{
  "index": "2020-B-4",
  "type": "COMB",
  "tag": [
    "COMB",
    "NT",
    "ALG"
  ],
  "difficulty": "",
  "question": "Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\\mathbf{v} = (s_0, s_1, \\cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\\cdots,2n$. Define\n\\[\nq(\\mathbf{v}) = 1 + \\sum_{j=1}^{2n-1} 3^{s_j},\n\\]\nand let $M(n)$ be the average of $\\frac{1}{q(\\mathbf{v})}$ over all $\\mathbf{v} \\in V_n$. Evaluate $M(2020)$.",
  "solution": "The answer is $\\frac{1}{4040}$. We will show the following more general fact. Let $a$ be any nonzero number and define $q(\\mathbf{v}) = 1+\\sum_{j=1}^{2n-1} a^{s_j}$; then the average of $\\frac{1}{q(\\mathbf{v})}$ over all $\\mathbf{v} \\in V_n$ is equal to $\\frac{1}{2n}$, independent of $a$.\n\nLet $W_n$ denote the set of $(2n)$-tuples $\\mathbf{w} = (w_1,\\ldots,w_{2n})$ such that $n$ of the $w_i$'s are equal to $+1$ and the other $n$ are equal to $-1$. Define a map $\\phi :\\thinspace W_n \\to W_n$ by $\\phi(w_1,w_2,\\ldots,w_{2n}) = (w_2,\\ldots,w_{2n},w_1)$; that is, $\\phi$ moves the first entry to the end. For $\\mathbf{w} \\in W_n$, define the \\textit{orbit} of $\\mathbf{w}$ to be the collection of elements of $W_n$ of the form $\\phi^k(\\mathbf{w})$, $k \\geq 1$, where $\\phi^k$ denotes the $k$-th iterate of $\\phi$, and note that $\\phi^{2n}(\\mathbf{w}) = \\mathbf{w}$. Then $W_n$ is a disjoint union of orbits. For a given $\\mathbf{w} \\in W_n$, the orbit of $\\mathbf{w}$ consists of $\\mathbf{w},\\phi(\\mathbf{w}),\\ldots,\\phi^{m-1}(\\mathbf{w})$, where $m$ is the smallest positive integer with $\\phi^m(\\mathbf{w}) = \\mathbf{w}$; the list $\\phi(\\mathbf{w}),\\ldots,\\phi^{2n}(\\mathbf{w})$ runs through the orbit of $\\mathbf{w}$ completely $2n/m$ times, with each element of the orbit appearing the same number of times.\n\nNow define the map $f :\\thinspace W_n \\to V_n$ by $f(\\mathbf{w}) = \\mathbf{v} = (s_0,\\ldots,s_{2n})$ with $s_j = \\sum_{i=1}^j w_i$; this is a one-to-one correspondence between $W_n$ and $V_n$, with the inverse map given by $w_j = s_j-s_{j-1}$ for $j=1,\\ldots,2n$. We claim that for any $\\mathbf{w} \\in W_n$, the average of $\\frac{1}{q(\\mathbf{v})}$, where $\\mathbf{v}$ runs over vectors in the image of the orbit of $\\mathbf{w}$ under $f$, is equal to $\\frac{1}{2n}$. Since $W_n$ is a disjoint union of orbits, $V_n$ is a disjoint union of the images of these orbits under $f$, and it then follows that the overall average of $\\frac{1}{q(\\mathbf{v})}$ over $\\mathbf{v} \\in V_n$ is $\\frac{1}{2n}$.\n\nTo prove the claim, we compute the average of $\\frac{1}{q(f(\\phi^k(\\mathbf{w})))}$ over $k=1,\\ldots,2n$; since $\\phi^k(\\mathbf{w})$ for $k=1,\\ldots,2n$ runs over the orbit of $\\mathbf{w}$ with each element in the orbit appearing equally, this is equal to the desired average. Now if we adopt the convention that the indices in $w_i$ are considered mod $2n$, so that $w_{2n+i} = w_i$ for all $i$, then the $i$-th entry of $\\phi^k(\\mathbf{w})$ is $w_{i+k}$; we can then define $s_j = \\sum_{i=1}^j w_i$ for all $j\\geq 1$, and $s_{2n+i}=s_i$ for all $i$ since $\\sum_{i=1}^{2n} w_i = 0$. We now have\n\\[\nq(f(\\phi^k(\\mathbf{w}))) = \\sum_{j=1}^{2n} a^{\\sum_{i=1}^j w_{i+k}} = \\sum_{j=1}^{2n} a^{s_{j+k}-s_k} = a^{-s_k} \\sum_{j=1}^{2n} a^{s_j}.\n\\]\n\nThus\n\\[\n\\sum_{k=1}^{2n} \\frac{1}{q(f(\\phi^k(\\mathbf{w})))} = \\sum_{k=1}^{2n} \\frac{a^{s_k}}{\\sum_{j=1}^{2n} a^{s_j}} = 1,\n\\]\nand the average of $\\frac{1}{q(f(\\phi^k(\\mathbf{w})))}$ over $k=1,\\ldots,2n$ is $\\frac{1}{2n}$, as desired.",
  "vars": [
    "n",
    "j",
    "k",
    "m",
    "i",
    "q",
    "s_j",
    "s_j-1",
    "s_0",
    "s_2n",
    "s_2n-1",
    "s_k",
    "s_j+k",
    "s_2n+i",
    "w_i",
    "w_i+k",
    "w_j",
    "w_1",
    "w_2n",
    "w_2n+i",
    "V_n",
    "W_n",
    "\\\\phi"
  ],
  "params": [
    "a"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "iterlen",
        "j": "indexvar",
        "k": "shiftvar",
        "m": "orbitlen",
        "i": "loopvar",
        "q": "sumexpr",
        "s_j": "partialsum",
        "s_j-1": "prevpart",
        "s_0": "zerostart",
        "s_2n": "finalsum",
        "s_2n-1": "finalminus",
        "s_k": "shiftpartial",
        "s_j+k": "forwardsum",
        "s_2n+i": "wrappedsum",
        "w_i": "stepval",
        "w_i+k": "shiftedstep",
        "w_j": "stepindex",
        "w_1": "firststep",
        "w_2n": "laststep",
        "w_2n+i": "wrappedstep",
        "V_n": "pathset",
        "W_n": "stepset",
        "\\phi": "rotation",
        "a": "basevalue"
      },
      "question": "Let $iterlen$ be a positive integer, and let $pathset$ be the set of integer $(2iterlen+1)$-tuples $\\mathbf{v} = (zerostart, s_1, \\cdots, finalminus, finalsum)$ for which $zerostart = finalsum = 0$ and $|partialsum - prevpart| = 1$ for $indexvar = 1,2,\\cdots,2iterlen$. Define\n\\[\nsumexpr(\\mathbf{v}) = 1 + \\sum_{indexvar=1}^{2iterlen-1} 3^{partialsum},\n\\]\nand let $M(iterlen)$ be the average of $\\frac{1}{sumexpr(\\mathbf{v})}$ over all $\\mathbf{v} \\in pathset$. Evaluate $M(2020)$.",
