $S$ of $G$, the set of elements $g \\in G$ which commute with all elements of $S$ forms a subgroup $Z(S)$ of $G$, called the \\emph{centralizer} (or \\emph{commutant}) of $S$. \nAt any given point in the game, the set $S$ of previously chosen elements is contained in $Z(S)$. Initially $S = \\emptyset$ and $Z(S) = G$;\nafter each turn, $S$ is increased by one element and $Z(S)$ is replaced by a subgroup.\nIn particular, if the order of $Z(S)$ is odd at some point, it remains odd thereafter;\nconversely, if $S$ contains an element of even order, then the order of $Z(S)$ remains even thereafter. Therefore, any element $g \\in G$ for which $Z(\\{g\\})$ has odd order is a winning first move for Patniss, while any other first move by Patniss loses if Keeta responds with some $h \\in Z(\\{g\\})$ of even order (e.g., an element of a 2-Sylow subgroup of $Z(\\{g\\})$). In both cases, the win is guaranteed no matter what moves follow.\n\nNow let $G$ be the group of invertible $n \\times n$ matrices with entries in $\\ZZ/p\\ZZ$.\nIf $p>2$, then $Z(S)$ will always contain the scalar matrix $-1$ of order 2, so the win for Keeta is guaranteed. (An explicit winning strategy is to answer any move $g$ with the move $-g$.)\n\nIf $p=2$, we establish the existence of $g \\in G$ such that $Z(\\{g\\})$ has odd order\nusing the existence of an irreducible polynomial $P(x)$ of degree $n$ over $\\ZZ/p\\ZZ$ (see remark). We construct an $n \\times n$ matrix over $\\ZZ/p\\ZZ$ with characteristic polynomial $P(x)$ by taking the \\emph{companion matrix} of $P(x)$: write $P(x) = x^n + P_{n-1} x^{n-1} + \\cdots + P_0$ and set\n\\[\ng = \\begin{pmatrix}\n0 & 0 & \\cdots & 0 & -P_0 \\\\\n1 & 0 & \\cdots & 0 & -P_1 \\\\\n0 & 1 & \\cdots & 0 & -P_2 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & 1 & -P_{n-1}\n\\end{pmatrix}.\n\\]\nIn particular, $\\det(g) = (-1)^n P_0 \\neq 0$, so $g \\in G$.\nOver an algebraic closure of $\\ZZ/p \\ZZ$, $g$ becomes diagonalizable with distinct eigenvalues, so any matrix commuting with $g$ must also be diagonalizable, and hence of odd order. In particular, $Z(\\{g\\})$ is of odd order, so Patniss has a winning strategy.\n\n\\noindent\n\\textbf{Remark:}\nIt can be shown that in the case $p=2$, the only elements $g \\in G$ for which $Z(\\{g\\})$ has odd order are those for which $g$ has distinct eigenvalues: in any other case, $Z(\\{g\\})$ contains a subgroup isomorphic to the group of $k \\times k$ invertible matrices over $\\ZZ/2\\ZZ$ for some $k>1$, and this group has order $(2^k-1)(2^k-2) \\cdots(2^k - 2^{k-1})$.\n\n\\noindent\n\\textbf{Remark:}\nWe sketch two ways to verify the existence of an irreducible polynomial of degree $n$ over $\\ZZ/p\\ZZ$ for any positive integer $n$ and any prime number $p$. One is to use M\\\"obius inversion to count the number of irreducible polynomials of degree $n$ over $\\ZZ/p\\ZZ$ and then give a positive lower bound for this count. The other is to first establish the existence of a finite field $\\FF$ of cardinality $p^n$, e.g., as the set of roots of the polynomial $x^{p^n}-1$ inside a splitting field, and then take the minimal polynomial of a nonzero element of $\\FF$ over $\\ZZ/p\\ZZ$ which is a primitive $(p^n-1)$-st root of unity in $\\FF$ (which exist because the multiplicative group of $\\FF$ contains at most one cyclic subgroup of any given order). One might be tempted to apply the primitive element theorem for $\\FF$ over $\\ZZ/p\\ZZ$, but in fact one of the preceding techniques is needed in order to verify this result for finite fields, as the standard argument that ``most'' elements of the upper field are primitive breaks down for finite fields.\n\nOne may also describe the preceding analysis in terms of an identification of $\\FF$ as a $\\ZZ/p\\ZZ$-vector space with the space of column vectors of length $n$. Under such an identification, if we take $g$ to be an element of $\\FF - \\{0\\}$ generating this group, then any element of $Z(\\{g\\})$ commutes with all of $\\FF- \\{0\\}$ and hence must define an $\\FF$-linear endomorphism of $\\FF$. Any such endomorphism is itself multiplication by an element of $\\FF$, so $Z(\\{g\\})$ is identified with the multiplicative group of $\\FF$, whose order is the odd number $2^n-1$.",
  "vars": [
    "F",
    "G",
    "S",
    "Z",
    "g",
    "h",
    "k",
    "P",
    "P_n-1",
    "P_0",
    "P_1",
    "P_2",
    "x"
  ],
  "params": [
    "n",
    "p"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "F": "fieldspace",
        "G": "groupspace",
        "S": "chosenlist",
        "Z": "centralizer",
        "g": "elementone",
        "h": "respondelem",
        "k": "dimindex",
        "P": "irreducpoly",
        "P_n-1": "polycoefnminusone",
        "P_0": "polycoefzero",
        "P_1": "polycoefone",
        "P_2": "polycoeftwo",
        "x": "variablex",
        "n": "dimension",
        "p": "primevalue"
      },
      "question": "In the 75th annual Putnam Games, participants compete at mathematical games.\nPatniss and Keeta play a game in which they take turns choosing an element \nfrom the group of invertible $dimension \\times dimension$ matrices with entries in the field\n$\\mathbb{Z}/primevalue \\mathbb{Z}$ of integers modulo $primevalue$, where $dimension$ is a fixed positive integer and $primevalue$ is a fixed prime number. The rules of the game are:\n\\begin{enumerate}\n\\item[(1)]\nA player cannot choose an element that has been chosen by either player on any previous turn.\n\\item[(2)]\nA player can only choose an element that commutes with all previously chosen elements.\n\\item[(3)]\nA player who cannot choose an element on his/her turn loses the game.\n\\end{enumerate}\nPatniss takes the first turn. Which player has a winning strategy?\n(Your answer may depend on $dimension$ and $primevalue$.)",
