{
  "index": "1969-A-2",
  "type": "ALG",
  "tag": [
    "ALG"
  ],
  "difficulty": "",
  "question": "A-2. Let \\( D_{n} \\) be the determinant of order \\( n \\) of which the element in the \\( i \\) th row and the \\( j \\) th column is the absolute value of the difference of \\( i \\) and \\( j \\). Show that \\( D_{n} \\) is equal to\n\\[\n(-1)^{n-1}(n-1) 2^{n-2}\n\\]",
  "solution": "A-2 Subtract the first column from every other column. Then add the first row to every other row. The last row now has all zeros except for ( \\( n-1 \\) ) in the first column. \\( D_{n} \\) is \\( (-1)^{n-1}(n-1) \\) times the minor formed by deleting the first column and last row from the transformed determinant. This minor has only zeros below the main diagonal and thus is equal to the product of its diagonal elements. Hence the minor has value \\( 2^{n-2} \\) and \\( D_{n}=(-1)^{n-1}(n-1) 2^{n-2} \\).\n\nAlternate Solution: From the bottom row of \\( D_{n+1} \\), subtract \\( 1 /(n-1) \\) times the first row and \\( n /(n-1) \\) times the \\( n \\)th row. This shows that \\( D_{n+1} \\) \\( =-[2 n /(n-1)] D_{n} \\), for \\( n>1 \\). The result follows easily by iteration and the observation that \\( D_{2}=-1 \\).",
  "vars": [
    "i",
    "j"
  ],
  "params": [
    "n",
    "D_n",
    "D_n+1",
    "D_2"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "i": "rowindex",
        "j": "colindex",
        "n": "ordersize",
        "D_n": "determinantn",
        "D_n+1": "determinantnplusone",
        "D_2": "determinanttwo"
      },
      "question": "A-2. Let \\( determinantn \\) be the determinant of order \\( ordersize \\) of which the element in the \\( rowindex \\) th row and the \\( colindex \\) th column is the absolute value of the difference of \\( rowindex \\) and \\( colindex \\). Show that \\( determinantn \\) is equal to\n\\[\n(-1)^{ordersize-1}(ordersize-1) 2^{ordersize-2}\n\\]\n",
      "solution": "A-2 Subtract the first column from every other column. Then add the first row to every other row. The last row now has all zeros except for ( \\( ordersize-1 \\) ) in the first column. \\( determinantn \\) is \\( (-1)^{ordersize-1}(ordersize-1) \\) times the minor formed by deleting the first column and last row from the transformed determinant. This minor has only zeros below the main diagonal and thus is equal to the product of its diagonal elements. Hence the minor has value \\( 2^{ordersize-2} \\) and \\( determinantn=(-1)^{ordersize-1}(ordersize-1) 2^{ordersize-2} \\).\n\nAlternate Solution: From the bottom row of \\( determinantnplusone \\), subtract \\( 1 /(ordersize-1) \\) times the first row and \\( ordersize /(ordersize-1) \\) times the \\( ordersize \\)th row. This shows that \\( determinantnplusone \\) \\( =-[2 ordersize /(ordersize-1)] determinantn \\), for \\( ordersize>1 \\). The result follows easily by iteration and the observation that \\( determinanttwo=-1 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "i": "sunflower",
        "j": "paintbrush",
        "n": "waterfall",
        "D_n": "keyboarder",
        "D_n+1": "sailboater",
        "D_2": "flashlight"
      },
      "question": "A-2. Let \\( keyboarder \\) be the determinant of order \\( waterfall \\) of which the element in the \\( sunflower \\) th row and the \\( paintbrush \\) th column is the absolute value of the difference of \\( sunflower \\) and \\( paintbrush \\). Show that \\( keyboarder \\) is equal to\n\\[\n(-1)^{waterfall-1}(waterfall-1) 2^{waterfall-2}\n\\]",