      "solution": "The answer is $\\frac{1}{4040}$. We will show the following more general fact. Let $basevalue$ be any nonzero number and define $sumexpr(\\mathbf{v}) = 1+\\sum_{indexvar=1}^{2iterlen-1} basevalue^{partialsum}$; then the average of $\\frac{1}{sumexpr(\\mathbf{v})}$ over all $\\mathbf{v} \\in pathset$ is equal to $\\frac{1}{2iterlen}$, independent of $basevalue$.\n\nLet $stepset$ denote the set of $(2iterlen)$-tuples $\\mathbf{w} = (firststep,\\ldots,laststep)$ such that $iterlen$ of the $stepval$'s are equal to $+1$ and the other $iterlen$ are equal to $-1$. Define a map $rotation :\\thinspace stepset \\to stepset$ by $rotation(firststep,w_2,\\ldots,laststep) = (w_2,\\ldots,laststep,firststep)$; that is, $rotation$ moves the first entry to the end. For $\\mathbf{w} \\in stepset$, define the \\textit{orbit} of $\\mathbf{w}$ to be the collection of elements of $stepset$ of the form $rotation^{shiftvar}(\\mathbf{w})$, $shiftvar \\ge 1$, where $rotation^{shiftvar}$ denotes the $shiftvar$-th iterate of $rotation$, and note that $rotation^{2iterlen}(\\mathbf{w}) = \\mathbf{w}$. Then $stepset$ is a disjoint union of orbits. For a given $\\mathbf{w} \\in stepset$, the orbit of $\\mathbf{w}$ consists of $\\mathbf{w}, rotation(\\mathbf{w}), \\ldots, rotation^{orbitlen-1}(\\mathbf{w})$, where $orbitlen$ is the smallest positive integer with $rotation^{orbitlen}(\\mathbf{w}) = \\mathbf{w}$; the list $rotation(\\mathbf{w}), \\ldots, rotation^{2iterlen}(\\mathbf{w})$ runs through the orbit of $\\mathbf{w}$ completely $2iterlen/orbitlen$ times, with each element of the orbit appearing the same number of times.\n\nNow define the map $f :\\thinspace stepset \\to pathset$ by $f(\\mathbf{w}) = \\mathbf{v} = (zerostart,\\ldots,finalsum)$ with $partialsum = \\sum_{loopvar=1}^{indexvar} stepval$; this is a one-to-one correspondence between $stepset$ and $pathset$, with the inverse map given by $stepindex = partialsum - prevpart$ for $indexvar = 1,\\ldots,2iterlen$. We claim that for any $\\mathbf{w} \\in stepset$, the average of $\\frac{1}{sumexpr(\\mathbf{v})}$, where $\\mathbf{v}$ runs over vectors in the image of the orbit of $\\mathbf{w}$ under $f$, is equal to $\\frac{1}{2iterlen}$. Since $stepset$ is a disjoint union of orbits, $pathset$ is a disjoint union of the images of these orbits under $f$, and it then follows that the overall average of $\\frac{1}{sumexpr(\\mathbf{v})}$ over $\\mathbf{v} \\in pathset$ is $\\frac{1}{2iterlen}$.\n\nTo prove the claim, we compute the average of $\\frac{1}{sumexpr(f(rotation^{shiftvar}(\\mathbf{w})))}$ over $shiftvar = 1,\\ldots,2iterlen$; since $rotation^{shiftvar}(\\mathbf{w})$ for $shiftvar = 1,\\ldots,2iterlen$ runs over the orbit of $\\mathbf{w}$ with each element in the orbit appearing equally, this is equal to the desired average. Now if we adopt the convention that the indices in $stepval$ are considered $\\bmod\\; 2iterlen$, so that $wrappedstep = stepval$ for all $loopvar$, then the $loopvar$-th entry of $rotation^{shiftvar}(\\mathbf{w})$ is $shiftedstep$; we can then define $partialsum = \\sum_{loopvar=1}^{indexvar} stepval$ for all $indexvar \\ge 1$, and $wrappedsum = s_i$ for all $loopvar$ since $\\sum_{loopvar=1}^{2iterlen} stepval = 0$. We now have\n\\[\nsumexpr(f(rotation^{shiftvar}(\\mathbf{w}))) = \\sum_{indexvar=1}^{2iterlen} basevalue^{\\sum_{loopvar=1}^{indexvar} shiftedstep} = \\sum_{indexvar=1}^{2iterlen} basevalue^{forwardsum - shiftpartial} = basevalue^{-shiftpartial} \\sum_{indexvar=1}^{2iterlen} basevalue^{partialsum}.\n\\]\n\nThus\n\\[\n\\sum_{shiftvar=1}^{2iterlen} \\frac{1}{sumexpr(f(rotation^{shiftvar}(\\mathbf{w})))} = \\sum_{shiftvar=1}^{2iterlen} \\frac{basevalue^{shiftpartial}}{\\sum_{indexvar=1}^{2iterlen} basevalue^{partialsum}} = 1,\n\\]\nand the average of $\\frac{1}{sumexpr(f(rotation^{shiftvar}(\\mathbf{w})))}$ over $shiftvar = 1,\\ldots,2iterlen$ is $\\frac{1}{2iterlen}$, as desired."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "marigolds",
        "j": "buttermilk",
        "k": "cheesecurd",
        "m": "dragonfly",
        "i": "i",
        "q": "bluewhale",
        "s_j": "peppermint",
        "s_j-1": "buttercup",
        "s_0": "boomerang",
        "s_2n": "chandelier",
        "s_2n-1": "watercress",
        "s_k": "lightning",
        "s_j+k": "snowflake",
        "s_2n+i": "rattlesnake",
        "w_i": "marshmallow",
        "w_i+k": "strawberry",
        "w_j": "blacksmith",
        "w_1": "hummingbird",
        "w_2n": "brainstorm",
        "w_2n+i": "gallbladder",
        "V_n": "adventure",
        "W_n": "continents",
        "\\\\phi": "labyrinth",
        "a": "carousel"
      },
      "question": "Let $marigolds$ be a positive integer, and let $adventure$ be the set of integer $(2marigolds+1)$-tuples $\\mathbf{v} = (boomerang, s_1, \\cdots, watercress, chandelier)$ for which $boomerang = chandelier = 0$ and $|peppermint - buttercup| = 1$ for $buttermilk=1,2,\\cdots,2marigolds$. Define\n\\[\nbluewhale(\\mathbf{v}) = 1 + \\sum_{buttermilk=1}^{2marigolds-1} 3^{peppermint},\n\\]\nand let $M(marigolds)$ be the average of $\\frac{1}{bluewhale(\\mathbf{v})}$ over all $\\mathbf{v} \\in adventure$. Evaluate $M(2020)$.",
