{
  "index": "2017-A-6",
  "type": "COMB",
  "tag": [
    "COMB",
    "ALG"
  ],
  "difficulty": "",
  "question": "The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways \nare there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]",
  "solution": "The number of such colorings is $2^{20} 3^{10} = 61917364224$.\n\n\\noindent\n\\textbf{First solution:}\nIdentify the three colors red, white, and blue with (in some order) the elements of the field $\\mathbb{F}_3$ of three elements (i.e., the ring of integers mod 3). \nThe set of colorings may then be identified with the $\\mathbb{F}_3$-vector space $\\mathbb{F}_3^E$\ngenerated by the set $E$ of edges. Let $F$ be the set of faces, and let $\\mathbb{F}_3^F$ be the $\\mathbb{F}_3$-vector space on the basis $F$; we may then define a linear transformation\n$T: \\mathbb{F}_3^E \\to \\mathbb{F}_3^F$ taking a coloring to the vector whose component corresponding to a given face equals the sum of the three edges of that face. The colorings we wish to count are the ones whose images under $T$ consist of vectors with no zero components.\n\nWe now show that $T$ is surjective. (There are many possible approaches to this step; for instance, see the following remark.) \nLet $\\Gamma$ be the dual graph of the icosahedron, that is, $\\Gamma$ has vertex set $F$ and two elements of $F$ are adjacent in $\\Gamma$ if they share an edge in the icosahedron. The graph $\\Gamma$ admits a hamiltonian path, that is, there exists an ordering\n$f_1,\\dots,f_{20}$ of the faces such that any two consecutive faces are adjacent in $\\Gamma$. \nFor example, such an ordering can be constructed with $f_1,\\dots,f_5$ being the five faces sharing a vertex of the icosahedron and $f_{16},\\dots,f_{20}$ being the five faces sharing the antipodal vertex.\n\nFor $i=1,\\dots,19$, let $e_i$ be the common edge of $f_i$ and $f_{i+1}$; these are obviously all distinct.\nBy prescribing components for $e_1,\\dots,e_{19}$ in turn and setting the others to zero,\nwe can construct an element of $\\mathbb{F}_3^E$ whose image under $T$ matches any given vector of $\\mathbb{F}_3^F$ in the components of $f_1,\\dots,f_{19}$. The vectors in $\\mathbb{F}_3^F$ obtained in this way thus form a 19-dimensional subspace; this subspace may also be described as the vectors for which the components of $f_1,\\dots,f_{19}$ have the same sum as the components of $f_{2},\\dots,f_{20}$. \n\nBy performing a mirror reflection, we can construct a second hamiltonian path $g_1,\\dots,g_{20}$ with the property that\n\\[\ng_1 = f_1, g_2 = f_5, g_3 = f_4, g_4 = f_3, g_5 = f_2.\n\\]\nRepeating the previous construction, we obtain a \\emph{different} 19-dimensional subspace of $\\mathbb{F}_3^F$ which is contained in the image of $T$. This implies that $T$ is surjective, as asserted earlier.\n\nSince $T$ is a surjective homomorphism from a 30-dimensional vector space to a 20-dimensional vector space, it has a 10-dimensional kernel. Each of the $2^{20}$ elements of $\\mathbb{F}_3^F$ with no zero components is then the image of exactly $3^{10}$ colorings of the desired form, yielding the result.\n\n\\noindent\n\\textbf{Remark:}\nThere are many ways to check that $T$ is surjective. One of the simplest is the following\n(from Art of Problem Solving, user \\texttt{Ravi12346}): form a vector in $\\mathbb{F}^E$ with components $2,1,2,1,2$ at the five edges around some vertex and all other components 0. This maps to a vector in $\\mathbb{F}^F$ with only a single nonzero component; by symmetry, every standard basis vector of $\\mathbb{F}^F$ arises in this way.\n\n\\noindent\n\\textbf{Second solution:}\n(from Bill Huang, via Art of Problem Solving user \\texttt{superpi83})\nLet $v$ and $w$ be two antipodal vertices of the icosahedron. Let $S_v$ (resp.\\ $S_w$) be the set of five edges incident to $v$ (resp.\\ $w$). Let $T_v$ (resp.\\ $T_w$) be the set of five edges of the pentagon formed by the opposite endpoints of the five edges in $S_v$ (resp. $S_w$). Let $U$ be the set of the ten remaining edges of the icosahedron.\n\nConsider any one of the $3^{10}$ possible colorings of $U$. The edges of $T_v \\cup U$ form the boundaries of five faces with no edges in common; thus each edge of $T_v$ can be colored in one of two ways consistent with the given condition,\nand similarly for $T_w$. That is, there are $3^{10} 2^{10}$ possible colorings of $T_v \\cup T_w \\cup U$ consistent with the given condition.\n\nTo complete the count, it suffices to check that there are exactly $2^5$ ways to color $S_v$ consistent with any given coloring of $T_v$. Using the linear-algebraic interpretation from the first solution, this follows by observing that\n(by the previous remark) the map from $\\mathbb{F}_3^{S_v}$ to the $\\mathbb{F}_3$-vector space on the faces incident to $v$ is surjective, and hence an isomorphism for dimensional reasons. A direct combinatorial proof is also possible.",
