{
  "index": "2010-B-2",
  "type": "GEO",
  "tag": [
    "GEO",
    "NT"
  ],
  "difficulty": "",
  "question": "Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$?",
  "solution": "The smallest distance is 3, achieved by $A = (0,0)$, $B = (3,0)$, $C = (0,4)$.\nTo check this, it suffices to check that $AB$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triangle inequality implies that $|AC - BC| \\leq AB$, with equality if and only if $A,B,C$\nare collinear. If $AB = 1$, we may assume without loss of generality that $A = (0,0)$, $B = (1,0)$.\nTo avoid collinearity, we must have $AC = BC$, but this forces $C = (1/2, y)$ for some $y \\in \\RR$,\na contradiction. (One can also treat this case by scaling by a factor of 2 to reduce to the case $AB=2$,\ntreated in the next paragraph.)\n\nIf $AB = 2$, then we may assume without loss of generality that $A = (0,0), B = (2,0)$.\nThe triangle inequality implies $|AC - BC| \\in \\{0,1\\}$.\nAlso, for $C = (x,y)$, $AC^2 = x^2 + y^2$ and $BC^2 = (2-x)^2 + y^2$ have the same parity;\nit follows that $AC = BC$. Hence $c = (1,y)$ for some $y \\in \\RR$, so $y^2$ and $y^2+1=BC^2$\nare consecutive perfect squares. This can only happen for $y = 0$, but then $A,B,C$ are collinear,\na contradiction again.\n\n\\textbf{Remark.} Manjul Bhargava points out that more generally, a \\emph{Heronian triangle}\n(a triangle with integer sides and rational area) cannot have a side of length 1 or 2 (and again\nit is enough to treat the case of length 2). The original\nproblem follows from this because a triangle whose vertices have integer coordinates has area\nequal to half an integer (by Pick's formula or the explicit formula for the area as a determinant).",
  "vars": [
    "A",
    "B",
    "C",
    "x",
    "y"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "A": "vertexalpha",
        "B": "vertexbeta",
        "C": "vertexgamma",
        "x": "abscissa",
        "y": "ordinate"
      },
      "question": "Given that $vertexalpha$, $vertexbeta$, and $vertexgamma$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $vertexalphavertexbeta$, $vertexalphavertexgamma$, and $vertexbetavertexgamma$ are integers, what is the smallest possible value of $vertexalphavertexbeta$?",
      "solution": "The smallest distance is 3, achieved by $vertexalpha = (0,0)$, $vertexbeta = (3,0)$, $vertexgamma = (0,4)$.\nTo check this, it suffices to check that $vertexalphavertexbeta$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triangle inequality implies that $|vertexalphavertexgamma - vertexbetavertexgamma| \\leq vertexalphavertexbeta$, with equality if and only if $vertexalpha,vertexbeta,vertexgamma$\nare collinear. If $vertexalphavertexbeta = 1$, we may assume without loss of generality that $vertexalpha = (0,0)$, $vertexbeta = (1,0)$.\nTo avoid collinearity, we must have $vertexalphavertexgamma = vertexbetavertexgamma$, but this forces $vertexgamma = (1/2, ordinate)$ for some $ordinate \\in \\RR$,\na contradiction. (One can also treat this case by scaling by a factor of 2 to reduce to the case $vertexalphavertexbeta=2$,\ntreated in the next paragraph.)\n\nIf $vertexalphavertexbeta = 2$, then we may assume without loss of generality that $vertexalpha = (0,0), vertexbeta = (2,0)$.\nThe triangle inequality implies $|vertexalphavertexgamma - vertexbetavertexgamma| \\in \\{0,1\\}$.\nAlso, for $vertexgamma = (abscissa,ordinate)$, $vertexalphavertexgamma^2 = abscissa^2 + ordinate^2$ and $vertexbetavertexgamma^2 = (2-abscissa)^2 + ordinate^2$ have the same parity;\nit follows that $vertexalphavertexgamma = vertexbetavertexgamma$. Hence $c = (1,ordinate)$ for some $ordinate \\in \\RR$, so $ordinate^2$ and $ordinate^2+1=vertexbetavertexgamma^2$\nare consecutive perfect squares. This can only happen for $ordinate = 0$, but then $vertexalpha,vertexbeta,vertexgamma$ are collinear,\na contradiction again.\n\n\\textbf{Remark.} Manjul Bhargava points out that more generally, a \\emph{Heronian triangle}\n(a triangle with integer sides and rational area) cannot have a side of length 1 or 2 (and again\nit is enough to treat the case of length 2). The original\nproblem follows from this because a triangle whose vertices have integer coordinates has area\nequal to half an integer (by Pick's formula or the explicit formula for the area as a determinant)."