      "solution": "The answer is $\\frac{1}{4040}$. We will show the following more general fact. Let carousel be any nonzero number and define $bluewhale(\\mathbf{v}) = 1+\\sum_{buttermilk=1}^{2marigolds-1} carousel^{peppermint}$; then the average of $\\frac{1}{bluewhale(\\mathbf{v})}$ over all $\\mathbf{v} \\in adventure$ is equal to $\\frac{1}{2marigolds}$, independent of carousel.\n\nLet continents denote the set of $(2marigolds)$-tuples $\\mathbf{w} = (w_1,\\ldots,w_{2marigolds})$ such that marigolds of the $w_i$'s are equal to $+1$ and the other marigolds are equal to $-1$. Define a map $labyrinth :\\thinspace continents \\to continents$ by $labyrinth(w_1,w_2,\\ldots,w_{2marigolds}) = (w_2,\\ldots,w_{2marigolds},w_1)$; that is, $labyrinth$ moves the first entry to the end. For $\\mathbf{w} \\in continents$, define the \\textit{orbit} of $\\mathbf{w}$ to be the collection of elements of continents of the form $labyrinth^{cheesecurd}(\\mathbf{w})$, $cheesecurd \\geq 1$, where $labyrinth^{cheesecurd}$ denotes the cheesecurd-th iterate of labyrinth, and note that $labyrinth^{2marigolds}(\\mathbf{w}) = \\mathbf{w}$. Then continents is a disjoint union of orbits. For a given $\\mathbf{w} \\in continents$, the orbit of $\\mathbf{w}$ consists of $\\mathbf{w},labyrinth(\\mathbf{w}),\\ldots,labyrinth^{dragonfly-1}(\\mathbf{w})$, where dragonfly is the smallest positive integer with $labyrinth^{dragonfly}(\\mathbf{w}) = \\mathbf{w}$; the list $labyrinth(\\mathbf{w}),\\ldots,labyrinth^{2marigolds}(\\mathbf{w})$ runs through the orbit of $\\mathbf{w}$ completely $2marigolds/dragonfly$ times, with each element of the orbit appearing the same number of times.\n\nNow define the map $f :\\thinspace continents \\to adventure$ by $f(\\mathbf{w}) = \\mathbf{v} = (boomerang,\\ldots,chandelier)$ with $peppermint = \\sum_{i=1}^{buttermilk} w_i$; this is a one-to-one correspondence between continents and adventure, with the inverse map given by $blacksmith = peppermint - buttercup$ for $buttermilk=1,\\ldots,2marigolds$. We claim that for any $\\mathbf{w} \\in continents$, the average of $\\frac{1}{bluewhale(\\mathbf{v})}$, where $\\mathbf{v}$ runs over vectors in the image of the orbit of $\\mathbf{w}$ under $f$, is equal to $\\frac{1}{2marigolds}$. Since continents is a disjoint union of orbits, adventure is a disjoint union of the images of these orbits under $f$, and it then follows that the overall average of $\\frac{1}{bluewhale(\\mathbf{v})}$ over $\\mathbf{v} \\in adventure$ is $\\frac{1}{2marigolds}$.\n\nTo prove the claim, we compute the average of $\\frac{1}{bluewhale(f(labyrinth^{cheesecurd}(\\mathbf{w})))}$ over $cheesecurd=1,\\ldots,2marigolds$; since $labyrinth^{cheesecurd}(\\mathbf{w})$ for $cheesecurd=1,\\ldots,2marigolds$ runs over the orbit of $\\mathbf{w}$ with each element in the orbit appearing equally, this is equal to the desired average. Now if we adopt the convention that the indices in $w_i$ are considered mod $2marigolds$, so that $w_{2marigolds+i} = w_i$ for all $i$, then the $i$-th entry of $labyrinth^{cheesecurd}(\\mathbf{w})$ is strawberry; we can then define $peppermint = \\sum_{i=1}^{buttermilk} marshmallow$ for all $buttermilk\\geq 1$, and $rattlesnake = s_i$ for all $i$ since $\\sum_{i=1}^{2marigolds} marshmallow = 0$. We now have\n\\[\nbluewhale(f(labyrinth^{cheesecurd}(\\mathbf{w}))) = \\sum_{buttermilk=1}^{2marigolds} carousel^{\\sum_{i=1}^{buttermilk} strawberry} = \\sum_{buttermilk=1}^{2marigolds} carousel^{snowflake - lightning} = carousel^{-lightning} \\sum_{buttermilk=1}^{2marigolds} carousel^{peppermint}.\n\\]\n\nThus\n\\[\n\\sum_{cheesecurd=1}^{2marigolds} \\frac{1}{bluewhale(f(labyrinth^{cheesecurd}(\\mathbf{w})))} = \\sum_{cheesecurd=1}^{2marigolds} \\frac{carousel^{lightning}}{\\sum_{buttermilk=1}^{2marigolds} carousel^{peppermint}} = 1,\n\\]\nand the average of $\\frac{1}{bluewhale(f(labyrinth^{cheesecurd}(\\mathbf{w})))}$ over $cheesecurd=1,\\ldots,2marigolds$ is $\\frac{1}{2marigolds}$, as desired."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "endlesscount",
        "j": "staticindex",
        "k": "steadystate",
        "m": "maximalnumber",
        "i": "completeunit",
        "q": "segregator",
        "s_j": "flatland",
        "s_j-1": "flatlandprevious",
        "s_0": "flatlandorigin",
        "s_2n": "flatlandterminal",
        "s_2n-1": "flatlandpenultimate",
        "s_k": "flatlandshifted",
        "s_j+k": "flatlandoffset",
        "s_2n+i": "flatlandbeyond",
        "w_i": "standstillunit",
        "w_i+k": "standstillshifted",
        "w_j": "standstillindex",
        "w_1": "standstillfirst",
        "w_2n": "standstilllast",
        "w_2n+i": "standstillbeyond",
        "V_n": "restcollection",
        "W_n": "motionlessgroup",
        "\\phi": "fixturemap",
        "a": "voidnumber"
      },