  "vars": [
    "i",
    "v",
    "w",
    "e_i",
    "f_i",
    "g_i"
  ],
  "params": [
    "F_3",
    "E",
    "F",
    "T",
    "\\\\Gamma",
    "S_v",
    "S_w",
    "T_v",
    "T_w",
    "U"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "i": "indexi",
        "v": "vertexv",
        "w": "vertexw",
        "e_i": "edgeidx",
        "f_i": "faceidx",
        "g_i": "pathidx",
        "F_3": "fieldthree",
        "E": "edgeset",
        "F": "faceset",
        "T": "lintrans",
        "\\Gamma": "dualgraph",
        "S_v": "starvset",
        "S_w": "starwset",
        "T_v": "ringvset",
        "T_w": "ringwset",
        "U": "midedges"
      },
      "question": "The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways \nare there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]",
      "solution": "The number of such colorings is $2^{20} 3^{10} = 61917364224$.\n\n\\noindent\n\\textbf{First solution:}\nIdentify the three colors red, white, and blue with (in some order) the elements of the field $\\mathbb{fieldthree}$ of three elements (i.e., the ring of integers mod 3). \nThe set of colorings may then be identified with the $\\mathbb{fieldthree}^{edgeset}$-vector space $\\mathbb{fieldthree}^{edgeset}$ generated by the set $edgeset$ of edges. Let $faceset$ be the set of faces, and let $\\mathbb{fieldthree}^{faceset}$ be the $\\mathbb{fieldthree}$-vector space on the basis $faceset$; we may then define a linear transformation\n$lintrans: \\mathbb{fieldthree}^{edgeset} \\to \\mathbb{fieldthree}^{faceset}$ taking a coloring to the vector whose component corresponding to a given face equals the sum of the three edges of that face. The colorings we wish to count are the ones whose images under $lintrans$ consist of vectors with no zero components.\n\nWe now show that $lintrans$ is surjective. (There are many possible approaches to this step; for instance, see the following remark.) \nLet $dualgraph$ be the dual graph of the icosahedron, that is, $dualgraph$ has vertex set $faceset$ and two elements of $faceset$ are adjacent in $dualgraph$ if they share an edge in the icosahedron. The graph $dualgraph$ admits a hamiltonian path, that is, there exists an ordering\n$faceidx_1,\\dots,faceidx_{20}$ of the faces such that any two consecutive faces are adjacent in $dualgraph$. \nFor example, such an ordering can be constructed with $f_1,\\dots,f_5$ being the five faces sharing a vertex of the icosahedron and $f_{16},\\dots,f_{20}$ being the five faces sharing the antipodal vertex.\n\nFor $indexi=1,\\dots,19$, let $edgeidx$ be the common edge of $faceidx$ and $f_{indexi+1}$; these are obviously all distinct.\nBy prescribing components for $edgeidx$ in turn and setting the others to zero,\nwe can construct an element of $\\mathbb{fieldthree}^{edgeset}$ whose image under $lintrans$ matches any given vector of $\\mathbb{fieldthree}^{faceset}$ in the components of $faceidx_1,\\dots,faceidx_{19}$. The vectors in $\\mathbb{fieldthree}^{faceset}$ obtained in this way thus form a 19-dimensional subspace; this subspace may also be described as the vectors for which the components of $faceidx_1,\\dots,faceidx_{19}$ have the same sum as the components of $f_{2},\\dots,f_{20}$. \n\nBy performing a mirror reflection, we can construct a second hamiltonian path $pathidx_1,\\dots,pathidx_{20}$ with the property that\n\\[\npathidx_1 = f_1, \\; pathidx_2 = f_5, \\; pathidx_3 = f_4, \\; pathidx_4 = f_3, \\; pathidx_5 = f_2.\n\\]\nRepeating the previous construction, we obtain a \\emph{different} 19-dimensional subspace of $\\mathbb{fieldthree}^{faceset}$ which is contained in the image of $lintrans$. This implies that $lintrans$ is surjective, as asserted earlier.