    },
    "descriptive_long_confusing": {
      "map": {
        "A": "notebook",
        "B": "lanterns",
        "C": "wildfire",
        "x": "bookcase",
        "y": "paintings"
      },
      "question": "Given that $notebook$, $lanterns$, and $wildfire$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $notebooklanterns$, $notebookwildfire$, and $lanternswildfire$ are integers, what is the smallest possible value of $notebooklanterns$?",
      "solution": "The smallest distance is 3, achieved by $notebook = (0,0)$, $lanterns = (3,0)$, $wildfire = (0,4)$.\nTo check this, it suffices to check that $notebooklanterns$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triangle inequality implies that $|notebookwildfire - lanternswildfire| \\leq notebooklanterns$, with equality if and only if $notebook,lanterns,wildfire$\nare collinear. If $notebooklanterns = 1$, we may assume without loss of generality that $notebook = (0,0)$, $lanterns = (1,0)$.\nTo avoid collinearity, we must have $notebookwildfire = lanternswildfire$, but this forces $wildfire = (1/2, paintings)$ for some $paintings \\in \\RR$,\na contradiction. (One can also treat this case by scaling by a factor of 2 to reduce to the case $notebooklanterns=2$,\ntreated in the next paragraph.)\n\nIf $notebooklanterns = 2$, then we may assume without loss of generality that $notebook = (0,0), lanterns = (2,0)$.\nThe triangle inequality implies $|notebookwildfire - lanternswildfire| \\in \\{0,1\\}$.\nAlso, for $wildfire = (bookcase,paintings)$, $notebookwildfire^2 = bookcase^2 + paintings^2$ and $lanternswildfire^2 = (2-bookcase)^2 + paintings^2$ have the same parity;\nit follows that $notebookwildfire = lanternswildfire$. Hence $wildfire = (1,paintings)$ for some $paintings \\in \\RR$, so $paintings^2$ and $paintings^2+1=lanternswildfire^2$\nare consecutive perfect squares. This can only happen for $paintings = 0$, but then $notebook,lanterns,wildfire$ are collinear,\na contradiction again.\n\n\\textbf{Remark.} Manjul Bhargava points out that more generally, a \\emph{Heronian triangle}\n(a triangle with integer sides and rational area) cannot have a side of length 1 or 2 (and again\nit is enough to treat the case of length 2). The original\nproblem follows from this because a triangle whose vertices have integer coordinates has area\nequal to half an integer (by Pick's formula or the explicit formula for the area as a determinant)."