      "question": "Let $endlesscount$ be a positive integer, and let $restcollection$ be the set of integer $(2endlesscount+1)$-tuples \\mathbf{v} = (flatlandorigin, s_1, \\cdots, flatlandpenultimate, flatlandterminal) for which $flatlandorigin = flatlandterminal = 0$ and $|flatland - flatlandprevious| = 1$ for $staticindex=1,2,\\cdots,2endlesscount$. Define\n\\[\nsegregator(\\mathbf{v}) = 1 + \\sum_{staticindex=1}^{2endlesscount-1} 3^{flatland},\n\\]\nand let $M(endlesscount)$ be the average of $\\frac{1}{segregator(\\mathbf{v})}$ over all $\\mathbf{v} \\in restcollection$. Evaluate $M(2020)$.",
      "solution": "The answer is $\\frac{1}{4040}$. We will show the following more general fact. Let $voidnumber$ be any nonzero number and define $segregator(\\mathbf{v}) = 1+\\sum_{staticindex=1}^{2endlesscount-1} voidnumber^{flatland}$; then the average of $\\frac{1}{segregator(\\mathbf{v})}$ over all $\\mathbf{v} \\in restcollection$ is equal to $\\frac{1}{2endlesscount}$, independent of $voidnumber$.\n\nLet $motionlessgroup$ denote the set of $(2endlesscount)$-tuples $\\mathbf{w} = (standstillfirst,w_2,\\ldots,standstilllast)$ such that $endlesscount$ of the standstillunit's are equal to $+1$ and the other $endlesscount$ are equal to $-1$. Define a map $fixturemap :\\thinspace motionlessgroup \\to motionlessgroup$ by\n\\[\nfixturemap(standstillfirst,w_2,\\ldots,standstilllast) = (w_2,\\ldots,standstilllast,standstillfirst);\n\\]\nthat is, $fixturemap$ moves the first entry to the end. For $\\mathbf{w} \\in motionlessgroup$, define the \\textit{orbit} of $\\mathbf{w}$ to be the collection of elements of motionlessgroup of the form $fixturemap^{steadystate}(\\mathbf{w})$, $steadystate \\geq 1$, where $fixturemap^{steadystate}$ denotes the $steadystate$-th iterate of $fixturemap$, and note that $fixturemap^{2endlesscount}(\\mathbf{w}) = \\mathbf{w}$. Then motionlessgroup is a disjoint union of orbits. For a given $\\mathbf{w} \\in motionlessgroup$, the orbit of $\\mathbf{w}$ consists of $\\mathbf{w},fixturemap(\\mathbf{w}),\\ldots,fixturemap^{maximalnumber-1}(\\mathbf{w})$, where $maximalnumber$ is the smallest positive integer with $fixturemap^{maximalnumber}(\\mathbf{w}) = \\mathbf{w}$; the list $fixturemap(\\mathbf{w}),\\ldots,fixturemap^{2endlesscount}(\\mathbf{w})$ runs through the orbit of $\\mathbf{w}$ completely $2endlesscount/maximalnumber$ times, with each element of the orbit appearing the same number of times.\n\nNow define the map $f :\\thinspace motionlessgroup \\to restcollection$ by\n\\[\nf(\\mathbf{w}) = \\mathbf{v} = (flatlandorigin,\\ldots,flatlandterminal), \\qquad flatland = \\sum_{completeunit=1}^{staticindex} standstillunit;\n\\]\nthis is a one-to-one correspondence between motionlessgroup and restcollection, with the inverse map given by\n\\[\nstandstillindex = flatland-flatlandprevious\n\\]\nfor $staticindex=1,\\ldots,2endlesscount$. We claim that for any $\\mathbf{w} \\in motionlessgroup$, the average of $\\frac{1}{segregator(\\mathbf{v})}$, where $\\mathbf{v}$ runs over vectors in the image of the orbit of $\\mathbf{w}$ under $f$, is equal to $\\frac{1}{2endlesscount}$. Since motionlessgroup is a disjoint union of orbits, restcollection is a disjoint union of the images of these orbits under $f$, and it then follows that the overall average of $\\frac{1}{segregator(\\mathbf{v})}$ over $\\mathbf{v} \\in restcollection$ is $\\frac{1}{2endlesscount}$.\n\nTo prove the claim, we compute the average of $\\frac{1}{segregator(f(fixturemap^{steadystate}(\\mathbf{w})))}$ over $steadystate=1,\\ldots,2endlesscount$; since $fixturemap^{steadystate}(\\mathbf{w})$ for $steadystate=1,\\ldots,2endlesscount$ runs over the orbit of $\\mathbf{w}$ with each element in the orbit appearing equally, this is equal to the desired average. Now if we adopt the convention that the indices in $standstillunit$ are considered mod $2endlesscount$, so that $standstillbeyond = standstillunit$ for all $completeunit$, then the $completeunit$-th entry of $fixturemap^{steadystate}(\\mathbf{w})$ is $standstillshifted$; we can then define\n\\[\nflatland = \\sum_{completeunit=1}^{staticindex} standstillunit\n\\]\nfor all $staticindex\\geq 1$, and\n\\[\nflatlandbeyond = flatland\n\\]\nfor all $completeunit$ since $\\sum_{completeunit=1}^{2endlesscount} standstillunit = 0$. We now have\n\\[\nsegregator(f(fixturemap^{steadystate}(\\mathbf{w}))) = \\sum_{staticindex=1}^{2endlesscount} voidnumber^{\\sum_{completeunit=1}^{staticindex} standstillshifted} = \\sum_{staticindex=1}^{2endlesscount} voidnumber^{flatlandoffset-flatlandshifted} = voidnumber^{-flatlandshifted} \\sum_{staticindex=1}^{2endlesscount} voidnumber^{flatland}.\n\\]\n\nThus\n\\[\n\\sum_{steadystate=1}^{2endlesscount} \\frac{1}{segregator(f(fixturemap^{steadystate}(\\mathbf{w})))} = \\sum_{steadystate=1}^{2endlesscount} \\frac{voidnumber^{flatlandshifted}}{\\sum_{staticindex=1}^{2endlesscount} voidnumber^{flatland}} = 1,\n\\]\nand the average of $\\frac{1}{segregator(f(fixturemap^{steadystate}(\\mathbf{w})))}$ over $steadystate=1,\\ldots,2endlesscount$ is $\\frac{1}{2endlesscount}$, as desired."