\n\nSince $lintrans$ is a surjective homomorphism from a 30-dimensional vector space to a 20-dimensional vector space, it has a 10-dimensional kernel. Each of the $2^{20}$ elements of $\\mathbb{fieldthree}^{faceset}$ with no zero components is then the image of exactly $3^{10}$ colorings of the desired form, yielding the result.\n\n\\noindent\n\\textbf{Remark:}\nThere are many ways to check that $lintrans$ is surjective. One of the simplest is the following\n(from Art of Problem Solving, user \\texttt{Ravi12346}): form a vector in $\\mathbb{fieldthree}^{edgeset}$ with components $2,1,2,1,2$ at the five edges around some vertex and all other components 0. This maps to a vector in $\\mathbb{fieldthree}^{faceset}$ with only a single nonzero component; by symmetry, every standard basis vector of $\\mathbb{fieldthree}^{faceset}$ arises in this way.\n\n\\noindent\n\\textbf{Second solution:}\n(from Bill Huang, via Art of Problem Solving user \\texttt{superpi83})\nLet $vertexv$ and $vertexw$ be two antipodal vertices of the icosahedron. Let $starvset$ (resp.\\ $starwset$) be the set of five edges incident to $vertexv$ (resp.\\ $vertexw$). Let $ringvset$ (resp.\\ $ringwset$) be the set of five edges of the pentagon formed by the opposite endpoints of the five edges in $starvset$ (resp. $starwset$). Let $midedges$ be the set of the ten remaining edges of the icosahedron.\n\nConsider any one of the $3^{10}$ possible colorings of $midedges$. The edges of $ringvset \\cup midedges$ form the boundaries of five faces with no edges in common; thus each edge of $ringvset$ can be colored in one of two ways consistent with the given condition,\nand similarly for $ringwset$. That is, there are $3^{10} 2^{10}$ possible colorings of $ringvset \\cup ringwset \\cup midedges$ consistent with the given condition.\n\nTo complete the count, it suffices to check that there are exactly $2^5$ ways to color $starvset$ consistent with any given coloring of $ringvset$. Using the linear-algebraic interpretation from the first solution, this follows by observing that\n(by the previous remark) the map from $\\mathbb{fieldthree}^{starvset}$ to the $\\mathbb{fieldthree}$-vector space on the faces incident to $vertexv$ is surjective, and hence an isomorphism for dimensional reasons. A direct combinatorial proof is also possible."
    },
    "descriptive_long_confusing": {
      "map": {
        "i": "marigold",
        "v": "sunflower",
        "w": "hydrangea",
        "e_i": "buttercup",
        "f_i": "hibiscus",
        "g_i": "dandelion",
        "F_3": "pomegranate",
        "E": "elderberry",
        "F": "gooseberry",
        "T": "persimmon",
        "\\\\Gamma": "boysenberry",
        "S_v": "strawberry",
        "S_w": "tayberry",
        "T_v": "mulberry",
        "T_w": "loganberry",
        "U": "cloudberry"
      },
      "question": "The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]",
      "solution": "The number of such colorings is $2^{20} 3^{10} = 61917364224$.\n\n\\noindent\n\\textbf{First solution:}\nIdentify the three colors red, white, and blue with (in some order) the elements of the field $\\mathbb{pomegranate}$ of three elements (i.e., the ring of integers mod 3).\nThe set of colorings may then be identified with the $\\mathbb{pomegranate}$-vector space $\\mathbb{pomegranate}^{\\elderberry}$ generated by the set $\\elderberry$ of edges. Let $\\gooseberry$ be the set of faces, and let $\\mathbb{pomegranate}^{\\gooseberry}$ be the $\\mathbb{pomegranate}$-vector space on the basis $\\gooseberry$; we may then define a linear transformation\n$\\persimmon: \\mathbb{pomegranate}^{\\elderberry} \\to \\mathbb{pomegranate}^{\\gooseberry}$ taking a coloring to the vector whose component corresponding to a given face equals the sum of the three edges of that face. The colorings we wish to count are the ones whose images under $\\persimmon$ consist of vectors with no zero components.\n\nWe now show that $\\persimmon$ is surjective. (There are many possible approaches to this step; for instance, see the following remark.)\nLet $\\boysenberry$ be the dual graph of the icosahedron, that is, $\\boysenberry$ has vertex set $\\gooseberry$ and two elements of $\\gooseberry$ are adjacent in $\\boysenberry$ if they share an edge in the icosahedron. The graph $\\boysenberry$ admits a hamiltonian path, that is, there exists an ordering\n$\\hibiscus_1,\\dots,\\hibiscus_{20}$ of the faces such that any two consecutive faces are adjacent in $\\boysenberry$.