      "solution": "A-2 Subtract the first column from every other column. Then add the first row to every other row. The last row now has all zeros except for ( \\( waterfall-1 \\) ) in the first column. \\( keyboarder \\) is \\( (-1)^{waterfall-1}(waterfall-1) \\) times the minor formed by deleting the first column and last row from the transformed determinant. This minor has only zeros below the main diagonal and thus is equal to the product of its diagonal elements. Hence the minor has value \\( 2^{waterfall-2} \\) and \\( keyboarder=(-1)^{waterfall-1}(waterfall-1) 2^{waterfall-2} \\).\n\nAlternate Solution: From the bottom row of \\( sailboater \\), subtract \\( 1 /(waterfall-1) \\) times the first row and \\( waterfall /(waterfall-1) \\) times the \\( waterfall \\)th row. This shows that \\( sailboater \\) \\( =-[2 waterfall /(waterfall-1)] keyboarder \\), for \\( waterfall>1 \\). The result follows easily by iteration and the observation that \\( flashlight=-1 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "j": "rowindex",
        "n": "emptysize",
        "D_n": "indeterminate",
        "D_n+1": "indeterminateplusone",
        "D_2": "indeterminatepair"
      },
      "question": "A-2. Let \\( indeterminate \\) be the determinant of order \\( emptysize \\) of which the element in the \\( i \\) th row and the \\( rowindex \\) th column is the absolute value of the difference of \\( i \\) and \\( rowindex \\). Show that \\( indeterminate \\) is equal to\n\\[\n(-1)^{emptysize-1}(emptysize-1) 2^{emptysize-2}\n\\]\n",
      "solution": "A-2 Subtract the first column from every other column. Then add the first row to every other row. The last row now has all zeros except for ( \\( emptysize-1 \\) ) in the first column. \\( indeterminate \\) is \\( (-1)^{emptysize-1}(emptysize-1) \\) times the minor formed by deleting the first column and last row from the transformed determinant. This minor has only zeros below the main diagonal and thus is equal to the product of its diagonal elements. Hence the minor has value \\( 2^{emptysize-2} \\) and \\( indeterminate=(-1)^{emptysize-1}(emptysize-1) 2^{emptysize-2} \\).\n\nAlternate Solution: From the bottom row of \\( indeterminateplusone \\), subtract \\( 1 /(emptysize-1) \\) times the first row and \\( emptysize /(emptysize-1) \\) times the \\( emptysize \\)th row. This shows that \\( indeterminateplusone \\) \\( =-[2 emptysize /(emptysize-1)] indeterminate \\), for \\( emptysize>1 \\). The result follows easily by iteration and the observation that \\( indeterminatepair=-1 \\)."
    },
    "garbled_string": {
      "map": {
        "i": "i",
        "j": "qhrbplns",
        "n": "tkavzpmq",
        "D_n": "blqrfjks",
        "D_n+1": "vzmtkspq",
        "D_2": "rmnqsplk"
      },
      "question": "A-2. Let \\( blqrfjks \\) be the determinant of order \\( tkavzpmq \\) of which the element in the \\( i \\) th row and the \\( qhrbplns \\) th column is the absolute value of the difference of \\( i \\) and \\( qhrbplns \\). Show that \\( blqrfjks \\) is equal to\n\\[\n(-1)^{tkavzpmq-1}(tkavzpmq-1) 2^{tkavzpmq-2}\n\\]",
      "solution": "A-2 Subtract the first column from every other column. Then add the first row to every other row. The last row now has all zeros except for ( \\( tkavzpmq-1 \\) ) in the first column. \\( blqrfjks \\) is \\( (-1)^{tkavzpmq-1}(tkavzpmq-1) \\) times the minor formed by deleting the first column and last row from the transformed determinant. This minor has only zeros below the main diagonal and thus is equal to the product of its diagonal elements. Hence the minor has value \\( 2^{tkavzpmq-2} \\) and \\( blqrfjks=(-1)^{tkavzpmq-1}(tkavzpmq-1) 2^{tkavzpmq-2} \\).\n\nAlternate Solution: From the bottom row of \\( vzmtkspq \\), subtract \\( 1 /(tkavzpmq-1) \\) times the first row and \\( tkavzpmq /(tkavzpmq-1) \\) times the \\( tkavzpmq \\)th row. This shows that \\( vzmtkspq \\) \\( =-[2 tkavzpmq /(tkavzpmq-1)] blqrfjks \\), for \\( tkavzpmq>1 \\). The result follows easily by iteration and the observation that \\( rmnqsplk=-1 \\)."