      "solution": "We show that Patniss wins if $primevalue=2$ and Keeta wins if $primevalue>2$ (for all $dimension$).\nWe first analyze the analogous game played using an arbitrary finite group $groupspace$.\nRecall that for any subset $chosenlist$ of $groupspace$, the set of elements $elementone \\in groupspace$ which commute with all elements of $chosenlist$ forms a subgroup $centralizer(chosenlist)$ of $groupspace$, called the \\emph{centralizer} (or \\emph{commutant}) of $chosenlist$. \nAt any given point in the game, the set $chosenlist$ of previously chosen elements is contained in $centralizer(chosenlist)$. Initially $chosenlist = \\emptyset$ and $centralizer(chosenlist) = groupspace$;\nafter each turn, $chosenlist$ is increased by one element and $centralizer(chosenlist)$ is replaced by a subgroup.\nIn particular, if the order of $centralizer(chosenlist)$ is odd at some point, it remains odd thereafter;\nconversely, if $chosenlist$ contains an element of even order, then the order of $centralizer(chosenlist)$ remains even thereafter. Therefore, any element $elementone \\in groupspace$ for which $centralizer(\\{elementone\\})$ has odd order is a winning first move for Patniss, while any other first move by Patniss loses if Keeta responds with some $respondelem \\in centralizer(\\{elementone\\})$ of even order (e.g., an element of a 2-Sylow subgroup of $centralizer(\\{elementone\\})$). In both cases, the win is guaranteed no matter what moves follow.\n\nNow let $groupspace$ be the group of invertible $dimension \\times dimension$ matrices with entries in $\\ZZ/primevalue\\ZZ$.\nIf $primevalue>2$, then $centralizer(chosenlist)$ will always contain the scalar matrix $-1$ of order 2, so the win for Keeta is guaranteed. (An explicit winning strategy is to answer any move $elementone$ with the move $-elementone$.)\n\nIf $primevalue=2$, we establish the existence of $elementone \\in groupspace$ such that $centralizer(\\{elementone\\})$ has odd order\nusing the existence of an irreducible polynomial $irreducpoly(variablex)$ of degree $dimension$ over $\\ZZ/primevalue\\ZZ$ (see remark). We construct a $dimension \\times dimension$ matrix over $\\ZZ/primevalue\\ZZ$ with characteristic polynomial $irreducpoly(variablex)$ by taking the \\emph{companion matrix} of $irreducpoly(variablex)$: write\n\\[\nirreducpoly(variablex) = variablex^{dimension} + polycoefnminusone \\, variablex^{dimension-1} + \\cdots + polycoefzero\n\\]\nand set\n\\[\nelementone = \\begin{pmatrix}\n0 & 0 & \\cdots & 0 & -polycoefzero \\\\\n1 & 0 & \\cdots & 0 & -polycoefone \\\\\n0 & 1 & \\cdots & 0 & -polycoeftwo \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & 1 & -polycoefnminusone\n\\end{pmatrix}.\n\\]\nIn particular, $\\det(elementone) = (-1)^{dimension} \\, polycoefzero \\neq 0$, so $elementone \\in groupspace$.\nOver an algebraic closure of $\\ZZ/primevalue \\ZZ$, $elementone$ becomes diagonalizable with distinct eigenvalues, so any matrix commuting with $elementone$ must also be diagonalizable, and hence of odd order. In particular, $centralizer(\\{elementone\\})$ is of odd order, so Patniss has a winning strategy.\n\n\\textbf{Remark:}\nIt can be shown that in the case $primevalue=2$, the only elements $elementone \\in groupspace$ for which $centralizer(\\{elementone\\})$ has odd order are those for which $elementone$ has distinct eigenvalues: in any other case, $centralizer(\\{elementone\\})$ contains a subgroup isomorphic to the group of $dimindex \\times dimindex$ invertible matrices over $\\ZZ/2\\ZZ$ for some $dimindex>1$, and this group has order $(2^{dimindex}-1)(2^{dimindex}-2) \\cdots(2^{dimindex} - 2^{dimindex-1})$.\n\n\\textbf{Remark:}\nWe sketch two ways to verify the existence of an irreducible polynomial of degree $dimension$ over $\\ZZ/primevalue\\ZZ$ for any positive integer $dimension$ and any prime number $primevalue$. One is to use M\"obius inversion to count the number of irreducible polynomials of degree $dimension$ over $\\ZZ/primevalue\\ZZ$ and then give a positive lower bound for this count. The other is to first establish the existence of a finite field $fieldspace$ of cardinality $primevalue^{dimension}$, e.g., as the set of roots of the polynomial $variablex^{primevalue^{dimension}}-1$ inside a splitting field, and then take the minimal polynomial of a nonzero element of $fieldspace$ over $\\ZZ/primevalue\\ZZ$ which is a primitive $(primevalue^{dimension}-1)$-st root of unity in $fieldspace$ (which exist because the multiplicative group of $fieldspace$ contains at most one cyclic subgroup of any given order). One might be tempted to apply the primitive element theorem for $fieldspace$ over $\\ZZ/primevalue\\ZZ$, but in fact one of the preceding techniques is needed in order to verify this result for finite fields, as the standard argument that ``most'' elements of the upper field are primitive breaks down for finite fields.\n\nOne may also describe the preceding analysis in terms of an identification of $fieldspace$ as a $\\ZZ/primevalue\\ZZ$-vector space with the space of column vectors of length $dimension$. Under such an identification, if we take $elementone$ to be an element of $fieldspace - \\{0\\}$ generating this group, then any element of $centralizer(\\{elementone\\})$ commutes with all of $fieldspace- \\{0\\}$ and hence must define an $fieldspace$-linear endomorphism of $fieldspace$. Any such endomorphism is itself multiplication by an element of $fieldspace$, so $centralizer(\\{elementone\\})$ is identified with the multiplicative group of $fieldspace$, whose order is the odd number $2^{dimension}-1$."