\nFor example, such an ordering can be constructed with $\\hibiscus_1,\\dots,\\hibiscus_5$ being the five faces sharing a vertex of the icosahedron and $\\hibiscus_{16},\\dots,\\hibiscus_{20}$ being the five faces sharing the antipodal vertex.\n\nFor $\\marigold = 1,\\dots,19$, let $\\buttercup_{\\marigold}$ be the common edge of $\\hibiscus_{\\marigold}$ and $\\hibiscus_{\\marigold+1}$; these are obviously all distinct.\nBy prescribing components for $\\buttercup_1,\\dots,\\buttercup_{19}$ in turn and setting the others to zero,\nwe can construct an element of $\\mathbb{pomegranate}^{\\elderberry}$ whose image under $\\persimmon$ matches any given vector of $\\mathbb{pomegranate}^{\\gooseberry}$ in the components of $\\hibiscus_1,\\dots,\\hibiscus_{19}$. The vectors in $\\mathbb{pomegranate}^{\\gooseberry}$ obtained in this way thus form a 19-dimensional subspace; this subspace may also be described as the vectors for which the components of $\\hibiscus_1,\\dots,\\hibiscus_{19}$ have the same sum as the components of $\\hibiscus_{2},\\dots,\\hibiscus_{20}$.\n\nBy performing a mirror reflection, we can construct a second hamiltonian path $\\dandelion_1,\\dots,\\dandelion_{20}$ with the property that\n\\[\n\\dandelion_1 = \\hibiscus_1,\\quad \\dandelion_2 = \\hibiscus_5,\\quad \\dandelion_3 = \\hibiscus_4,\\quad \\dandelion_4 = \\hibiscus_3,\\quad \\dandelion_5 = \\hibiscus_2.\n\\]\nRepeating the previous construction, we obtain a \\emph{different} 19-dimensional subspace of $\\mathbb{pomegranate}^{\\gooseberry}$ which is contained in the image of $\\persimmon$. This implies that $\\persimmon$ is surjective, as asserted earlier.\n\nSince $\\persimmon$ is a surjective homomorphism from a 30-dimensional vector space to a 20-dimensional vector space, it has a 10-dimensional kernel. Each of the $2^{20}$ elements of $\\mathbb{pomegranate}^{\\gooseberry}$ with no zero components is then the image of exactly $3^{10}$ colorings of the desired form, yielding the result.\n\n\\noindent\n\\textbf{Remark:}\nThere are many ways to check that $\\persimmon$ is surjective. One of the simplest is the following\n(from Art of Problem Solving, user \\texttt{Ravi12346}): form a vector in $\\mathbb{pomegranate}^{\\elderberry}$ with components $2,1,2,1,2$ at the five edges around some vertex and all other components 0. This maps to a vector in $\\mathbb{pomegranate}^{\\gooseberry}$ with only a single nonzero component; by symmetry, every standard basis vector of $\\mathbb{pomegranate}^{\\gooseberry}$ arises in this way.\n\n\\noindent\n\\textbf{Second solution:}\n(from Bill Huang, via Art of Problem Solving user \\texttt{superpi83})\nLet $\\sunflower$ and $\\hydrangea$ be two antipodal vertices of the icosahedron. Let $\\strawberry$ (resp.\\ $\\tayberry$) be the set of five edges incident to $\\sunflower$ (resp.\\ $\\hydrangea$). Let $\\mulberry$ (resp.\\ $\\loganberry$) be the set of five edges of the pentagon formed by the opposite endpoints of the five edges in $\\strawberry$ (resp.\\ $\\tayberry$). Let $\\cloudberry$ be the set of the ten remaining edges of the icosahedron.\n\nConsider any one of the $3^{10}$ possible colorings of $\\cloudberry$. The edges of $\\mulberry \\cup \\cloudberry$ form the boundaries of five faces with no edges in common; thus each edge of $\\mulberry$ can be colored in one of two ways consistent with the given condition,\nand similarly for $\\loganberry$. That is, there are $3^{10} 2^{10}$ possible colorings of $\\mulberry \\cup \\loganberry \\cup \\cloudberry$ consistent with the given condition.\n\nTo complete the count, it suffices to check that there are exactly $2^5$ ways to color $\\strawberry$ consistent with any given coloring of $\\mulberry$. Using the linear-algebraic interpretation from the first solution, this follows by observing that\n(by the previous remark) the map from $\\mathbb{pomegranate}^{\\strawberry}$ to the $\\mathbb{pomegranate}$-vector space on the faces incident to $\\sunflower$ is surjective, and hence an isomorphism for dimensional reasons. A direct combinatorial proof is also possible."