    },
    "kernel_variant": {
      "question": "Let $n\\ge 2$ and let  \n\\[\nx_{1}<x_{2}<\\dots <x_{n}\n\\]\nbe arbitrary distinct real numbers.  Put  \n\\[\n\\Delta_{k}=x_{k+1}-x_{k}\\qquad (1\\le k\\le n-1), \\qquad \\Delta_{k}>0 .\n\\]\n\nForm the $n\\times n$ “one–dimensional distance matrix’’  \n\\[\nA_{n}=\\bigl(a_{ij}\\bigr)_{1\\le i,j\\le n},\\qquad \na_{ij}=|\\,x_{i}-x_{j}\\,|.\n\\]\n\nProve the closed-form determinant identity\n\\[\n\\boxed{\\;\n\\det A_{n}=(-1)^{\\,n-1}\\,2^{\\,n-2}\\,(x_{n}-x_{1})\\,\n        \\prod_{k=1}^{n-1}\\Delta_{k}\\;}\n\\tag{$\\star$}\n\\]\n\n(In particular, for equally spaced points $x_{k}=k$ one obtains the classical value  \n$\\det A_{n}=(-1)^{\\,n-1}(n-1)\\,2^{\\,n-2}$.)",
      "solution": "We emulate the “column–then–row’’ trick, but perform all differences \\emph{simultaneously} to avoid order-dependence.  Bidiagonal matrices with determinant $1$ make the bookkeeping transparent.\n\n--------------------------------------------------------------------\n1.  Simultaneous column differences\n--------------------------------------------------------------------\nLet\n\\[\nT\\;=\\;\n\\begin{pmatrix}\n1 & -1 &        &        & 0\\\\\n  &  1 & -1     &        &  \\\\\n  &    & \\ddots & \\ddots &  \\\\\n  &    &        &   1    & -1\\\\\n  &    &        &        &  1\n\\end{pmatrix}\n\\quad(\\text{upper bidiagonal}),\\qquad \\det T=1 .\n\\]\nThe super-diagonal entries $-1$ ensure that right multiplication\n$A_{n}T$ replaces every column $C_{j}$ ($j\\ge 2$) by $C_{j}-C_{j-1}$ while leaving $C_{1}$ unchanged.\nSet\n\\[\nB:=A_{n}T=\\bigl(b_{ij}\\bigr).\n\\]\nBecause the $x$’s are strictly increasing only two cases arise.  For fixed $j\\ge 2$,\n\\[\nb_{ij}=|x_{i}-x_{j}|-|x_{i}-x_{j-1}|\n      =\n\\begin{cases}\n+\\Delta_{\\,j-1}, & i\\le j-1,\\\\[4pt]\n-\\Delta_{\\,j-1}, & i\\ge j .\n\\end{cases}\\tag{1}\n\\]\nColumn $1$ is unaffected:\n\\[\nb_{i1}=|x_{i}-x_{1}|=x_{i}-x_{1}.\\tag{2}\n\\]\n\n--------------------------------------------------------------------\n2.  Simultaneous row differences\n--------------------------------------------------------------------\nLeft-multiply $B$ by $T^{\\!T}$ (now \\emph{lower} bidiagonal; still $\\det T^{\\!T}=1$):\n\\[\nM:=T^{\\!T}B=\\bigl(m_{ij}\\bigr).\n\\]\nEach row except the first is replaced by its difference from the preceding row, while the first row remains intact.\n\n•  Column $1$.  \nFrom (2),\n\\[\nm_{i1}=b_{i1}-b_{i-1,1}\n      =(x_{i}-x_{1})-(x_{i-1}-x_{1})\n      =x_{i}-x_{i-1}=\\Delta_{\\,i-1}\\quad (i\\ge 2),\n\\]\nand $m_{11}=0$.  Hence\n\\[\nM_{\\bullet 1}^{\\;T}=(0,\\;\\Delta_{1},\\;\\Delta_{2},\\dots,\\;\\Delta_{n-1}).\\tag{3}\n\\]\n\n•  Columns $2,\\dots,n$.  \nFix $j\\ge 2$.  