    },
    "descriptive_long_misleading": {
      "map": {
        "A": "emptiness",
        "B": "vacuumspot",
        "C": "nullplace",
        "x": "verticality",
        "y": "laterality"
      },
      "question": "Given that $emptiness$, $vacuumspot$, and $nullplace$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $emptinessvacuumspot$, $emptinessnullplace$, and $vacuumspotnullplace$ are integers, what is the smallest possible value of $emptinessvacuumspot$?",
      "solution": "The smallest distance is 3, achieved by $emptiness = (0,0)$, $vacuumspot = (3,0)$, $nullplace = (0,4)$.\nTo check this, it suffices to check that $emptinessvacuumspot$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triangle inequality implies that $|emptinessnullplace - vacuumspotnullplace| \\leq emptinessvacuumspot$, with equality if and only if $emptiness,vacuumspot,nullplace$\nare collinear. If $emptinessvacuumspot = 1$, we may assume without loss of generality that $emptiness = (0,0)$, $vacuumspot = (1,0)$.\nTo avoid collinearity, we must have $emptinessnullplace = vacuumspotnullplace$, but this forces $nullplace = (1/2, laterality)$ for some $laterality \\in \\RR$,\na contradiction. (One can also treat this case by scaling by a factor of 2 to reduce to the case $emptinessvacuumspot=2$,\ntreated in the next paragraph.)\n\nIf $emptinessvacuumspot = 2$, then we may assume without loss of generality that $emptiness = (0,0), vacuumspot = (2,0)$.\nThe triangle inequality implies $|emptinessnullplace - vacuumspotnullplace| \\in \\{0,1\\}$.\nAlso, for $nullplace = (verticality,laterality)$, $emptinessnullplace^2 = verticality^2 + laterality^2$ and $vacuumspotnullplace^2 = (2-verticality)^2 + laterality^2$ have the same parity;\nit follows that $emptinessnullplace = vacuumspotnullplace$. Hence $c = (1,laterality)$ for some $laterality \\in \\RR$, so $laterality^2$ and $laterality^2+1=vacuumspotnullplace^2$\nare consecutive perfect squares. This can only happen for $laterality = 0$, but then $emptiness,vacuumspot,nullplace$ are collinear,\na contradiction again.\n\n\\textbf{Remark.} Manjul Bhargava points out that more generally, a \\emph{Heronian triangle}\n(a triangle with integer sides and rational area) cannot have a side of length 1 or 2 (and again\nit is enough to treat the case of length 2). The original\nproblem follows from this because a triangle whose vertices have integer coordinates has area\nequal to half an integer (by Pick's formula or the explicit formula for the area as a determinant)."
    },
    "garbled_string": {
      "map": {
        "A": "ruvfgbzq",
        "B": "opsnltka",
        "C": "qdgehrmn",
        "x": "mbvceokl",
        "y": "sldirwza"
      },
      "question": "Given that $ruvfgbzq$, $opsnltka$, and $qdgehrmn$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $ruvfgbzqopsnltka$, $ruvfgbzqqdgehrmn$, and $opsnltkaqdgehrmn$ are integers, what is the smallest possible value of $ruvfgbzqopsnltka$?",
      "solution": "The smallest distance is 3, achieved by $ruvfgbzq = (0,0)$, $opsnltka = (3,0)$, $qdgehrmn = (0,4)$.\nTo check this, it suffices to check that $ruvfgbzqopsnltka$ cannot equal 1 or 2. (It cannot equal 0\nbecause if two of the points were to coincide, the three points would be collinear.)\n\nThe triangle inequality implies that $|ruvfgbzqqdgehrmn - opsnltkaqdgehrmn| \\leq ruvfgbzqopsnltka$, with equality if and only if $ruvfgbzq,opsnltka,qdgehrmn$\nare collinear. If $ruvfgbzqopsnltka = 1$, we may assume without loss of generality that $ruvfgbzq = (0,0)$, $opsnltka = (1,0)$.\nTo avoid collinearity, we must have $ruvfgbzqqdgehrmn = opsnltkaqdgehrmn$, but this forces $qdgehrmn = (1/2, sldirwza)$ for some $sldirwza \\in \\RR$,\na contradiction. (One can also treat this case by scaling by a factor of 2 to reduce to the case $ruvfgbzqopsnltka=2$,\ntreated in the next paragraph.)\n\nIf $ruvfgbzqopsnltka = 2$, then we may assume without loss of generality that $ruvfgbzq = (0,0), opsnltka = (2,0)$.\nThe triangle inequality implies $|ruvfgbzqqdgehrmn - opsnltkaqdgehrmn| \\in \\{0,1\\}$.\nAlso, for $qdgehrmn = (mbvceokl,sldirwza)$, $ruvfgbzqqdgehrmn^2 = mbvceokl^2 + sldirwza^2$ and $opsnltkaqdgehrmn^2 = (2-mbvceokl)^2 + sldirwza^2$ have the same parity;\nit follows that $ruvfgbzqqdgehrmn = opsnltkaqdgehrmn$. Hence $c = (1,sldirwza)$ for some $sldirwza \\in \\RR$, so $sldirwza^2$ and $sldirwza^2+1=opsnltkaqdgehrmn^2$\nare consecutive perfect squares. This can only happen for $sldirwza = 0$, but then $ruvfgbzq,opsnltka,qdgehrmn$ are collinear,\na contradiction again.\n\n\\textbf{Remark.} Manjul Bhargava points out that more generally, a \\emph{Heronian triangle}\n(a triangle with integer sides and rational area) cannot have a side of length 1 or 2 (and again\nit is enough to treat the case of length 2). The original\nproblem follows from this because a triangle whose vertices have integer coordinates has area\nequal to half an integer (by Pick's formula or the explicit formula for the area as a determinant)."