    },
    "descriptive_long_misleading": {
      "map": {
        "i": "collective",
        "v": "midcentre",
        "w": "middlepoint",
        "e_i": "surfaceindex",
        "f_i": "interiorindex",
        "g_i": "stationaryindex",
        "F_3": "infinitefield",
        "E": "corepoints",
        "F": "voidregion",
        "T": "stasismap",
        "\\Gamma": "antigraph",
        "S_v": "hollowregion",
        "S_w": "solidregion",
        "T_v": "innerpentagon",
        "T_w": "outerpentagon",
        "U": "wholespace"
      },
      "question": "The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways \nare there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]",
      "solution": "The number of such colorings is $2^{20} 3^{10} = 61917364224$.\n\n\\noindent\n\\textbf{First solution:}\nIdentify the three colors red, white, and blue with (in some order) the elements of the field $\\mathbb{infinitefield}$ of three elements (i.e., the ring of integers mod 3).\nThe set of colorings may then be identified with the $\\mathbb{infinitefield}^{corepoints}$-vector space generated by the set $corepoints$ of edges. Let $voidregion$ be the set of faces, and let $\\mathbb{infinitefield}^{voidregion}$ be the $\\mathbb{infinitefield}$-vector space on the basis $voidregion$; we may then define a linear transformation\n$stasismap: \\mathbb{infinitefield}^{corepoints} \\to \\mathbb{infinitefield}^{voidregion}$ taking a coloring to the vector whose component corresponding to a given face equals the sum of the three edges of that face. The colorings we wish to count are the ones whose images under $stasismap$ consist of vectors with no zero components.\n\nWe now show that $stasismap$ is surjective. (There are many possible approaches to this step; for instance, see the following remark.)\nLet $antigraph$ be the dual graph of the icosahedron, that is, $antigraph$ has vertex set $voidregion$ and two elements of $voidregion$ are adjacent in $antigraph$ if they share an edge in the icosahedron. The graph $antigraph$ admits a hamiltonian path, that is, there exists an ordering\n$interiorindex_1,\\dots,interiorindex_{20}$ of the faces such that any two consecutive faces are adjacent in $antigraph$.\nFor example, such an ordering can be constructed with $interiorindex_1,\\dots,interiorindex_5$ being the five faces sharing a vertex of the icosahedron and $interiorindex_{16},\\dots,interiorindex_{20}$ being the five faces sharing the antipodal vertex.\n\nFor $collective = 1,\\dots,19$, let $surfaceindex$ be the common edge of $interiorindex_{collective}$ and $interiorindex_{collective+1}$; these are obviously all distinct.\nBy prescribing components for $surfaceindex_1,\\dots,surfaceindex_{19}$ in turn and setting the others to zero,\nwe can construct an element of $\\mathbb{infinitefield}^{corepoints}$ whose image under $stasismap$ matches any given vector of $\\mathbb{infinitefield}^{voidregion}$ in the components of $interiorindex_1,\\dots,interiorindex_{19}$. The vectors in $\\mathbb{infinitefield}^{voidregion}$ obtained in this way thus form a 19-dimensional subspace; this subspace may also be described as the vectors for which the components of $interiorindex_1,\\dots,interiorindex_{19}$ have the same sum as the components of $interiorindex_{2},\\dots,interiorindex_{20}$.\n\nBy performing a mirror reflection, we can construct a second hamiltonian path $stationaryindex_1,\\dots,stationaryindex_{20}$ with the property that\n\\[\nstationaryindex_1 = interiorindex_1, \\quad stationaryindex_2 = interiorindex_5, \\quad stationaryindex_3 = interiorindex_4, \\quad stationaryindex_4 = interiorindex_3, \\quad stationaryindex_5 = interiorindex_2.\n\\]\nRepeating the previous construction, we obtain a \\emph{different} 19-dimensional subspace of $\\mathbb{infinitefield}^{voidregion}$ which is contained in the image of $stasismap$. This implies that $stasismap$ is surjective, as asserted earlier.\n\nSince $stasismap$ is a surjective homomorphism from a 30-dimensional vector space to a 20-dimensional vector space, it has a 10-dimensional kernel. Each of the $2^{20}$ elements of $\\mathbb{infinitefield}^{voidregion}$ with no zero components is then the image of exactly $3^{10}$ colorings of the desired form, yielding the result.