    },
    "descriptive_long_confusing": {
      "map": {
        "F": "watermelon",
        "G": "submarine",
        "S": "kangaroo",
        "Z": "telescope",
        "g": "marigolds",
        "h": "pinecones",
        "k": "snowflake",
        "P": "blueberry",
        "P_n-1": "papayafrost",
        "P_0": "coconutbay",
        "P_1": "dragonfruit",
        "P_2": "elderflower",
        "x": "juiceroad",
        "n": "cloudscape",
        "p": "starlights"
      },
      "question": "In the 75th annual Putnam Games, participants compete at mathematical games.\nPatniss and Keeta play a game in which they take turns choosing an element \nfrom the group of invertible $cloudscape \\times cloudscape$ matrices with entries in the field\n$\\mathbb{Z}/starlights \\mathbb{Z}$ of integers modulo $starlights$, where $cloudscape$ is a fixed positive integer and $starlights$ is a fixed prime number. The rules of the game are:\n\\begin{enumerate}\n\\item[(1)]\nA player cannot choose an element that has been chosen by either player on any previous turn.\n\\item[(2)]\nA player can only choose an element that commutes with all previously chosen elements.\n\\item[(3)]\nA player who cannot choose an element on his/her turn loses the game.\n\\end{enumerate}\nPatniss takes the first turn. Which player has a winning strategy?\n(Your answer may depend on $cloudscape$ and $starlights$.)",
      "solution": "We show that Patniss wins if $starlights=2$ and Keeta wins if $starlights>2$ (for all $cloudscape$).\nWe first analyze the analogous game played using an arbitrary finite group $submarine$.\nRecall that for any subset $kangaroo$ of $submarine$, the set of elements $marigolds \\in submarine$ which commute with all elements of $kangaroo$ forms a subgroup $telescope(kangaroo)$ of $submarine$, called the \\emph{centralizer} (or \\emph{commutant}) of $kangaroo$.\nAt any given point in the game, the set $kangaroo$ of previously chosen elements is contained in $telescope(kangaroo)$. Initially $kangaroo = \\emptyset$ and $telescope(kangaroo) = submarine$;\nafter each turn, $kangaroo$ is increased by one element and $telescope(kangaroo)$ is replaced by a subgroup.\nIn particular, if the order of $telescope(kangaroo)$ is odd at some point, it remains odd thereafter;\nconversely, if $kangaroo$ contains an element of even order, then the order of $telescope(kangaroo)$ remains even thereafter. Therefore, any element $marigolds \\in submarine$ for which $telescope(\\{marigolds\\})$ has odd order is a winning first move for Patniss, while any other first move by Patniss loses if Keeta responds with some $pinecones \\in telescope(\\{marigolds\\})$ of even order (e.g., an element of a 2-Sylow subgroup of $telescope(\\{marigolds\\})$). In both cases, the win is guaranteed no matter what moves follow.\n\nNow let $submarine$ be the group of invertible $cloudscape \\times cloudscape$ matrices with entries in $\\ZZ/starlights\\ZZ$.\nIf $starlights>2$, then $telescope(kangaroo)$ will always contain the scalar matrix $-1$ of order 2, so the win for Keeta is guaranteed. (An explicit winning strategy is to answer any move $marigolds$ with the move $-marigolds$.)\n\nIf $starlights=2$, we establish the existence of $marigolds \\in submarine$ such that $telescope(\\{marigolds\\})$ has odd order\nusing the existence of an irreducible polynomial $blueberry(juiceroad)$ of degree $cloudscape$ over $\\ZZ/starlights\\ZZ$ (see remark). We construct an $cloudscape \\times cloudscape$ matrix over $\\ZZ/starlights\\ZZ$ with characteristic polynomial $blueberry(juiceroad)$ by taking the \\emph{companion matrix} of $blueberry(juiceroad)$: write $blueberry(juiceroad) = juiceroad^{cloudscape} + papayafrost\\, juiceroad^{cloudscape-1} + \\cdots + coconutbay$ and set\n\\[\nmarigolds = \\begin{pmatrix}\n0 & 0 & \\cdots & 0 & -coconutbay \\\\\n1 & 0 & \\cdots & 0 & -dragonfruit \\\\\n0 & 1 & \\cdots & 0 & -elderflower \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & 1 & -papayafrost\n\\end{pmatrix}.\n\\]\nIn particular, $\\det(marigolds) = (-1)^{cloudscape}\\,coconutbay \\neq 0$, so $marigolds \\in submarine$.\nOver an algebraic closure of $\\ZZ/starlights \\ZZ$, $marigolds$ becomes diagonalizable with distinct eigenvalues, so any matrix commuting with $marigolds$ must also be diagonalizable, and hence of odd order. In particular, $telescope(\\{marigolds\\})$ is of odd order, so Patniss has a winning strategy.\n\n\\noindent\n\\textbf{Remark:}\nIt can be shown that in the case $starlights=2$, the only elements $marigolds \\in submarine$ for which $telescope(\\{marigolds\\})$ has odd order are those for which $marigolds$ has distinct eigenvalues: in any other case, $telescope(\\{marigolds\\})$ contains a subgroup isomorphic to the group of $snowflake \\times snowflake$ invertible matrices over $\\ZZ/2\\ZZ$ for some $snowflake>1$, and this group has order $(2^{snowflake}-1)(2^{snowflake}-2) \\cdots(2^{snowflake} - 2^{snowflake-1})$.\n\n\\noindent\n\\textbf{Remark:}\nWe sketch two ways to verify the existence of an irreducible polynomial of degree $cloudscape$ over $\\ZZ/starlights\\ZZ$ for any positive integer $cloudscape$ and any prime number $starlights$. One is to use M\"obius inversion to count the number of irreducible polynomials of degree $cloudscape$ over $\\ZZ/starlights\\ZZ$ and then give a positive lower bound for this count. The other is to first establish the existence of a finite field $\\FF$ of cardinality $starlights^{cloudscape}$, e.g., as the set of roots of the polynomial $juiceroad^{starlights^{cloudscape}}-1$ inside a splitting field, and then take the minimal polynomial of a nonzero element of $\\FF$ over $\\ZZ/starlights\\ZZ$ which is a primitive $(starlights^{cloudscape}-1)$-st root of unity in $\\FF$ (which exist because the multiplicative group of $\\FF$ contains at most one cyclic subgroup of any given order). One might be tempted to apply the primitive element theorem for $\\FF$ over $\\ZZ/starlights\\ZZ$, but in fact one of the preceding techniques is needed in order to verify this result for finite fields, as the standard argument that ``most'' elements of the upper field are primitive breaks down for finite fields.\n\nOne may also describe the preceding analysis in terms of an identification of $\\FF$ as a $\\ZZ/starlights\\ZZ$-vector space with the space of column vectors of length $cloudscape$. Under such an identification, if we take $marigolds$ to be an element of $\\FF - \\{0\\}$ generating this group, then any element of $telescope(\\{marigolds\\})$ commutes with all of $\\FF- \\{0\\}$ and hence must define an $\\FF$-linear endomorphism of $\\FF$. Any such endomorphism is itself multiplication by an element of $\\FF$, so $telescope(\\{marigolds\\})$ is identified with the multiplicative group of $\\FF$, whose order is the odd number $2^{cloudscape}-1$. "