    },
    "kernel_variant": {
      "question": "Four distinct lattice points — denote them $P_1,P_2,P_3,P_4$ — are placed in the Euclidean plane subject to the two conditions  \n\n1. no three of the four points are collinear,  \n2. every one of the six mutual distances $P_iP_j$ is an integer.  \n\n(The four points are thus an integral lattice 4-set.)  \nThe diameter of the set is the largest of those six distances.\n\nDetermine the least possible diameter of such a set and exhibit an explicit configuration that attains this minimum.",
      "solution": "Throughout “lattice point’’ means a point with integer coordinates.\n\nStep 1.  A preparatory lemma on small sums of two squares.  \nLemma 1.  Let $d\\in\\{1,2,3,4\\}$.  If integers $x,y$ satisfy  \n  $x^{2}+y^{2}=d^{2}$,  \nthen one of $x,y$ equals $0$.\n\nProof.  A direct enumeration of the at most $9$ possibilities for $|x|,|y|\\le4$ suffices:\n\nd = 1 : possible squares 0,1 → only $(1,0)$ or $(0,1)$;  \n\nd = 2 : $d^{2}=4$; among $\\{0,1,4\\}$ the only decomposition of $4$ is $4+0$;  \n\nd = 3 : $d^{2}=9$; possible decompositions with squares $\\le9$ are $9+0$;  \n\nd = 4 : $d^{2}=16$; admissible squares $\\le16$ are $0,1,4,9,16$; the only way to hit $16$ is $16+0$.  \n\nHence in every case $x\\,y=0$. ∎\n\n(One can also give a modular proof: if both $x,y$ were odd, then $x^{2}+y^{2}\\equiv2\\pmod4$; if both were even, $x^{2}+y^{2}$ is divisible by $4$ but not by $16$; either situation contradicts $d^{2}$ being a square $\\le16$.)\n\nConsequences of Lemma 1.  Whenever two lattice points are at integral distance $1,2,3,$ or $4$, the segment joining them is axis-parallel: their difference vector has one zero coordinate.\n\nStep 2.  What would happen if the diameter were at most 4?  \nTranslate the configuration so that $P_1=(0,0)$.  Because every distance $P_1P_j\\;(j=2,3,4)$ is an integer not exceeding the putative diameter $4$, Lemma 1 forces each $P_j$ to lie on one of the two coordinate axes through $P_1$.  Concretely,\n $P_j=(\\pm d_j,0)$  or  $P_j=(0,\\pm d_j)$\nwith $d_j\\in\\{1,2,3,4\\}$ for $j=2,3,4$.\n\n(a) If all three points $P_2,P_3,P_4$ share the $x$-axis with $P_1$, then the four points are collinear; this is forbidden.\n\n(b) Consequently at least one of $P_2,P_3,P_4$ lies on the $y$-axis.  Without loss of generality suppose $P_2$ is on the $x$-axis and $P_3$ is on the $y$-axis; write\n $P_2=(\\varepsilon_2 d_2,0)$,  $P_3=(0,\\varepsilon_3 d_3)$,  $\\varepsilon_2,\\varepsilon_3\\in\\{\\pm1\\}$.\n\nStep 3.  The inter-axis distance cannot stay below 5.  \nThe distance between $P_2$ and $P_3$ is\n $P_2P_3=\\sqrt{d_2^{2}+d_3^{2}}\\ge\\sqrt{1^{2}+1^{2}}=\\sqrt2.$\nFor this distance to be an integer $\\le4$, the integer $d_2^{2}+d_3^{2}$ would itself have to be a perfect square not exceeding $16$.  