\n\n\\noindent\n\\textbf{Remark:}\nThere are many ways to check that $stasismap$ is surjective. One of the simplest is the following\n(from Art of Problem Solving, user \\texttt{Ravi12346}): form a vector in $\\mathbb{voidregion}^{corepoints}$ with components $2,1,2,1,2$ at the five edges around some vertex and all other components 0. This maps to a vector in $\\mathbb{voidregion}^{voidregion}$ with only a single nonzero component; by symmetry, every standard basis vector of $\\mathbb{voidregion}^{voidregion}$ arises in this way.\n\n\\noindent\n\\textbf{Second solution:}\n(from Bill Huang, via Art of Problem Solving user \\texttt{superpi83})\nLet $midcentre$ and $middlepoint$ be two antipodal vertices of the icosahedron. Let $hollowregion$ (resp.\\ $solidregion$) be the set of five edges incident to $midcentre$ (resp.\\ $middlepoint$). Let $innerpentagon$ (resp.\\ $outerpentagon$) be the set of five edges of the pentagon formed by the opposite endpoints of the five edges in $hollowregion$ (resp. $solidregion$). Let $wholespace$ be the set of the ten remaining edges of the icosahedron.\n\nConsider any one of the $3^{10}$ possible colorings of $wholespace$. The edges of $innerpentagon \\cup wholespace$ form the boundaries of five faces with no edges in common; thus each edge of $innerpentagon$ can be colored in one of two ways consistent with the given condition,\nand similarly for $outerpentagon$. That is, there are $3^{10} 2^{10}$ possible colorings of $innerpentagon \\cup outerpentagon \\cup wholespace$ consistent with the given condition.\n\nTo complete the count, it suffices to check that there are exactly $2^5$ ways to color $hollowregion$ consistent with any given coloring of $innerpentagon$. Using the linear-algebraic interpretation from the first solution, this follows by observing that\n(by the previous remark) the map from $\\mathbb{infinitefield}^{hollowregion}$ to the $\\mathbb{infinitefield}$-vector space on the faces incident to $midcentre$ is surjective, and hence an isomorphism for dimensional reasons. A direct combinatorial proof is also possible."
    },
    "garbled_string": {
      "map": {
        "v": "qzxwvtnp",
        "w": "hjgrksla",
        "e_i": "mnbvcxzq",
        "f_i": "plmoknij",
        "g_i": "qazwsxed",
        "F_3": "ugykflpn",
        "E": "zmxncbvd",
        "F": "aslkdjfh",
        "T": "weruiops",
        "\\\\Gamma": "oiuytrew",
        "S_v": "poilkjmn",
        "S_w": "mnbvfdsa",
        "T_v": "qazplmok",
        "T_w": "wsxedcrf",
        "U": "kjhgfdsa"
      },
      "question": "The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\\dots,30$. How many different ways \nare there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? [Note: the top matter on each exam paper included the logo of the Mathematical Association of America, which is itself an icosahedron.]",
      "solution": "The number of such colorings is $2^{20} 3^{10} = 61917364224$.\n\n\\noindent\n\\textbf{First solution:}\nIdentify the three colors red, white, and blue with (in some order) the elements of the field $\\mathbb{ugykflpn}$ of three elements (i.e., the ring of integers mod 3). \nThe set of colorings may then be identified with the $\\mathbb{ugykflpn}$-vector space $\\mathbb{ugykflpn}^{zmxncbvd}$\ngenerated by the set $zmxncbvd$ of edges. Let $aslkdjfh$ be the set of faces, and let $\\mathbb{ugykflpn}^{aslkdjfh}$ be the $\\mathbb{ugykflpn}$-vector space on the basis $aslkdjfh$; we may then define a linear transformation\n$weruiops: \\mathbb{ugykflpn^{zmxncbvd}} \\to \\mathbb{ugykflpn^{aslkdjfh}}$ taking a coloring to the vector whose component corresponding to a given face equals the sum of the three edges of that face. The colorings we wish to count are the ones whose images under $weruiops$ consist of vectors with no zero components.