    },
    "descriptive_long_misleading": {
      "map": {
        "F": "densewood",
        "G": "chaosland",
        "S": "totality",
        "Z": "periphery",
        "g": "voiditem",
        "h": "nullatom",
        "k": "negindex",
        "P": "flatvalue",
        "P_{n-1}": "stillabove",
        "P_n-1": "stillabove",
        "P_0": "stillzero",
        "P_1": "stillone",
        "P_2": "stilltwo",
        "x": "fixpoint",
        "n": "zeroness",
        "p": "composite"
      },
      "question": "In the 75th annual Putnam Games, participants compete at mathematical games.\nPatniss and Keeta play a game in which they take turns choosing an element \nfrom the group of invertible $zeroness \\times zeroness$ matrices with entries in the field\n$\\mathbb{Z}/composite \\mathbb{Z}$ of integers modulo $composite$, where $zeroness$ is a fixed positive integer and $composite$ is a fixed prime number. The rules of the game are:\n\\begin{enumerate}\n\\item[(1)]\nA player cannot choose an element that has been chosen by either player on any previous turn.\n\\item[(2)]\nA player can only choose an element that commutes with all previously chosen elements.\n\\item[(3)]\nA player who cannot choose an element on his/her turn loses the game.\n\\end{enumerate}\nPatniss takes the first turn. Which player has a winning strategy?\n(Your answer may depend on $zeroness$ and $composite$.)",
      "solution": "We show that Patniss wins if $composite=2$ and Keeta wins if $composite>2$ (for all $zeroness$).\nWe first analyze the analogous game played using an arbitrary finite group $chaosland$.\nRecall that for any subset $totality$ of $chaosland$, the set of elements $voiditem \\in chaosland$ which commute with all elements of $totality$ forms a subgroup $\\periphery(totality)$ of $chaosland$. \nAt any given point in the game, the set $totality$ of previously chosen elements is contained in $\\periphery(totality)$. Initially $totality = \\emptyset$ and $\\periphery(totality) = chaosland$;\nafter each turn, $totality$ is increased by one element and $\\periphery(totality)$ is replaced by a subgroup.\nIn particular, if the order of $\\periphery(totality)$ is odd at some point, it remains odd thereafter;\nconversely, if $totality$ contains an element of even order, then the order of $\\periphery(totality)$ remains even thereafter. Therefore, any element $voiditem \\in chaosland$ for which $\\periphery(\\{voiditem\\})$ has odd order is a winning first move for Patniss, while any other first move by Patniss loses if Keeta responds with some $nullatom \\in \\periphery(\\{voiditem\\})$ of even order (e.g., an element of a 2-Sylow subgroup of $\\periphery(\\{voiditem\\})$). In both cases, the win is guaranteed no matter what moves follow.\n\nNow let $chaosland$ be the group of invertible $zeroness \\times zeroness$ matrices with entries in $\\ZZ/composite\\ZZ$.\nIf $composite>2$, then $\\periphery(totality)$ will always contain the scalar matrix $-1$ of order 2, so the win for Keeta is guaranteed. (An explicit winning strategy is to answer any move $voiditem$ with the move $-voiditem$.)\n\nIf $composite=2$, we establish the existence of $voiditem \\in chaosland$ such that $\\periphery(\\{voiditem\\})$ has odd order\nusing the existence of an irreducible polynomial $flatvalue(fixpoint)$ of degree $zeroness$ over $\\ZZ/composite\\ZZ$ (see remark). We construct an $zeroness \\times zeroness$ matrix over $\\ZZ/composite\\ZZ$ with characteristic polynomial $flatvalue(fixpoint)$ by taking the \\emph{companion matrix} of $flatvalue(fixpoint)$: write $flatvalue(fixpoint) = fixpoint^{zeroness} + stillabove \\, fixpoint^{zeroness-1} + \\cdots + stillzero$ and set\n\\[\nvoiditem = \\begin{pmatrix}\n0 & 0 & \\cdots & 0 & -stillzero \\\\\n1 & 0 & \\cdots & 0 & -stillone \\\\\n0 & 1 & \\cdots & 0 & -stilltwo \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & 1 & -stillabove\n\\end{pmatrix}.\n\\]\nIn particular, $\\det(voiditem) = (-1)^{zeroness} \\, stillzero \\neq 0$, so $voiditem \\in chaosland$.\nOver an algebraic closure of $\\ZZ/composite \\ZZ$, $voiditem$ becomes diagonalizable with distinct eigenvalues, so any matrix commuting with $voiditem$ must also be diagonalizable, and hence of odd order. In particular, $\\periphery(\\{voiditem\\})$ is of odd order, so Patniss has a winning strategy.\n\n\\noindent\n\\textbf{Remark:}\nIt can be shown that in the case $composite=2$, the only elements $voiditem \\in chaosland$ for which $\\periphery(\\{voiditem\\})$ has odd order are those for which $voiditem$ has distinct eigenvalues: in any other case, $\\periphery(\\{voiditem\\})$ contains a subgroup isomorphic to the group of $negindex \\times negindex$ invertible matrices over $\\ZZ/2\\ZZ$ for some $negindex>1$, and this group has order $(2^{negindex}-1)(2^{negindex}-2) \\cdots(2^{negindex} - 2^{negindex-1})$.\n\n\\noindent\n\\textbf{Remark:}\nWe sketch two ways to verify the existence of an irreducible polynomial of degree $zeroness$ over $\\ZZ/composite\\ZZ$ for any positive integer $zeroness$ and any prime number $composite$. One is to use M\"obius inversion to count the number of irreducible polynomials of degree $zeroness$ over $\\ZZ/composite\\ZZ$ and then give a positive lower bound for this count. The other is to first establish the existence of a finite field $\\FF$ of cardinality $composite^{zeroness}$, e.g., as the set of roots of the polynomial $fixpoint^{composite^{zeroness}}-1$ inside a splitting field, and then take the minimal polynomial of a nonzero element of $\\FF$ over $\\ZZ/composite\\ZZ$ which is a primitive $(composite^{zeroness}-1)$-st root of unity in $\\FF$ (which exist because the multiplicative group of $\\FF$ contains at most one cyclic subgroup of any given order). One might be tempted to apply the primitive element theorem for $\\FF$ over $\\ZZ/composite\\ZZ$, but in fact one of the preceding techniques is needed in order to verify this result for finite fields, as the standard argument that ``most'' elements of the upper field are primitive breaks down for finite fields.