The map $i\\mapsto b_{ij}$ in (1) is constant for\n$i\\le j-1$ and for $i\\ge j$, so the forward difference vanishes everywhere except at $i=j$:\n\\[\nm_{ij}=b_{ij}-b_{i-1,j}=\n\\begin{cases}\n-2\\Delta_{\\,j-1}, & i=j,\\\\[4pt]\n0, & i\\ne j\\;(i\\ge 2).\n\\end{cases}\\tag{4}\n\\]\nFor $i=1$ the original first row survives, giving\n$m_{1j}=+\\Delta_{\\,j-1}$.\n\n--------------------------------------------------------------------\n3.  Explicit block form of $M$\n--------------------------------------------------------------------\nWrite $v:=(\\Delta_{1},\\dots,\\Delta_{n-1})^{T}\\in\\mathbb{R}^{\\,n-1}$ and  \n\\[\nD:=\\operatorname{diag}\\bigl(-2\\Delta_{1},\\dots,-2\\Delta_{\\,n-1}\\bigr)\\in\\mathbb{R}^{(n-1)\\times(n-1)}.\n\\]\nEquations (3)–(4) assemble to\n\\[\nM=\\begin{pmatrix}\n0 & v^{T}\\\\[4pt]\nv & D\n\\end{pmatrix}.\\tag{5}\n\\]\n\n--------------------------------------------------------------------\n4.  Determinant via the Schur complement\n--------------------------------------------------------------------\nBecause $D$ is invertible, the Schur complement of the $0$ in the\nupper-left corner yields\n\\[\n\\det M=\\det D\\;\\det\\bigl(-v^{T}D^{-1}v\\bigr)\n      =-\\det D\\;v^{T}D^{-1}v.\\tag{6}\n\\]\n\nCompute the two factors.\n\n•  The diagonal matrix gives\n\\[\n\\det D=\\prod_{k=1}^{n-1}(-2\\Delta_{k})\n      =(-1)^{\\,n-1}\\,2^{\\,n-1}\\prod_{k=1}^{n-1}\\Delta_{k}.\\tag{7}\n\\]\n\n•  As $D^{-1}=\\operatorname{diag}\\bigl(-\\tfrac{1}{2\\Delta_{1}},\\dots,-\\tfrac{1}{2\\Delta_{\\,n-1}}\\bigr)$,\n\\[\nv^{T}D^{-1}v=\\sum_{k=1}^{n-1}\\Delta_{k}\\Bigl(-\\frac{1}{2\\Delta_{k}}\\Bigr)\\Delta_{k}\n            =-\\frac12\\sum_{k=1}^{n-1}\\Delta_{k}\n            =-\\frac12\\bigl(x_{n}-x_{1}\\bigr).\\tag{8}\n\\]\n\nInsert (7) and (8) into (6):\n\\[\n\\det M\n=-\\,\\bigl((-1)^{\\,n-1}2^{\\,n-1}\\!\\prod\\Delta_{k}\\bigr)\\;\n   \\bigl(-\\tfrac12\\,(x_{n}-x_{1})\\bigr)\n= (-1)^{\\,n-1}\\,2^{\\,n-2}\\,(x_{n}-x_{1})\\!\\prod_{k=1}^{n-1}\\!\\Delta_{k}.\\tag{9}\n\\]\n\n--------------------------------------------------------------------\n5.  Undoing the transformations\n--------------------------------------------------------------------\nSince $M=T^{\\!T}A_{n}T$ and $\\det T=\\det T^{\\!T}=1$, we have\n$\\det M=\\det A_{n}$.  Identity (9) is therefore exactly the claimed\nformula $(\\star)$.\n\n--------------------------------------------------------------------\n6.  Check for equally spaced points\n--------------------------------------------------------------------\nIf $x_{k}=k$, then $\\Delta_{k}=1$ and $x_{n}-x_{1}=n-1$, so $(\\star)$ reduces to\n\\[\n\\det A_{n}=(-1)^{\\,n-1}(n-1)\\,2^{\\,n-2},\n\\]\nthe well-known value for the integer lattice.\n\n\\hfill$\\blacksquare$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.582462",