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "j": "hjgrksla",
        "k": "fdlqvmpo",
        "m": "zycpruox",
        "i": "vstienal",
        "q": "lfunvdse",
        "s_j": "kjdhrmwe",
        "s_j-1": "agswpntc",
        "s_0": "rqkbalwo",
        "s_2n": "xzlefdmn",
        "s_2n-1": "mrgshdpe",
        "s_k": "ltqcvaeb",
        "s_j+k": "pzvtemah",
        "s_2n+i": "ykfhzqsu",
        "w_i": "sfraokdn",
        "w_i+k": "hqntlspe",
        "w_j": "tbmfsgai",
        "w_1": "clxrmeag",
        "w_2n": "nvdpfzqo",
        "w_2n+i": "dwtcekhs",
        "V_n": "gzwoxnle",
        "W_n": "rsyeabvk",
        "\\\\phi": "umyxkrta",
        "a": "vcmorhgl"
      },
      "question": "Let $qzxwvtnp$ be a positive integer, and let $gzwoxnle$ be the set of integer $(2qzxwvtnp+1)$-tuples $\\mathbf{v} = (rqkbalwo, s_1, \\cdots, mrgshdpe, xzlefdmn)$ for which $rqkbalwo = xzlefdmn = 0$ and $|kjdhrmwe - agswpntc| = 1$ for $hjgrksla=1,2,\\cdots,2qzxwvtnp$. Define\n\\[\nlfunvdse(\\mathbf{v}) = 1 + \\sum_{hjgrksla=1}^{2qzxwvtnp-1} 3^{kjdhrmwe},\n\\]\nand let $M(qzxwvtnp)$ be the average of $\\frac{1}{lfunvdse(\\mathbf{v})}$ over all $\\mathbf{v} \\in gzwoxnle$. Evaluate $M(2020)$.",
      "solution": "The answer is $\\frac{1}{4040}$. We will show the following more general fact. Let $vcmorhgl$ be any nonzero number and define $lfunvdse(\\mathbf{v}) = 1+\\sum_{hjgrksla=1}^{2qzxwvtnp-1} vcmorhgl^{kjdhrmwe}$; then the average of $\\frac{1}{lfunvdse(\\mathbf{v})}$ over all $\\mathbf{v} \\in gzwoxnle$ is equal to $\\frac{1}{2qzxwvtnp}$, independent of $vcmorhgl$.\n\nLet $rsyeabvk$ denote the set of $(2qzxwvtnp)$-tuples $\\mathbf{w} = (clxrmeag,\\ldots,nvdpfzqo)$ such that $qzxwvtnp$ of the $sfraokdn$'s are equal to $+1$ and the other $qzxwvtnp$ are equal to $-1$. Define a map $umyxkrta :\\thinspace rsyeabvk \\to rsyeabvk$ by $umyxkrta(clxrmeag,w_2,\\ldots,nvdpfzqo) = (w_2,\\ldots,nvdpfzqo,clxrmeag)$; that is, $umyxkrta$ moves the first entry to the end. For $\\mathbf{w} \\in rsyeabvk$, define the \\textit{orbit} of $\\mathbf{w}$ to be the collection of elements of $rsyeabvk$ of the form $umyxkrta^{fdlqvmpo}(\\mathbf{w})$, $fdlqvmpo \\ge 1$, where $umyxkrta^{fdlqvmpo}$ denotes the $fdlqvmpo$-th iterate of $umyxkrta$, and note that $umyxkrta^{2qzxwvtnp}(\\mathbf{w}) = \\mathbf{w}$. Then $rsyeabvk$ is a disjoint union of orbits. For a given $\\mathbf{w} \\in rsyeabvk$, the orbit of $\\mathbf{w}$ consists of $\\mathbf{w},umyxkrta(\\mathbf{w}),\\ldots,umyxkrta^{zycpruox-1}(\\mathbf{w})$, where $zycpruox$ is the smallest positive integer with $umyxkrta^{zycpruox}(\\mathbf{w}) = \\mathbf{w}$; the list $umyxkrta(\\mathbf{w}),\\ldots,umyxkrta^{2qzxwvtnp}(\\mathbf{w})$ runs through the orbit of $\\mathbf{w}$ completely $2qzxwvtnp/zycpruox$ times, with each element of the orbit appearing the same number of times.\n\nNow define the map $f :\\thinspace rsyeabvk \\to gzwoxnle$ by $f(\\mathbf{w}) = \\mathbf{v} = (rqkbalwo,\\ldots,xzlefdmn)$ with $kjdhrmwe = \\sum_{vstienal=1}^{hjgrksla} sfraokdn$; this is a one-to-one correspondence between $rsyeabvk$ and $gzwoxnle$, with the inverse map given by $tbmfsgai = kjdhrmwe-agswpntc$ for $hjgrksla=1,\\ldots,2qzxwvtnp$. We claim that for any $\\mathbf{w} \\in rsyeabvk$, the average of $\\frac{1}{lfunvdse(\\mathbf{v})}$, where $\\mathbf{v}$ runs over vectors in the image of the orbit of $\\mathbf{w}$ under $f$, is equal to $\\frac{1}{2qzxwvtnp}$. Since $rsyeabvk$ is a disjoint union of orbits, $gzwoxnle$ is a disjoint union of the images of these orbits under $f$, and it then follows that the overall average of $\\frac{1}{lfunvdse(\\mathbf{v})}$ over $\\mathbf{v} \\in gzwoxnle$ is $\\frac{1}{2qzxwvtnp}$.\n\nTo prove the claim, we compute the average of $\\frac{1}{lfunvdse(f(umyxkrta^{fdlqvmpo}(\\mathbf{w})))}$ over $fdlqvmpo=1,\\ldots,2qzxwvtnp$; since $umyxkrta^{fdlqvmpo}(\\mathbf{w})$ for $fdlqvmpo=1,\\ldots,2qzxwvtnp$ runs over the orbit of $\\mathbf{w}$ with each element in the orbit appearing equally, this is equal to the desired average. Now if we adopt the convention that the indices in $sfraokdn$ are considered mod $2qzxwvtnp$, so that $dwtcekhs = sfraokdn$ for all $vstienal$, then the $vstienal$-th entry of $umyxkrta^{fdlqvmpo}(\\mathbf{w})$ is $hqntlspe$; we can then define $kjdhrmwe = \\sum_{vstienal=1}^{hjgrksla} sfraokdn$ for all $hjgrksla\\ge 1$, and $ykfhzqsu = s_{vstienal}$ for all $vstienal$ since $\\sum_{vstienal=1}^{2qzxwvtnp} sfraokdn = 0$. We now have\n\\[\nlfunvdse(f(umyxkrta^{fdlqvmpo}(\\mathbf{w}))) = \\sum_{hjgrksla=1}^{2qzxwvtnp} vcmorhgl^{\\sum_{vstienal=1}^{hjgrksla} hqntlspe} = \\sum_{hjgrksla=1}^{2qzxwvtnp} vcmorhgl^{pzvtemah - ltqcvaeb} = vcmorhgl^{-ltqcvaeb} \\sum_{hjgrksla=1}^{2qzxwvtnp} vcmorhgl^{kjdhrmwe}.\n\\]\n\nThus\n\\[\n\\sum_{fdlqvmpo=1}^{2qzxwvtnp} \\frac{1}{lfunvdse(f(umyxkrta^{fdlqvmpo}(\\mathbf{w})))} = \\sum_{fdlqvmpo=1}^{2qzxwvtnp} \\frac{vcmorhgl^{ltqcvaeb}}{\\sum_{hjgrksla=1}^{2qzxwvtnp} vcmorhgl^{kjdhrmwe}} = 1,\n\\]\nand the average of $\\frac{1}{lfunvdse(f(umyxkrta^{fdlqvmpo}(\\mathbf{w})))}$ over $fdlqvmpo=1,\\ldots,2qzxwvtnp$ is $\\frac{1}{2qzxwvtnp}$, as desired."