\n\nOne may also describe the preceding analysis in terms of an identification of $\\FF$ as a $\\ZZ/composite\\ZZ$-vector space with the space of column vectors of length $zeroness$. Under such an identification, if we take $voiditem$ to be an element of $\\FF - \\{0\\}$ generating this group, then any element of $\\periphery(\\{voiditem\\})$ commutes with all of $\\FF- \\{0\\}$ and hence must define an $\\FF$-linear endomorphism of $\\FF$. Any such endomorphism is itself multiplication by an element of $\\FF$, so $\\periphery(\\{voiditem\\})$ is identified with the multiplicative group of $\\FF$, whose order is the odd number $2^{zeroness}-1$.}",
      "confidence": "0.14"
    },
    "garbled_string": {
      "map": {
        "F": "lkjweprq",
        "G": "vbhtsdml",
        "S": "qertmzph",
        "Z": "asdfjklq",
        "g": "mzcylpwo",
        "h": "nbxqrevi",
        "k": "pojgnsal",
        "P": "wudkqzcn",
        "P_n-1": "rnmdkgqe",
        "P_0": "bzxmskqf",
        "P_1": "ldawoepr",
        "P_2": "hvjksnqe",
        "x": "hvrplwqe",
        "n": "yzptxwvl",
        "p": "churnadk"
      },
      "question": "In the 75th annual Putnam Games, participants compete at mathematical games.\nPatniss and Keeta play a game in which they take turns choosing an element \nfrom the group of invertible $yzptxwvl \\times yzptxwvl$ matrices with entries in the field\n$\\mathbb{Z}/churnadk \\mathbb{Z}$ of integers modulo $churnadk$, where $yzptxwvl$ is a fixed positive integer and $churnadk$ is a fixed prime number. The rules of the game are:\n\\begin{enumerate}\n\\item[(1)]\nA player cannot choose an element that has been chosen by either player on any previous turn.\n\\item[(2)]\nA player can only choose an element that commutes with all previously chosen elements.\n\\item[(3)]\nA player who cannot choose an element on his/her turn loses the game.\n\\end{enumerate}\nPatniss takes the first turn. Which player has a winning strategy?\n(Your answer may depend on $yzptxwvl$ and $churnadk$.)",
      "solution": "We show that Patniss wins if $churnadk=2$ and Keeta wins if $churnadk>2$ (for all $yzptxwvl$).\nWe first analyze the analogous game played using an arbitrary finite group $vbhtsdml$.\nRecall that for any subset $qertmzph$ of $vbhtsdml$, the set of elements $mzcylpwo \\in vbhtsdml$ which commute with all elements of $qertmzph$ forms a subgroup $asdfjklq(qertmzph)$ of $vbhtsdml$, called the \\emph{centralizer} (or \\emph{commutant}) of $qertmzph$.\nAt any given point in the game, the set $qertmzph$ of previously chosen elements is contained in $asdfjklq(qertmzph)$. Initially $qertmzph = \\emptyset$ and $asdfjklq(qertmzph) = vbhtsdml$;\nafter each turn, $qertmzph$ is increased by one element and $asdfjklq(qertmzph)$ is replaced by a subgroup.\nIn particular, if the order of $asdfjklq(qertmzph)$ is odd at some point, it remains odd thereafter;\nconversely, if $qertmzph$ contains an element of even order, then the order of $asdfjklq(qertmzph)$ remains even thereafter. Therefore, any element $mzcylpwo \\in vbhtsdml$ for which $asdfjklq(\\{mzcylpwo\\})$ has odd order is a winning first move for Patniss, while any other first move by Patniss loses if Keeta responds with some $nbxqrevi \\in asdfjklq(\\{mzcylpwo\\})$ of even order (e.g., an element of a 2-Sylow subgroup of $asdfjklq(\\{mzcylpwo\\})$). In both cases, the win is guaranteed no matter what moves follow.\n\nNow let $vbhtsdml$ be the group of invertible $yzptxwvl \\times yzptxwvl$ matrices with entries in $\\ZZ/churnadk\\ZZ$.\nIf $churnadk>2$, then $asdfjklq(qertmzph)$ will always contain the scalar matrix $-1$ of order 2, so the win for Keeta is guaranteed. (An explicit winning strategy is to answer any move $mzcylpwo$ with the move $-mzcylpwo$.)\n\nIf $churnadk=2$, we establish the existence of $mzcylpwo \\in vbhtsdml$ such that $asdfjklq(\\{mzcylpwo\\})$ has odd order\nusing the existence of an irreducible polynomial $wudkqzcn(hvrplwqe)$ of degree $yzptxwvl$ over $\\ZZ/churnadk\\ZZ$ (see remark). We construct an $yzptxwvl \\times yzptxwvl$ matrix over $\\ZZ/churnadk\\ZZ$ with characteristic polynomial $wudkqzcn(hvrplwqe)$ by taking the \\emph{companion matrix} of $wudkqzcn(hvrplwqe)$: write \n\\[\nwudkqzcn(hvrplwqe) = hvrplwqe^{yzptxwvl} + rnmdkgqe\\, hvrplwqe^{yzptxwvl-1} + \\cdots + bzxmskqf\n\\]\nand set\n\\[\nmzcylpwo = \\begin{pmatrix}\n0 & 0 & \\cdots & 0 & -bzxmskqf \\\\\n1 & 0 & \\cdots & 0 & -ldawoepr \\\\\n0 & 1 & \\cdots & 0 & -hvjksnqe \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & 1 & -rnmdkgqe\n\\end{pmatrix}.\n\\]\nIn particular, $\\det(mzcylpwo) = (-1)^{yzptxwvl} bzxmskqf \\neq 0$, so $mzcylpwo \\in vbhtsdml$.\nOver an algebraic closure of $\\ZZ/churnadk \\ZZ$, $mzcylpwo$ becomes diagonalizable with distinct eigenvalues, so any matrix commuting with $mzcylpwo$ must also be diagonalizable, and hence of odd order. In particular, $asdfjklq(\\{mzcylpwo\\})$ is of odd order, so Patniss has a winning strategy.\n\n\\textbf{Remark:}\nIt can be shown that in the case $churnadk=2$, the only elements $mzcylpwo \\in vbhtsdml$ for which $asdfjklq(\\{mzcylpwo\\})$ has odd order are those for which $mzcylpwo$ has distinct eigenvalues: in any other case, $asdfjklq(\\{mzcylpwo\\})$ contains a subgroup isomorphic to the group of $pojgnsal \\times pojgnsal$ invertible matrices over $\\ZZ/2\\ZZ$ for some $pojgnsal>1$, and this group has order $(2^{pojgnsal}-1)(2^{pojgnsal}-2) \\cdots(2^{pojgnsal} - 2^{pojgnsal-1})$.