\n\nWe now show that $weruiops$ is surjective. (There are many possible approaches to this step; for instance, see the following remark.) \nLet $oiuytrew$ be the dual graph of the icosahedron, that is, $oiuytrew$ has vertex set $aslkdjfh$ and two elements of $aslkdjfh$ are adjacent in $oiuytrew$ if they share an edge in the icosahedron. The graph $oiuytrew$ admits a hamiltonian path, that is, there exists an ordering\n$plmoknij_1,\\dots,plmoknij_{20}$ of the faces such that any two consecutive faces are adjacent in $oiuytrew$. \nFor example, such an ordering can be constructed with $plmoknij_1,\\dots,plmoknij_5$ being the five faces sharing a vertex of the icosahedron and $plmoknij_{16},\\dots,plmoknij_{20}$ being the five faces sharing the antipodal vertex.\n\nFor $i=1,\\dots,19$, let $mnbvcxzq_i$ be the common edge of $plmoknij_i$ and $plmoknij_{i+1}$; these are obviously all distinct.\nBy prescribing components for $mnbvcxzq_1,\\dots,mnbvcxzq_{19}$ in turn and setting the others to zero,\nwe can construct an element of $\\mathbb{ugykflpn^{zmxncbvd}}$ whose image under $weruiops$ matches any given vector of $\\mathbb{ugykflpn^{aslkdjfh}}$ in the components of $plmoknij_1,\\dots,plmoknij_{19}$. The vectors in $\\mathbb{ugykflpn^{aslkdjfh}}$ obtained in this way thus form a 19-dimensional subspace; this subspace may also be described as the vectors for which the components of $plmoknij_1,\\dots,plmoknij_{19}$ have the same sum as the components of $plmoknij_{2},\\dots,plmoknij_{20}$. \n\nBy performing a mirror reflection, we can construct a second hamiltonian path $qazwsxed_1,\\dots,qazwsxed_{20}$ with the property that\n\\[\nqazwsxed_1 = plmoknij_1, \\quad qazwsxed_2 = plmoknij_5, \\quad qazwsxed_3 = plmoknij_4, \\quad qazwsxed_4 = plmoknij_3, \\quad qazwsxed_5 = plmoknij_2.\n\\]\nRepeating the previous construction, we obtain a \\emph{different} 19-dimensional subspace of $\\mathbb{ugykflpn^{aslkdjfh}}$ which is contained in the image of $weruiops$. This implies that $weruiops$ is surjective, as asserted earlier.\n\nSince $weruiops$ is a surjective homomorphism from a 30-dimensional vector space to a 20-dimensional vector space, it has a 10-dimensional kernel. Each of the $2^{20}$ elements of $\\mathbb{ugykflpn^{aslkdjfh}}$ with no zero components is then the image of exactly $3^{10}$ colorings of the desired form, yielding the result.\n\n\\noindent\n\\textbf{Remark:}\nThere are many ways to check that $weruiops$ is surjective. One of the simplest is the following\n(from Art of Problem Solving, user \\texttt{Ravi12346}): form a vector in $\\mathbb{aslkdjfh^{zmxncbvd}}$ with components $2,1,2,1,2$ at the five edges around some vertex and all other components 0. This maps to a vector in $\\mathbb{aslkdjfh^{aslkdjfh}}$ with only a single nonzero component; by symmetry, every standard basis vector of $\\mathbb{aslkdjfh^{aslkdjfh}}$ arises in this way.\n\n\\noindent\n\\textbf{Second solution:}\n(from Bill Huang, via Art of Problem Solving user \\texttt{superpi83})\nLet $qzxwvtnp$ and $hjgrksla$ be two antipodal vertices of the icosahedron. Let $poilkjmn$ (resp.\\ $mnbvfdsa$) be the set of five edges incident to $qzxwvtnp$ (resp.\\ $hjgrksla$). Let $qazplmok$ (resp.\\ $wsxedcrf$) be the set of five edges of the pentagon formed by the opposite endpoints of the five edges in $poilkjmn$ (resp. $mnbvfdsa$). Let $kjhgfdsa$ be the set of the ten remaining edges of the icosahedron.\n\nConsider any one of the $3^{10}$ possible colorings of $kjhgfdsa$. The edges of $qazplmok \\cup kjhgfdsa$ form the boundaries of five faces with no edges in common; thus each edge of $qazplmok$ can be colored in one of two ways consistent with the given condition,\nand similarly for $wsxedcrf$. That is, there are $3^{10} 2^{10}$ possible colorings of $qazplmok \\cup wsxedcrf \\cup kjhgfdsa$ consistent with the given condition.\n\nTo complete the count, it suffices to check that there are exactly $2^5$ ways to color $poilkjmn$ consistent with any given coloring of $qazplmok$. Using the linear-algebraic interpretation from the first solution, this follows by observing that\n(by the previous remark) the map from $\\mathbb{ugykflpn^{poilkjmn}}$ to the $\\mathbb{ugykflpn}$-vector space on the faces incident to $qzxwvtnp$ is surjective, and hence an isomorphism for dimensional reasons. A direct combinatorial proof is also possible."