        "was_fixed": false,
        "difficulty_analysis": "1. More variables & parameters:  The original matrix involved the fixed integers 1,…,n; the enhanced variant allows an arbitrary strictly increasing n-tuple of real parameters, introducing n−1 independent spacings Δ₁,…,Δ_{n−1}.  \n2. Additional combinatorial cases:  The proof now requires a careful case split for |xᵢ−xⱼ|−|xᵢ−x₁| that depends on the relative order of i and j, rather than the automatic {−1,0,1} pattern of the integral version.  \n3. Structural insight:  One must recognise that after the two rounds of operations the upper-triangular minor acquires heterogeneous diagonal entries Δ₁, 2Δ₂, …, 2Δ_{n−1}, then track how many factors 2 actually occur.  \n4. Parameter tracking:  Sign bookkeeping and the precise placement of the (xₙ−x₁) factor are subtler than in the integer case, since every Δ_k contributes differently.  \n5. Broader applicability:  The solver has to show that the resultant formula gracefully specialises to the classical one; this ensures the computation did not lose any hidden restrictions.\n\nBecause of these added layers—variable spacings, more elaborate case distinctions, modified diagonal structure, and careful parameter bookkeeping—the enhanced kernel variant is substantially harder than both the original problem and the current kernel version."
      }
    },
    "original_kernel_variant": {
      "question": "Let $n\\ge 2$ and let  \n\\[\nx_{1}<x_{2}<\\dots <x_{n}\n\\]\nbe arbitrary distinct real numbers.  Put  \n\\[\n\\Delta_{k}=x_{k+1}-x_{k}\\qquad (1\\le k\\le n-1), \\qquad \\Delta_{k}>0 .\n\\]\n\nForm the $n\\times n$ “one–dimensional distance matrix’’  \n\\[\nA_{n}=\\bigl(a_{ij}\\bigr)_{1\\le i,j\\le n},\\qquad \na_{ij}=|\\,x_{i}-x_{j}\\,|.\n\\]\n\nProve the closed-form determinant identity\n\\[\n\\boxed{\\;\n\\det A_{n}=(-1)^{\\,n-1}\\,2^{\\,n-2}\\,(x_{n}-x_{1})\\,\n        \\prod_{k=1}^{n-1}\\Delta_{k}\\;}\n\\tag{$\\star$}\n\\]\n\n(In particular, for equally spaced points $x_{k}=k$ one obtains the classical value  \n$\\det A_{n}=(-1)^{\\,n-1}(n-1)\\,2^{\\,n-2}$.)",
      "solution": "We emulate the “column–then–row’’ trick, but perform all differences \\emph{simultaneously} to avoid order-dependence.  Bidiagonal matrices with determinant $1$ make the bookkeeping transparent.\n\n--------------------------------------------------------------------\n1.  Simultaneous column differences\n--------------------------------------------------------------------\nLet\n\\[\nT\\;=\\;\n\\begin{pmatrix}\n1 & -1 &        &        & 0\\\\\n  &  1 & -1     &        &  \\\\\n  &    & \\ddots & \\ddots &  \\\\\n  &    &        &   1    & -1\\\\\n  &    &        &        &  1\n\\end{pmatrix}\n\\quad(\\text{upper bidiagonal}),\\qquad \\det T=1 .