\n\n\\textbf{Remark:}\nWe sketch two ways to verify the existence of an irreducible polynomial of degree $yzptxwvl$ over $\\ZZ/churnadk\\ZZ$ for any positive integer $yzptxwvl$ and any prime number $churnadk$. One is to use M\"obius inversion to count the number of irreducible polynomials of degree $yzptxwvl$ over $\\ZZ/churnadk\\ZZ$ and then give a positive lower bound for this count. The other is to first establish the existence of a finite field $\\FF$ of cardinality $churnadk^{yzptxwvl}$, e.g., as the set of roots of the polynomial $hvrplwqe^{churnadk^{yzptxwvl}}-1$ inside a splitting field, and then take the minimal polynomial of a nonzero element of $\\FF$ over $\\ZZ/churnadk\\ZZ$ which is a primitive $(churnadk^{yzptxwvl}-1)$-st root of unity in $\\FF$ (which exist because the multiplicative group of $\\FF$ contains at most one cyclic subgroup of any given order). One might be tempted to apply the primitive element theorem for $\\FF$ over $\\ZZ/churnadk\\ZZ$, but in fact one of the preceding techniques is needed in order to verify this result for finite fields, as the standard argument that ``most\" elements of the upper field are primitive breaks down for finite fields.\n\nOne may also describe the preceding analysis in terms of an identification of $\\FF$ as a $\\ZZ/churnadk\\ZZ$-vector space with the space of column vectors of length $yzptxwvl$. Under such an identification, if we take $mzcylpwo$ to be an element of $\\FF - \\{0\\}$ generating this group, then any element of $asdfjklq(\\{mzcylpwo\\})$ commutes with all of $\\FF- \\{0\\}$ and hence must define an $\\FF$-linear endomorphism of $\\FF$. Any such endomorphism is itself multiplication by an element of $\\FF$, so $asdfjklq(\\{mzcylpwo\\})$ is identified with the multiplicative group of $\\FF$, whose order is the odd number $2^{yzptxwvl}-1$.}",
      "confidence": 0.11
    },
    "kernel_variant": {
      "question": "Let $n\\ge 2$ be an \\emph{even} integer and let $p$ be a prime number.  Denote  \n\\[\nG_{n,p}\\;:=\\;\\bigl\\{A\\in SL_{n}(\\mathbb{F}_{p})\\ \\bigm|\\ \\operatorname{tr}A=0\\bigr\\}.\n\\tag{$\\dagger$}\n\\]\nTwo players, \\textbf{Alice} and \\textbf{Bob}, alternately pick matrices subject to  \n\\begin{enumerate}\n\\item the chosen matrix lies in $G_{n,p}$;\n\\item the same matrix may not be chosen twice;\n\\item each new matrix must commute with every matrix selected earlier.\n\\end{enumerate}\nA player who has no legal move on her/his turn loses.\n\nFor every pair $(n,p)$ with $n$ even, determine which player possesses a winning strategy.",
      "solution": "Throughout put $\\mathbb{F}:=\\mathbb{F}_{p}$.  \nFor every finite subset $S\\subseteq G_{n,p}$ write  \n\\[\nC(S):=\\bigl\\{A\\in G_{n,p}\\,\\bigm|\\,AB=BA\\text{ for all }B\\in S\\bigr\\},\n\\]\nthe current \\emph{legal–move set}.  For a finite set $X$, let $|X|$ denote its cardinality.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n0.  Two general principles\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Fact 1 (Odd–centralizer parity principle).}  \nLet $g\\in G_{n,p}$ such that  \n(i) $C\\bigl(\\{g\\}\\bigr)$ is abelian, and  \n(ii) $|C\\bigl(\\{g\\}\\bigr)|$ is \\emph{odd}.  \n\nIf Alice opens with $g$, then for every subsequent move $h\\in C(S)$ we have\n$C(S\\cup\\{h\\})=C(S)$ because $C\\bigl(\\{g\\}\\bigr)$ is abelian.  \nConsequently each turn removes exactly one element from the fixed finite\nset $C\\bigl(\\{g\\}\\bigr)$.  \nSince its size is odd, Alice (who plays at the odd–numbered turns)\ninevitably makes the \\emph{very last} move; Bob then has no legal move and loses.\nHence Alice wins.\n\n\\smallskip\n\\textbf{Fact 2 (Pairing principle).}  \nLet $\\tau:G_{n,p}\\longrightarrow G_{n,p}$ be a fixed–point–free involution\npreserving commutation:\n\\[\nAB=BA\\;\\Longleftrightarrow\\;\n\\tau(A)\\tau(B)=\\tau(B)\\tau(A).\n\\]\nIf the starting position is $\\tau$–stable and $|G_{n,p}|$ is even, the\nsecond player Bob can answer every move $X$ with $\\tau(X)$.\nThe two elements of such a $\\tau$–orbit disappear together,\nso Bob never runs out of moves and therefore wins.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n1.  The easy half: odd prime $p$  \\;(Bob wins)\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nDefine the involution\n\\[\n\\tau:G_{n,p}\\longrightarrow G_{n,p},\\qquad X\\longmapsto -X.\n\\tag{1}\n\\]\nBecause $n$ is even, $\\det(-X)=\\det X=1$ and $\\operatorname{tr}(-X)=\n-\\operatorname{tr}X=0$, hence $-X\\in G_{n,p}$.  \nFor $p$ odd the equality $-X=X$ would force $2X=0$, hence $X=0$,\ncontradicting $\\det X=1$; thus $\\tau$ is fixed–point–free.\nIt obviously preserves commutation, so Fact 2 applies.\nSince every element belongs to a $\\tau$–pair, $|G_{n,p}|$ is even, and Bob wins.\n\n\\[\n\\boxed{\\text{Bob wins for every odd prime }p\\text{ and all even }n.}\n\\tag{2}\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n2.  The subtle half: $p=2$\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFor $p=2$ the involution \\eqref{1} collapses to the identity,\nnecessitating a new analysis.  Separate $n=2$ and $n\\ge 4$.\n\n--------------------------------------------------------\n2.1.  The miniature arena $G_{2,2}$ \\;(Bob wins)\n--------------------------------------------------------\nRecall $|SL_{2}(\\mathbb{F}_{2})|=6$ and list  \n\\[\nI=\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix},\\;\nU=\\begin{bmatrix}1&1\\\\0&1\\end{bmatrix},\\;\nS=\\begin{bmatrix}0&1\\\\1&0\\end{bmatrix},\\;\nL=\\begin{bmatrix}1&0\\\\1&1\\end{bmatrix}.\n\\]\nAll four non–scalar matrices have trace $0$, hence\n\\[\nG_{2,2}=\\{I,U,S,L\\}.\n\\tag{3}\n\\]\nCommutation relations: $I$ commutes with everyone, while the three\nnon–identity elements do \\emph{not} commute mutually.  Thus for\n$X\\in\\{U,S,L\\}$,\n\\[\nC\\bigl(\\{X\\}\\bigr)=\\{I,X\\}.\n\\tag{4}\n\\]\nGame tree.  \nIf Alice begins with $I$, Bob plays any of $\\{U,S,L\\}$ and no moves remain.\nIf Alice starts with $U,S$, or $L$, Bob replies $I$; again no move is left.\nHence Bob wins when $(n,p)=(2,2)$.\n\n\\[\n\\boxed{\\text{Bob wins for }(n,p)=(2,2).}\n\\tag{5}\n\\]\n\n--------------------------------------------------------\n2.2.  Even dimensions $n\\ge 4$ \\;(Alice wins)\n--------------------------------------------------------\nWe exhibit an opening move $g$ whose centralizer inside $G_{n,2}$ is\nabelian of \\emph{odd} size and then invoke Fact 1.\n\n\\emph{Step 1.  Field model.}\nIdentify the $\\mathbb{F}_{2}$–space $\\mathbb{F}_{2}^{\\,n}$ with the\nfinite field $\\mathbb{F}:=\\mathbb{F}_{2^{n}}$.\nFor $\\beta\\in\\mathbb{F}$ define $T_{\\beta}:x\\mapsto\\beta x$.\nRelative to the standard basis, $T_{\\beta}$ is an $n\\times n$ matrix over\n$\\mathbb{F}_{2}$.  