    },
    "kernel_variant": {
      "question": "Let $n$ be a positive integer and put $L:=12n$.  \nWork in $\\mathbb{Z}^{3}$ with the twelve step vectors  \n\n\\[\n\\pm e_{1}=(\\pm1,0,0),\\qquad\n\\pm e_{2}=(0,\\pm1,0),\\qquad\n\\pm e_{3}=(0,0,\\pm1),\n\\]\n\\[\n\\pm(e_{1}+e_{2})=(\\pm1,\\pm1,0),\\qquad\n\\pm(e_{2}+e_{3})=(0,\\pm1,\\pm1),\\qquad\n\\pm(e_{3}+e_{1})=(\\pm1,0,\\pm1).\n\\]\n\nDefine $V_{n}$ to be the set of lattice walks  \n\n\\[\nv=(s_{0},s_{1},\\dots ,s_{L}),\\qquad s_{j}\\in\\mathbb{Z}^{3},\n\\]\n\nthat satisfy  \n\n(a) $s_{0}=s_{L}=(0,0,0)$;  \n\n(b) for each $1\\le j\\le L$ the step $w_{j}:=s_{j}-s_{j-1}$ is one of the above\ntwelve vectors;  \n\n(c) every one of the twelve step types is taken **exactly** $n$ times in the\nwalk.\n\nClearly $\\sum_{j=1}^{L}w_{j}=0$, so condition (a) is in fact forced by (c),\nbut it is recorded for emphasis.\n\nFor arbitrary positive real numbers $\\alpha,\\beta,\\gamma$ define the weight  \n\n\\[\nq(v):=\\sum_{j=0}^{L-1}\\alpha^{x_{j}}\\beta^{y_{j}}\\gamma^{z_{j}}\\qquad\n\\bigl(s_{j}=(x_{j},y_{j},z_{j})\\bigr).\n\\]\n\nLet  \n\n\\[\nM(n):=\\frac1{|V_{n}|}\\sum_{v\\in V_{n}}\\frac1{q(v)}\n\\]\n\nbe the mean value of $1/q(v)$ over all admissible walks.\n\n(i) Prove that $M(n)$ is independent of the particular positive parameters\n$\\alpha,\\beta,\\gamma$.\n\n(ii) Determine the exact value of $M(2023)$.",
      "solution": "Throughout let $n\\in\\mathbb{N}$, $L=12n$ and $\\alpha,\\beta,\\gamma>0$ be fixed\nbut otherwise arbitrary.\n\nStep 1.  Encoding a walk by its steps.  \nFor $1\\le j\\le L$ put  \n\n\\[\nw_{j}:=s_{j}-s_{j-1}\\in\\Bigl\\{\\pm e_{1},\\pm e_{2},\\pm e_{3},\n\\pm(e_{1}+e_{2}),\\pm(e_{2}+e_{3}),\\pm(e_{3}+e_{1})\\Bigr\\}.\n\\]\n\nCondition (c) states that each of these twelve vectors occurs exactly\n$n$ times.  Consequently  \n\n\\[\n\\sum_{j=1}^{L}w_{j}=n\n\\Bigl[(e_{1}+(-e_{1}))+(e_{2}+(-e_{2}))+(e_{3}+(-e_{3}))\n+(e_{1}+e_{2}+(-e_{1}-e_{2}))+\\cdots\\Bigr]=0,\n\\]\n\nso indeed $s_{L}=s_{0}$, justifying the periodicity that will be used\nlater.  Denote by $W_{n}$ the set of all such step sequences\n$w=(w_{1},\\dots ,w_{L})$.  The map  \n\n\\[\nf:W_{n}\\longrightarrow V_{n},\\qquad\nf(w)=(s_{0},\\dots ,s_{L}),\\quad s_{0}=0,\\;\ns_{j}=\\sum_{i=1}^{j}w_{i},\n\\]\n\nis a bijection with inverse $w_{j}=s_{j}-s_{j-1}$.\n\nStep 2.  The cyclic left–shift.  \nDefine  \n\n\\[\n\\varphi:W_{n}\\longrightarrow W_{n},\\qquad\n\\varphi(w_{1},w_{2},\\dots ,w_{L})=(w_{2},w_{3},\\dots ,w_{L},w_{1}).\n\\]\n\nBecause $\\varphi^{L}=\\mathrm{id}$, every $w\\in W_{n}$ lies in a finite\n$\\varphi$–orbit.  For $k\\in\\{0,\\dots ,L-1\\}$ write\n$w^{(k)}:=\\varphi^{k}(w)$ and $v^{(k)}:=f\\bigl(w^{(k)}\\bigr)$.\n\nStep 3.  Behaviour of $q$ under a shift.  \nIntroduce the shorthand\n$A^{(x,y,z)}:=\\alpha^{x}\\beta^{y}\\gamma^{z}$.  For the partial sums of\n$v^{(k)}$ one has\n\n\\[\ns^{(k)}_{j}=s_{j+k}-s_{k}\\qquad(0\\le j\\le L),\n\\]\n\nwhere indices of the form $j+k$ are read modulo $L$; the identity\nfollows from the definition of $\\varphi$ and the fact (proved above)\nthat $s_{L}=s_{0}$.  Hence\n\n\\[\n\\begin{aligned}\nq\\bigl(v^{(k)}\\bigr)\n  &=\\sum_{j=0}^{L-1}A^{s^{(k)}_{j}}\n    =\\sum_{j=0}^{L-1}A^{s_{j+k}-s_{k}}\n    =A^{-s_{k}}\\sum_{j=0}^{L-1}A^{s_{j+k}}                              \\\\\n  &=A^{-s_{k}}\\sum_{j=0}^{L-1}A^{s_{j}}\n    =A^{-s_{k}}\\,q(v).\n\\end{aligned}\n\\]\n\nConsequently  \n\n\\[\n\\frac1{q\\bigl(v^{(k)}\\bigr)}=\\frac{A^{s_{k}}}{q(v)}\\qquad(0\\le k<L).\n\\tag{1}\n\\]\n\n(The fourth equality uses the periodicity $s_{j+L}=s_{j}$\nestablished in Step 1.)\n\nStep 4.  Summing over a complete period.  \nAdding (1) for $k=0,\\dots ,L-1$ gives  \n\n\\[\n\\sum_{k=0}^{L-1}\\frac1{q\\bigl(v^{(k)}\\bigr)}\n    =\\frac1{q(v)}\\sum_{k=0}^{L-1}A^{s_{k}}\n    =\\frac{q(v)}{q(v)}=1.\n\\tag{2}\n\\]\n\nStep 5.  Orbit–stabiliser bookkeeping.  \nLet $d$ be the length of the $\\varphi$–orbit $\\mathcal{O}(w)$.  Among\nthe $L$ indices $k$ each of the $d$ distinct walks\n$v^{(k)}\\;(k\\in\\{0,\\dots ,d-1\\})$ appears exactly $L/d$ times in the\nlist $\\{v^{(0)},\\dots ,v^{(L-1)}\\}$.  Therefore\n\n\\[\n\\sum_{u\\in\\mathcal{O}(w)}\\frac1{q(u)}\n     =\\frac{d}{L}\\sum_{k=0}^{L-1}\\frac1{q\\bigl(v^{(k)}\\bigr)}\n     =\\frac{d}{L}\\quad\\text{by (2).