\n\\]\nThe super-diagonal entries $-1$ ensure that right multiplication\n$A_{n}T$ replaces every column $C_{j}$ ($j\\ge 2$) by $C_{j}-C_{j-1}$ while leaving $C_{1}$ unchanged.\nSet\n\\[\nB:=A_{n}T=\\bigl(b_{ij}\\bigr).\n\\]\nBecause the $x$’s are strictly increasing only two cases arise.  For fixed $j\\ge 2$,\n\\[\nb_{ij}=|x_{i}-x_{j}|-|x_{i}-x_{j-1}|\n      =\n\\begin{cases}\n+\\Delta_{\\,j-1}, & i\\le j-1,\\\\[4pt]\n-\\Delta_{\\,j-1}, & i\\ge j .\n\\end{cases}\\tag{1}\n\\]\nColumn $1$ is unaffected:\n\\[\nb_{i1}=|x_{i}-x_{1}|=x_{i}-x_{1}.\\tag{2}\n\\]\n\n--------------------------------------------------------------------\n2.  Simultaneous row differences\n--------------------------------------------------------------------\nLeft-multiply $B$ by $T^{\\!T}$ (now \\emph{lower} bidiagonal; still $\\det T^{\\!T}=1$):\n\\[\nM:=T^{\\!T}B=\\bigl(m_{ij}\\bigr).\n\\]\nEach row except the first is replaced by its difference from the preceding row, while the first row remains intact.\n\n•  Column $1$.  \nFrom (2),\n\\[\nm_{i1}=b_{i1}-b_{i-1,1}\n      =(x_{i}-x_{1})-(x_{i-1}-x_{1})\n      =x_{i}-x_{i-1}=\\Delta_{\\,i-1}\\quad (i\\ge 2),\n\\]\nand $m_{11}=0$.  Hence\n\\[\nM_{\\bullet 1}^{\\;T}=(0,\\;\\Delta_{1},\\;\\Delta_{2},\\dots,\\;\\Delta_{n-1}).\\tag{3}\n\\]\n\n•  Columns $2,\\dots,n$.  \nFix $j\\ge 2$.  The map $i\\mapsto b_{ij}$ in (1) is constant for\n$i\\le j-1$ and for $i\\ge j$, so the forward difference vanishes everywhere except at $i=j$:\n\\[\nm_{ij}=b_{ij}-b_{i-1,j}=\n\\begin{cases}\n-2\\Delta_{\\,j-1}, & i=j,\\\\[4pt]\n0, & i\\ne j\\;(i\\ge 2).\n\\end{cases}\\tag{4}\n\\]\nFor $i=1$ the original first row survives, giving\n$m_{1j}=+\\Delta_{\\,j-1}$.\n\n--------------------------------------------------------------------\n3.  Explicit block form of $M$\n--------------------------------------------------------------------\nWrite $v:=(\\Delta_{1},\\dots,\\Delta_{n-1})^{T}\\in\\mathbb{R}^{\\,n-1}$ and  \n\\[\nD:=\\operatorname{diag}\\bigl(-2\\Delta_{1},\\dots,-2\\Delta_{\\,n-1}\\bigr)\\in\\mathbb{R}^{(n-1)\\times(n-1)}.\n\\]\nEquations (3)–(4) assemble to\n\\[\nM=\\begin{pmatrix}\n0 & v^{T}\\\\[4pt]\nv & D\n\\end{pmatrix}.\\tag{5}\n\\]\n\n--------------------------------------------------------------------\n4.  Determinant via the Schur complement\n--------------------------------------------------------------------\nBecause $D$ is invertible, the Schur complement of the $0$ in the\nupper-left corner yields\n\\[\n\\det M=\\det D\\;\\det\\bigl(-v^{T}D^{-1}v\\bigr)\n      =-\\det D\\;v^{T}D^{-1}v.\\tag{6}\n\\]\n\nCompute the two factors.\n\n•  The diagonal matrix gives\n\\[\n\\det D=\\prod_{k=1}^{n-1}(-2\\Delta_{k})\n      =(-1)^{\\,n-1}\\,2^{\\,n-1}\\prod_{k=1}^{n-1}\\Delta_{k}.