One has\n\\[\n\\det T_{\\beta}=N(\\beta),\\qquad\n\\operatorname{tr}T_{\\beta}=\\operatorname{Tr}(\\beta),\n\\]\nwhere $N$ and $\\operatorname{Tr}$ are the absolute\nnorm and trace $\\mathbb{F}_{2^{n}}\\to\\mathbb{F}_{2}$.\n\n\\emph{Step 2.  Choosing $\\beta$.}\nBecause $n\\ge 4$, the hyperplane $\\ker(\\operatorname{Tr})$\ncontains $2^{\\,n-1}>2$ elements, so pick $\\beta\\in\\mathbb{F}^{\\times}$\nwith $\\operatorname{Tr}\\beta=0$ and $\\beta\\neq 1$.\nSince $\\mathbb{F}_{2}^{\\times}=\\{1\\}$, every non–zero element has norm $1$,\nhence $\\det T_{\\beta}=1$.  Set\n\\[\ng:=T_{\\beta}\\;\\in\\;SL_{n}(\\mathbb{F}_{2}), \\qquad\\operatorname{tr}g=0,\n\\]\nso $g\\in G_{n,2}$.\n\nBecause $\\beta$ is separable, $g$ is regular semisimple, and its\ncentralizer in $GL_{n}(\\mathbb{F}_{2})$ is the torus\n$\\{T_{\\gamma}\\mid\\gamma\\in\\mathbb{F}^{\\times}\\}$, which is abelian.\n\n\\emph{Step 3.  Centralizer of $g$ inside $G_{n,2}$.}\nA matrix commuting with $g$ is $T_{\\gamma}$ for a unique $\\gamma\\in\\mathbb{F}^{\\times}$.\nNow $T_{\\gamma}\\in SL_{n}(\\mathbb{F}_{2})$ automatically, and\n$T_{\\gamma}\\in G_{n,2}$ exactly when $\\operatorname{Tr}\\gamma=0$.\nTherefore\n\\[\nC\\bigl(\\{g\\}\\bigr)=\n\\bigl\\{T_{\\gamma}\\bigm|\\gamma\\in\\mathbb{F}^{\\times},\\;\n\\operatorname{Tr}\\gamma=0\\bigr\\},\\qquad\n|C\\bigl(\\{g\\}\\bigr)|=2^{\\,n-1}-1.\n\\tag{6}\n\\]\nBecause $n$ is even $\\ge 4$, the exponent $n-1$ is odd, so\n$|C\\bigl(\\{g\\}\\bigr)|$ is \\emph{odd}.  The set is abelian by construction.\n\n\\emph{Step 4.  Parity–win.}\nAll hypotheses of Fact 1 are met; hence Alice, opening with $g$,\nforces the win.\n\n\\[\n\\boxed{\\text{Alice wins for }p=2\\text{ and even }n\\ge 4.}\n\\tag{7}\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n3.  Final classification\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nCollecting \\eqref{2}, \\eqref{5}, and \\eqref{7}:\n\n\\[\n\\begin{aligned}\n&\\text{Alice wins}\\quad\\Longleftrightarrow\\quad p=2\\text{ and }n\\ge 4\\text{ (even)},\\\\\n&\\text{Bob wins}\\quad\\Longleftrightarrow\\quad\n\\bigl(p\\text{ odd}\\bigr)\\ \\text{or}\\ \\bigl(n,p\\bigr)=(2,2).\n\\end{aligned}\n\\]\n\nHence the game has been completely solved.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.834972",
        "was_fixed": false,
        "difficulty_analysis": "1. Added structural constraints: players may pick only matrices that are simultaneously (i) traceless and (ii) determinant $1$.  These two *non-central* linear/ multiplicative conditions interact and obstruct the classical “pick the negative” strategy except in a very narrow arithmetic window.\n\n2. Required a blend of:\n   • linear algebra over finite fields,  \n   • field–theoretic representation of $\\operatorname{GL}_{n}(\\mathbb F_p)$ via $\\mathbb F_{p^{n}}$,  \n   • existence of elements with prescribed trace and norm (Hilbert’s Theorem 90 flavor),  \n   • structure of centralisers of regular semisimple elements,  \n   • cyclic-group arithmetic and 2-adic valuations (lifting-the-exponent lemma).\n\n3. The parity analysis now depends on three independent arithmetic\n   parameters (the prime $p$, the parity of $n$, and the divisibility\n   relation $p\\mid n$) instead of the single dichotomy “$p=2$ or not”.\n\n4. Constructing the winning first move for Alice is *much* subtler; it\n   uses trace-and-norm conditions inside a cyclic group of size $p^{n}-1$,\n   whereas the original problem needed only an element with distinct\n   eigenvalues when $p=2$.\n\n5. Bob’s sole winning situation had to be isolated by\n   simultaneously analysing determinant,\n   trace, order, and centrality—far more delicate than noticing $-I$ is\n   always present.  \n\nCollectively these features force the solver to employ field theory,\ncyclic-group valuation calculations and fine centraliser structure,\nmaking the variant decisively harder than the original."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n\\ge 2$ be an \\emph{even} integer and let $p$ be a prime number.  Denote  \n\\[\nG_{n,p}\\;:=\\;\\bigl\\{A\\in SL_{n}(\\mathbb{F}_{p})\\ \\bigm|\\ \\operatorname{tr}A=0\\bigr\\}.\n\\tag{$\\dagger$}\n\\]\nTwo players, \\textbf{Alice} and \\textbf{Bob}, alternately pick matrices subject to  \n\\begin{enumerate}\n\\item the chosen matrix lies in $G_{n,p}$;\n\\item the same matrix may not be chosen twice;\n\\item each new matrix must commute with every matrix selected earlier.\n\\end{enumerate}\nA player who has no legal move on her/his turn loses.\n\nFor every pair $(n,p)$ with $n$ even, determine which player possesses a winning strategy.",
      "solution": "Throughout put $\\mathbb{F}:=\\mathbb{F}_{p}$.  \nFor every finite subset $S\\subseteq G_{n,p}$ write  \n\\[\nC(S):=\\bigl\\{A\\in G_{n,p}\\,\\bigm|\\,AB=BA\\text{ for all }B\\in S\\bigr\\},\n\\]\nthe current \\emph{legal–move set}.  For a finite set $X$, let $|X|$ denote its cardinality.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n0.  Two general principles\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Fact 1 (Odd–centralizer parity principle).}  \nLet $g\\in G_{n,p}$ such that  \n(i) $C\\bigl(\\{g\\}\\bigr)$ is abelian, and  \n(ii) $|C\\bigl(\\{g\\}\\bigr)|$ is \\emph{odd}.  \n\nIf Alice opens with $g$, then for every subsequent move $h\\in C(S)$ we have\n$C(S\\cup\\{h\\})=C(S)$ because $C\\bigl(\\{g\\}\\bigr)$ is abelian.  \nConsequently each turn removes exactly one element from the fixed finite\nset $C\\bigl(\\{g\\}\\bigr)$.  \nSince its size is odd, Alice (who plays at the odd–numbered turns)\ninevitably makes the \\emph{very last} move; Bob then has no legal move and loses.\nHence Alice wins.\n\n\\smallskip\n\\textbf{Fact 2 (Pairing principle).}  \nLet $\\tau:G_{n,p}\\longrightarrow G_{n,p}$ be a fixed–point–free involution\npreserving commutation:\n\\[\nAB=BA\\;\\Longleftrightarrow\\;\n\\tau(A)\\tau(B)=\\tau(B)\\tau(A).\n\\]\nIf the starting position is $\\tau$–stable and $|G_{n,p}|$ is even, the\nsecond player Bob can answer every move $X$ with $\\tau(X)$.\nThe two elements of such a $\\tau$–orbit disappear together,\nso Bob never runs out of moves and therefore wins.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n1.  The easy half: odd prime $p$  \\;(Bob wins)\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nDefine the involution\n\\[\n\\tau:G_{n,p}\\longrightarrow G_{n,p},\\qquad X\\longmapsto -X.