}\n\\tag{3}\n\\]\n\nStep 6.  Averaging over all walks.  \nSince the $\\varphi$–orbits form a partition of $W_{n}$ and $f$ is a\nbijection,\n\n\\[\n\\begin{aligned}\n\\sum_{v\\in V_{n}}\\frac1{q(v)}\n      &=\\sum_{\\mathcal{O}}\\;\\sum_{u\\in\\mathcal{O}}\\frac1{q(u)}\n        =\\sum_{\\mathcal{O}}\\frac{|\\mathcal{O}|}{L}\n        =\\frac1{L}\\sum_{\\mathcal{O}}|\\mathcal{O}|                      \\\\\n      &=\\frac1{L}\\bigl|V_{n}\\bigr|,\n\\end{aligned}\n\\]\n\nwhence  \n\n\\[\nM(n)=\\frac1{L}=\\frac1{12n}.\n\\tag{4}\n\\]\n\nRelation (4) shows that $M(n)$ is independent of\n$\\alpha,\\beta,\\gamma$, completing part (i).\n\nStep 7.  Numerical evaluation.  \nFor $n=2023$ we have $L=12\\cdot2023=24\\,276$, so\n\n\\[\n\\boxed{\\,M(2023)=\\dfrac1{24\\,276}\\,}.\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.872022",
        "was_fixed": false,
        "difficulty_analysis": "• Higher dimension – the original one-dimensional walk has been replaced by a 3-dimensional walk; the step set now has six elements instead of two.  \n• Global balancing constraint – the requirement that every one of the six step types appears exactly n times couples the three coordinates in a non-trivial way and greatly enlarges the combinatorial complexity of the state space.  \n• Non-scalar weight – the summand uses the multiplicatively independent weights α, β, γ, so the exponent at time j is the full dot-product (log α,log β,log γ)·𝐬_j.  Simple scalar tricks no longer suffice.  \n• Orbit-decomposition in a high-symmetry setting – the proof demands an understanding of group actions (here a cyclic action on a very large and complicated set) and exploits multiplicative factorisation in ℝ⁺³.  \n• Independence of parameters – despite the complicated weights, the expected value miraculously collapses to 1/(6n); recognising and justifying this invariance adds conceptual depth.\n\nThese enhancements introduce multidimensional combinatorics, group-action arguments and parameter-independence phenomena that are absent from the original kernel problem, making the variant markedly more challenging."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n$ be a positive integer and put $L:=12n$.  \nWork in $\\mathbb{Z}^{3}$ with the twelve step vectors  \n\n\\[\n\\pm e_{1}=(\\pm1,0,0),\\qquad\n\\pm e_{2}=(0,\\pm1,0),\\qquad\n\\pm e_{3}=(0,0,\\pm1),\n\\]\n\\[\n\\pm(e_{1}+e_{2})=(\\pm1,\\pm1,0),\\qquad\n\\pm(e_{2}+e_{3})=(0,\\pm1,\\pm1),\\qquad\n\\pm(e_{3}+e_{1})=(\\pm1,0,\\pm1).\n\\]\n\nDefine $V_{n}$ to be the set of lattice walks  \n\n\\[\nv=(s_{0},s_{1},\\dots ,s_{L}),\\qquad s_{j}\\in\\mathbb{Z}^{3},\n\\]\n\nthat satisfy  \n\n(a) $s_{0}=s_{L}=(0,0,0)$;  \n\n(b) for each $1\\le j\\le L$ the step $w_{j}:=s_{j}-s_{j-1}$ is one of the above\ntwelve vectors;  \n\n(c) every one of the twelve step types is taken **exactly** $n$ times in the\nwalk.\n\nClearly $\\sum_{j=1}^{L}w_{j}=0$, so condition (a) is in fact forced by (c),\nbut it is recorded for emphasis.\n\nFor arbitrary positive real numbers $\\alpha,\\beta,\\gamma$ define the weight  \n\n\\[\nq(v):=\\sum_{j=0}^{L-1}\\alpha^{x_{j}}\\beta^{y_{j}}\\gamma^{z_{j}}\\qquad\n\\bigl(s_{j}=(x_{j},y_{j},z_{j})\\bigr).\n\\]\n\nLet  \n\n\\[\nM(n):=\\frac1{|V_{n}|}\\sum_{v\\in V_{n}}\\frac1{q(v)}\n\\]\n\nbe the mean value of $1/q(v)$ over all admissible walks.\n\n(i) Prove that $M(n)$ is independent of the particular positive parameters\n$\\alpha,\\beta,\\gamma$.\n\n(ii) Determine the exact value of $M(2023)$.",
      "solution": "Throughout let $n\\in\\mathbb{N}$, $L=12n$ and $\\alpha,\\beta,\\gamma>0$ be fixed\nbut otherwise arbitrary.\n\nStep 1.  Encoding a walk by its steps.  \nFor $1\\le j\\le L$ put  \n\n\\[\nw_{j}:=s_{j}-s_{j-1}\\in\\Bigl\\{\\pm e_{1},\\pm e_{2},\\pm e_{3},\n\\pm(e_{1}+e_{2}),\\pm(e_{2}+e_{3}),\\pm(e_{3}+e_{1})\\Bigr\\}.\n\\]\n\nCondition (c) states that each of these twelve vectors occurs exactly\n$n$ times.  Consequently  \n\n\\[\n\\sum_{j=1}^{L}w_{j}=n\n\\Bigl[(e_{1}+(-e_{1}))+(e_{2}+(-e_{2}))+(e_{3}+(-e_{3}))\n+(e_{1}+e_{2}+(-e_{1}-e_{2}))+\\cdots\\Bigr]=0,\n\\]\n\nso indeed $s_{L}=s_{0}$, justifying the periodicity that will be used\nlater.  Denote by $W_{n}$ the set of all such step sequences\n$w=(w_{1},\\dots ,w_{L})$.  