\\tag{7}\n\\]\n\n•  As $D^{-1}=\\operatorname{diag}\\bigl(-\\tfrac{1}{2\\Delta_{1}},\\dots,-\\tfrac{1}{2\\Delta_{\\,n-1}}\\bigr)$,\n\\[\nv^{T}D^{-1}v=\\sum_{k=1}^{n-1}\\Delta_{k}\\Bigl(-\\frac{1}{2\\Delta_{k}}\\Bigr)\\Delta_{k}\n            =-\\frac12\\sum_{k=1}^{n-1}\\Delta_{k}\n            =-\\frac12\\bigl(x_{n}-x_{1}\\bigr).\\tag{8}\n\\]\n\nInsert (7) and (8) into (6):\n\\[\n\\det M\n=-\\,\\bigl((-1)^{\\,n-1}2^{\\,n-1}\\!\\prod\\Delta_{k}\\bigr)\\;\n   \\bigl(-\\tfrac12\\,(x_{n}-x_{1})\\bigr)\n= (-1)^{\\,n-1}\\,2^{\\,n-2}\\,(x_{n}-x_{1})\\!\\prod_{k=1}^{n-1}\\!\\Delta_{k}.\\tag{9}\n\\]\n\n--------------------------------------------------------------------\n5.  Undoing the transformations\n--------------------------------------------------------------------\nSince $M=T^{\\!T}A_{n}T$ and $\\det T=\\det T^{\\!T}=1$, we have\n$\\det M=\\det A_{n}$.  Identity (9) is therefore exactly the claimed\nformula $(\\star)$.\n\n--------------------------------------------------------------------\n6.  Check for equally spaced points\n--------------------------------------------------------------------\nIf $x_{k}=k$, then $\\Delta_{k}=1$ and $x_{n}-x_{1}=n-1$, so $(\\star)$ reduces to\n\\[\n\\det A_{n}=(-1)^{\\,n-1}(n-1)\\,2^{\\,n-2},\n\\]\nthe well-known value for the integer lattice.\n\n\\hfill$\\blacksquare$",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.469224",
        "was_fixed": false,
        "difficulty_analysis": "1. More variables & parameters:  The original matrix involved the fixed integers 1,…,n; the enhanced variant allows an arbitrary strictly increasing n-tuple of real parameters, introducing n−1 independent spacings Δ₁,…,Δ_{n−1}.  \n2. Additional combinatorial cases:  The proof now requires a careful case split for |xᵢ−xⱼ|−|xᵢ−x₁| that depends on the relative order of i and j, rather than the automatic {−1,0,1} pattern of the integral version.  \n3. Structural insight:  One must recognise that after the two rounds of operations the upper-triangular minor acquires heterogeneous diagonal entries Δ₁, 2Δ₂, …, 2Δ_{n−1}, then track how many factors 2 actually occur.  \n4. Parameter tracking:  Sign bookkeeping and the precise placement of the (xₙ−x₁) factor are subtler than in the integer case, since every Δ_k contributes differently.  \n5. Broader applicability:  The solver has to show that the resultant formula gracefully specialises to the classical one; this ensures the computation did not lose any hidden restrictions.\n\nBecause of these added layers—variable spacings, more elaborate case distinctions, modified diagonal structure, and careful parameter bookkeeping—the enhanced kernel variant is substantially harder than both the original problem and the current kernel version."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}