\n\\tag{1}\n\\]\nBecause $n$ is even, $\\det(-X)=\\det X=1$ and $\\operatorname{tr}(-X)=\n-\\operatorname{tr}X=0$, hence $-X\\in G_{n,p}$.  \nFor $p$ odd the equality $-X=X$ would force $2X=0$, hence $X=0$,\ncontradicting $\\det X=1$; thus $\\tau$ is fixed–point–free.\nIt obviously preserves commutation, so Fact 2 applies.\nSince every element belongs to a $\\tau$–pair, $|G_{n,p}|$ is even, and Bob wins.\n\n\\[\n\\boxed{\\text{Bob wins for every odd prime }p\\text{ and all even }n.}\n\\tag{2}\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n2.  The subtle half: $p=2$\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nFor $p=2$ the involution \\eqref{1} collapses to the identity,\nnecessitating a new analysis.  Separate $n=2$ and $n\\ge 4$.\n\n--------------------------------------------------------\n2.1.  The miniature arena $G_{2,2}$ \\;(Bob wins)\n--------------------------------------------------------\nRecall $|SL_{2}(\\mathbb{F}_{2})|=6$ and list  \n\\[\nI=\\begin{bmatrix}1&0\\\\0&1\\end{bmatrix},\\;\nU=\\begin{bmatrix}1&1\\\\0&1\\end{bmatrix},\\;\nS=\\begin{bmatrix}0&1\\\\1&0\\end{bmatrix},\\;\nL=\\begin{bmatrix}1&0\\\\1&1\\end{bmatrix}.\n\\]\nAll four non–scalar matrices have trace $0$, hence\n\\[\nG_{2,2}=\\{I,U,S,L\\}.\n\\tag{3}\n\\]\nCommutation relations: $I$ commutes with everyone, while the three\nnon–identity elements do \\emph{not} commute mutually.  Thus for\n$X\\in\\{U,S,L\\}$,\n\\[\nC\\bigl(\\{X\\}\\bigr)=\\{I,X\\}.\n\\tag{4}\n\\]\nGame tree.  \nIf Alice begins with $I$, Bob plays any of $\\{U,S,L\\}$ and no moves remain.\nIf Alice starts with $U,S$, or $L$, Bob replies $I$; again no move is left.\nHence Bob wins when $(n,p)=(2,2)$.\n\n\\[\n\\boxed{\\text{Bob wins for }(n,p)=(2,2).}\n\\tag{5}\n\\]\n\n--------------------------------------------------------\n2.2.  Even dimensions $n\\ge 4$ \\;(Alice wins)\n--------------------------------------------------------\nWe exhibit an opening move $g$ whose centralizer inside $G_{n,2}$ is\nabelian of \\emph{odd} size and then invoke Fact 1.\n\n\\emph{Step 1.  Field model.}\nIdentify the $\\mathbb{F}_{2}$–space $\\mathbb{F}_{2}^{\\,n}$ with the\nfinite field $\\mathbb{F}:=\\mathbb{F}_{2^{n}}$.\nFor $\\beta\\in\\mathbb{F}$ define $T_{\\beta}:x\\mapsto\\beta x$.\nRelative to the standard basis, $T_{\\beta}$ is an $n\\times n$ matrix over\n$\\mathbb{F}_{2}$.  One has\n\\[\n\\det T_{\\beta}=N(\\beta),\\qquad\n\\operatorname{tr}T_{\\beta}=\\operatorname{Tr}(\\beta),\n\\]\nwhere $N$ and $\\operatorname{Tr}$ are the absolute\nnorm and trace $\\mathbb{F}_{2^{n}}\\to\\mathbb{F}_{2}$.\n\n\\emph{Step 2.  Choosing $\\beta$.}\nBecause $n\\ge 4$, the hyperplane $\\ker(\\operatorname{Tr})$\ncontains $2^{\\,n-1}>2$ elements, so pick $\\beta\\in\\mathbb{F}^{\\times}$\nwith $\\operatorname{Tr}\\beta=0$ and $\\beta\\neq 1$.\nSince $\\mathbb{F}_{2}^{\\times}=\\{1\\}$, every non–zero element has norm $1$,\nhence $\\det T_{\\beta}=1$.  Set\n\\[\ng:=T_{\\beta}\\;\\in\\;SL_{n}(\\mathbb{F}_{2}), \\qquad\\operatorname{tr}g=0,\n\\]\nso $g\\in G_{n,2}$.\n\nBecause $\\beta$ is separable, $g$ is regular semisimple, and its\ncentralizer in $GL_{n}(\\mathbb{F}_{2})$ is the torus\n$\\{T_{\\gamma}\\mid\\gamma\\in\\mathbb{F}^{\\times}\\}$, which is abelian.\n\n\\emph{Step 3.  Centralizer of $g$ inside $G_{n,2}$.}\nA matrix commuting with $g$ is $T_{\\gamma}$ for a unique $\\gamma\\in\\mathbb{F}^{\\times}$.\nNow $T_{\\gamma}\\in SL_{n}(\\mathbb{F}_{2})$ automatically, and\n$T_{\\gamma}\\in G_{n,2}$ exactly when $\\operatorname{Tr}\\gamma=0$.\nTherefore\n\\[\nC\\bigl(\\{g\\}\\bigr)=\n\\bigl\\{T_{\\gamma}\\bigm|\\gamma\\in\\mathbb{F}^{\\times},\\;\n\\operatorname{Tr}\\gamma=0\\bigr\\},\\qquad\n|C\\bigl(\\{g\\}\\bigr)|=2^{\\,n-1}-1.\n\\tag{6}\n\\]\nBecause $n$ is even $\\ge 4$, the exponent $n-1$ is odd, so\n$|C\\bigl(\\{g\\}\\bigr)|$ is \\emph{odd}.  The set is abelian by construction.\n\n\\emph{Step 4.  Parity–win.}\nAll hypotheses of Fact 1 are met; hence Alice, opening with $g$,\nforces the win.\n\n\\[\n\\boxed{\\text{Alice wins for }p=2\\text{ and even }n\\ge 4.}\n\\tag{7}\n\\]\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n3.  Final classification\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\nCollecting \\eqref{2}, \\eqref{5}, and \\eqref{7}:\n\n\\[\n\\begin{aligned}\n&\\text{Alice wins}\\quad\\Longleftrightarrow\\quad p=2\\text{ and }n\\ge 4\\text{ (even)},\\\\\n&\\text{Bob wins}\\quad\\Longleftrightarrow\\quad\n\\bigl(p\\text{ odd}\\bigr)\\ \\text{or}\\ \\bigl(n,p\\bigr)=(2,2).\n\\end{aligned}\n\\]\n\nHence the game has been completely solved.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.637594",
        "was_fixed": false,
        "difficulty_analysis": "1. Added structural constraints: players may pick only matrices that are simultaneously (i) traceless and (ii) determinant $1$.  These two *non-central* linear/ multiplicative conditions interact and obstruct the classical “pick the negative” strategy except in a very narrow arithmetic window.\n\n2. Required a blend of:\n   • linear algebra over finite fields,  \n   • field–theoretic representation of $\\operatorname{GL}_{n}(\\mathbb F_p)$ via $\\mathbb F_{p^{n}}$,  \n   • existence of elements with prescribed trace and norm (Hilbert’s Theorem 90 flavor),  \n   • structure of centralisers of regular semisimple elements,  \n   • cyclic-group arithmetic and 2-adic valuations (lifting-the-exponent lemma).\n\n3. The parity analysis now depends on three independent arithmetic\n   parameters (the prime $p$, the parity of $n$, and the divisibility\n   relation $p\\mid n$) instead of the single dichotomy “$p=2$ or not”.\n\n4. Constructing the winning first move for Alice is *much* subtler; it\n   uses trace-and-norm conditions inside a cyclic group of size $p^{n}-1$,\n   whereas the original problem needed only an element with distinct\n   eigenvalues when $p=2$.\n\n5. Bob’s sole winning situation had to be isolated by\n   simultaneously analysing determinant,\n   trace, order, and centrality—far more delicate than noticing $-I$ is\n   always present.  \n\nCollectively these features force the solver to employ field theory,\ncyclic-group valuation calculations and fine centraliser structure,\nmaking the variant decisively harder than the original."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}