The map  \n\n\\[\nf:W_{n}\\longrightarrow V_{n},\\qquad\nf(w)=(s_{0},\\dots ,s_{L}),\\quad s_{0}=0,\\;\ns_{j}=\\sum_{i=1}^{j}w_{i},\n\\]\n\nis a bijection with inverse $w_{j}=s_{j}-s_{j-1}$.\n\nStep 2.  The cyclic left–shift.  \nDefine  \n\n\\[\n\\varphi:W_{n}\\longrightarrow W_{n},\\qquad\n\\varphi(w_{1},w_{2},\\dots ,w_{L})=(w_{2},w_{3},\\dots ,w_{L},w_{1}).\n\\]\n\nBecause $\\varphi^{L}=\\mathrm{id}$, every $w\\in W_{n}$ lies in a finite\n$\\varphi$–orbit.  For $k\\in\\{0,\\dots ,L-1\\}$ write\n$w^{(k)}:=\\varphi^{k}(w)$ and $v^{(k)}:=f\\bigl(w^{(k)}\\bigr)$.\n\nStep 3.  Behaviour of $q$ under a shift.  \nIntroduce the shorthand\n$A^{(x,y,z)}:=\\alpha^{x}\\beta^{y}\\gamma^{z}$.  For the partial sums of\n$v^{(k)}$ one has\n\n\\[\ns^{(k)}_{j}=s_{j+k}-s_{k}\\qquad(0\\le j\\le L),\n\\]\n\nwhere indices of the form $j+k$ are read modulo $L$; the identity\nfollows from the definition of $\\varphi$ and the fact (proved above)\nthat $s_{L}=s_{0}$.  Hence\n\n\\[\n\\begin{aligned}\nq\\bigl(v^{(k)}\\bigr)\n  &=\\sum_{j=0}^{L-1}A^{s^{(k)}_{j}}\n    =\\sum_{j=0}^{L-1}A^{s_{j+k}-s_{k}}\n    =A^{-s_{k}}\\sum_{j=0}^{L-1}A^{s_{j+k}}                              \\\\\n  &=A^{-s_{k}}\\sum_{j=0}^{L-1}A^{s_{j}}\n    =A^{-s_{k}}\\,q(v).\n\\end{aligned}\n\\]\n\nConsequently  \n\n\\[\n\\frac1{q\\bigl(v^{(k)}\\bigr)}=\\frac{A^{s_{k}}}{q(v)}\\qquad(0\\le k<L).\n\\tag{1}\n\\]\n\n(The fourth equality uses the periodicity $s_{j+L}=s_{j}$\nestablished in Step 1.)\n\nStep 4.  Summing over a complete period.  \nAdding (1) for $k=0,\\dots ,L-1$ gives  \n\n\\[\n\\sum_{k=0}^{L-1}\\frac1{q\\bigl(v^{(k)}\\bigr)}\n    =\\frac1{q(v)}\\sum_{k=0}^{L-1}A^{s_{k}}\n    =\\frac{q(v)}{q(v)}=1.\n\\tag{2}\n\\]\n\nStep 5.  Orbit–stabiliser bookkeeping.  \nLet $d$ be the length of the $\\varphi$–orbit $\\mathcal{O}(w)$.  Among\nthe $L$ indices $k$ each of the $d$ distinct walks\n$v^{(k)}\\;(k\\in\\{0,\\dots ,d-1\\})$ appears exactly $L/d$ times in the\nlist $\\{v^{(0)},\\dots ,v^{(L-1)}\\}$.  Therefore\n\n\\[\n\\sum_{u\\in\\mathcal{O}(w)}\\frac1{q(u)}\n     =\\frac{d}{L}\\sum_{k=0}^{L-1}\\frac1{q\\bigl(v^{(k)}\\bigr)}\n     =\\frac{d}{L}\\quad\\text{by (2).}\n\\tag{3}\n\\]\n\nStep 6.  Averaging over all walks.  \nSince the $\\varphi$–orbits form a partition of $W_{n}$ and $f$ is a\nbijection,\n\n\\[\n\\begin{aligned}\n\\sum_{v\\in V_{n}}\\frac1{q(v)}\n      &=\\sum_{\\mathcal{O}}\\;\\sum_{u\\in\\mathcal{O}}\\frac1{q(u)}\n        =\\sum_{\\mathcal{O}}\\frac{|\\mathcal{O}|}{L}\n        =\\frac1{L}\\sum_{\\mathcal{O}}|\\mathcal{O}|                      \\\\\n      &=\\frac1{L}\\bigl|V_{n}\\bigr|,\n\\end{aligned}\n\\]\n\nwhence  \n\n\\[\nM(n)=\\frac1{L}=\\frac1{12n}.\n\\tag{4}\n\\]\n\nRelation (4) shows that $M(n)$ is independent of\n$\\alpha,\\beta,\\gamma$, completing part (i).\n\nStep 7.  Numerical evaluation.  \nFor $n=2023$ we have $L=12\\cdot2023=24\\,276$, so\n\n\\[\n\\boxed{\\,M(2023)=\\dfrac1{24\\,276}\\,}.\n\\]",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.660605",
        "was_fixed": false,
        "difficulty_analysis": "• Higher dimension – the original one-dimensional walk has been replaced by a 3-dimensional walk; the step set now has six elements instead of two.  \n• Global balancing constraint – the requirement that every one of the six step types appears exactly n times couples the three coordinates in a non-trivial way and greatly enlarges the combinatorial complexity of the state space.  \n• Non-scalar weight – the summand uses the multiplicatively independent weights α, β, γ, so the exponent at time j is the full dot-product (log α,log β,log γ)·𝐬_j.  Simple scalar tricks no longer suffice.  \n• Orbit-decomposition in a high-symmetry setting – the proof demands an understanding of group actions (here a cyclic action on a very large and complicated set) and exploits multiplicative factorisation in ℝ⁺³.  \n• Independence of parameters – despite the complicated weights, the expected value miraculously collapses to 1/(6n); recognising and justifying this invariance adds conceptual depth.\n\nThese enhancements introduce multidimensional combinatorics, group-action arguments and parameter-independence phenomena that are absent from the original kernel problem, making the variant markedly more challenging."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation",
  "iteratively_fixed": true
}