Exhausting the small squares,\n $1,4,9,16$,\none checks at once that none can be written as a sum of two positive squares (the smallest such sum is $1^{2}+1^{2}=2$ and the next is $1^{2}+2^{2}=5$).  The first perfect square that *is* a sum of two positive squares is\n $3^{2}+4^{2}=25=5^{2}$.\n\nHence any pair of lattice points that lie on distinct coordinate axes and are not at the origin are at distance at least 5.\n\nStep 4.  Contradiction for diameter ≤ 4, therefore a lower bound.  \nBecause the configuration is non-collinear, case (b) of Step 2 occurs, which by Step 3 forces a distance $\\ge5$ inside the set.  Thus no integral lattice 4-set can have diameter $\\le4$ and we obtain the lower bound\n $D_{\\min}\\ge5$.\n\nStep 5.  A configuration whose diameter is exactly 5.  \nConsider the $3\\times4$ rectangle\n $P_1=(0,0),\\;P_2=(3,0),\\;P_3=(0,4),\\;P_4=(3,4).$\nThe six pairwise distances are\n $P_1P_2=3,\\;P_1P_3=4,\\;P_1P_4=5,\\;P_2P_3=5,\\;P_2P_4=4,\\;P_3P_4=3$,\nall integers.  No three of the four vertices are collinear, and the largest distance equals $5$.  Hence $\\operatorname{diam}\\{P_1,P_2,P_3,P_4\\}=5$.\n\nStep 6.  Conclusion.  \nA diameter of $5$ is attainable, and a diameter $\\le4$ is impossible.  Therefore\n\n  least possible diameter $=\\boxed{5}$.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.817786",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original triangle problem, the enhanced variant adds several layers of complexity.\n\n1. More objects and constraints.  \n   The configuration involves four points instead of three, which increases the number of distance conditions from three to six and introduces the additional “no three collinear’’ constraint.\n\n2. Global-versus-local reasoning.  \n   One must consider all pairwise distances simultaneously and understand how short lattice distances constrain the \\emph{entire} layout, not just a single side.\n\n3. Number-theoretic classification.  \n   The key lemma classifies sums of two squares up to 16, invoking modular arithmetic (work modulo 4) and elementary facts about representations as sums of squares.\n\n4. Geometric synthesis.  \n   Translating the numerical lemma into geometric language (all very short integral vectors are axis-parallel) and then using it to rule out a whole class of configurations requires a blend of combinatorial geometry and arithmetic.\n\n5. Exhaustion of small cases without brute force.  \n   Instead of enumerating hundreds of point sets, the argument shows abstractly that diameter ≤ 4 cannot occur, a more delicate task than the original “length 1 or 2 is impossible’’ proof.\n\nAltogether these elements demand a broader toolkit—modular arithmetic, properties of sums of squares, and global geometric reasoning—making the new kernel variant substantially harder than the original."