    },
    "kernel_variant": {
      "question": "The 12 edges of a regular octahedron are distinguished by the labels $A,B,\\dots ,L$.  Each edge is to be coloured in one of three shades—teal, amber, or lilac—so that on every one of the octahedron’s eight triangular faces exactly two of its edges share a colour while the third edge is of the other colour.  How many admissible colourings are there?",
      "solution": "Corrected solution:  Identify the three colours teal, amber, lilac with 0,1,2 in F₃, and let E be the 12 edges, F the 8 triangular faces.  A colouring is c∈F₃^E, and define the linear map T:F₃^E→F₃^F by\n  (T(c))_f = ∑_{e⊂f} c_e  (sum in F₃).\nOn a face whose three edge‐colours are {c,c,d} with c≠d, the sum is 2c+d≡±1≠0, so the condition “exactly two edges match” is equivalent to requiring T(c)_f≠0 for every face f.\n\n1.  Rank and kernel of T.  A linear relation ∑_{f∈F}a_f·(T(c))_f≡0 for all c forces for each edge e (which lies on two faces f,g) the coefficient of c_e to vanish: a_f+a_g=0 in F₃.  Since the dual graph is the cube (bipartite), if we partition faces into two sets A,B of size 4, then a_f must be constant x on A and –x on B.  Thus there is a 1‐dimensional space of nonzero relations, so rank T=8−1=7 and nullity T=12−7=5.\n\n2.  Image hyperplane.  Concretely, for any c one checks\n   ∑_{f∈A}(T(c))_f − ∑_{f∈B}(T(c))_f = 0\nin F₃, so Im T is the 7‐dimensional hyperplane H = {v∈F₃^8: ∑_{A} v_f = ∑_{B} v_f}.\n\n3.  Counting colourings.  Since ker T has size 3^5=243, each v∈Im T has 243 preimages.  We must count those v∈H with all 8 coordinates in F₃^*={1,2}.  There are 2^8=256 vectors in {1,2}^8, but the constraint ∑_A v_f=∑_B v_f (mod 3) cuts this to\n   (1/3)(256 + 2·1) = 86\nby a short character sum or by elementary case‐work on the number of +1’s in A and B.  Hence the total number of admissible c is\n   86 × 3^5 = 86 × 243 = 20898.",
      "_meta": {
        "core_steps": [
          "Encode every edge–colour assignment as a vector in F3^E (3 colours → field of 3 elements).",
          "Define the linear map T : F3^E → F3^F that sends a colouring to the sum of the three edge-values around each triangular face; admissible colourings are those whose image has no zero entry.",
          "Show T is surjective, e.g. by exhibiting a Hamiltonian path in the dual graph and assigning edge values successively to hit any prescribed face-vector.",
          "Use rank–nullity: dim Ker T = |E| − |F|.",
          "Count: (3−1)^{|F|} · 3^{dim Ker T} admissible colourings."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Actual names/words chosen for the three colours",
            "original": [
              "red",
              "white",
              "blue"
            ]
          },
          "slot2": {
            "description": "Labels attached to the edges",
            "original": "1,2,…,30"
          },
          "slot3": {
            "description": "Which field element (0,1,2) is paired with which colour",
            "original": "Any of the 3! possible assignments"
          },
          "slot4": {
            "description": "Concrete Hamiltonian path of faces used to prove surjectivity",
            "original": "f1,…,f20 as described in the solution"
          },
          "slot5": {
            "description": "Mirror-reflected second Hamiltonian path invoked for surjectivity",
            "original": "g1,…,g20 with g1=f1, g2=f5, …"
          },
          "slot6": {
            "description": "Choice of vertex/face around which the single-face vector of the remark is built",
            "original": "An arbitrary vertex of the icosahedron"
          },
          "slot7": {
            "description": "Number of edges of the polyhedron (|E|)",
            "original": 30
          },
          "slot8": {
            "description": "Number of triangular faces (|F|)",
            "original": 20
          },
          "slot9": {
            "description": "Kernel dimension |E|−|F| derived from rank–nullity",
            "original": 10
          },
          "slot10": {
            "description": "Final numeric count produced after the calculation",
            "original": "2^{20} · 3^{10} = 61917364224"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}