      }
    },
    "original_kernel_variant": {
      "question": "Four distinct lattice points — denote them $P_1,P_2,P_3,P_4$ — are placed in the Euclidean plane subject to the two conditions  \n\n1. no three of the four points are collinear,  \n2. every one of the six mutual distances $P_iP_j$ is an integer.  \n\n(The four points are thus an integral lattice 4-set.)  \nThe diameter of the set is the largest of those six distances.\n\nDetermine the least possible diameter of such a set and exhibit an explicit configuration that attains this minimum.",
      "solution": "Throughout “lattice point’’ means a point with integer coordinates.\n\nStep 1.  A preparatory lemma on small sums of two squares.  \nLemma 1.  Let $d\\in\\{1,2,3,4\\}$.  If integers $x,y$ satisfy  \n  $x^{2}+y^{2}=d^{2}$,  \nthen one of $x,y$ equals $0$.\n\nProof.  A direct enumeration of the at most $9$ possibilities for $|x|,|y|\\le4$ suffices:\n\nd = 1 : possible squares 0,1 → only $(1,0)$ or $(0,1)$;  \n\nd = 2 : $d^{2}=4$; among $\\{0,1,4\\}$ the only decomposition of $4$ is $4+0$;  \n\nd = 3 : $d^{2}=9$; possible decompositions with squares $\\le9$ are $9+0$;  \n\nd = 4 : $d^{2}=16$; admissible squares $\\le16$ are $0,1,4,9,16$; the only way to hit $16$ is $16+0$.  \n\nHence in every case $x\\,y=0$. ∎\n\n(One can also give a modular proof: if both $x,y$ were odd, then $x^{2}+y^{2}\\equiv2\\pmod4$; if both were even, $x^{2}+y^{2}$ is divisible by $4$ but not by $16$; either situation contradicts $d^{2}$ being a square $\\le16$.)\n\nConsequences of Lemma 1.  Whenever two lattice points are at integral distance $1,2,3,$ or $4$, the segment joining them is axis-parallel: their difference vector has one zero coordinate.\n\nStep 2.  What would happen if the diameter were at most 4?  \nTranslate the configuration so that $P_1=(0,0)$.  Because every distance $P_1P_j\\;(j=2,3,4)$ is an integer not exceeding the putative diameter $4$, Lemma 1 forces each $P_j$ to lie on one of the two coordinate axes through $P_1$.  Concretely,\n $P_j=(\\pm d_j,0)$  or  $P_j=(0,\\pm d_j)$\nwith $d_j\\in\\{1,2,3,4\\}$ for $j=2,3,4$.\n\n(a) If all three points $P_2,P_3,P_4$ share the $x$-axis with $P_1$, then the four points are collinear; this is forbidden.\n\n(b) Consequently at least one of $P_2,P_3,P_4$ lies on the $y$-axis.  Without loss of generality suppose $P_2$ is on the $x$-axis and $P_3$ is on the $y$-axis; write\n $P_2=(\\varepsilon_2 d_2,0)$,  $P_3=(0,\\varepsilon_3 d_3)$,  $\\varepsilon_2,\\varepsilon_3\\in\\{\\pm1\\}$.\n\nStep 3.  The inter-axis distance cannot stay below 5.  \nThe distance between $P_2$ and $P_3$ is\n $P_2P_3=\\sqrt{d_2^{2}+d_3^{2}}\\ge\\sqrt{1^{2}+1^{2}}=\\sqrt2.$\nFor this distance to be an integer $\\le4$, the integer $d_2^{2}+d_3^{2}$ would itself have to be a perfect square not exceeding $16$.  Exhausting the small squares,\n $1,4,9,16$,\none checks at once that none can be written as a sum of two positive squares (the smallest such sum is $1^{2}+1^{2}=2$ and the next is $1^{2}+2^{2}=5$).  The first perfect square that *is* a sum of two positive squares is\n $3^{2}+4^{2}=25=5^{2}$.\n\nHence any pair of lattice points that lie on distinct coordinate axes and are not at the origin are at distance at least 5.\n\nStep 4.  Contradiction for diameter ≤ 4, therefore a lower bound.  \nBecause the configuration is non-collinear, case (b) of Step 2 occurs, which by Step 3 forces a distance $\\ge5$ inside the set.  Thus no integral lattice 4-set can have diameter $\\le4$ and we obtain the lower bound\n $D_{\\min}\\ge5$.\n\nStep 5.  A configuration whose diameter is exactly 5.  \nConsider the $3\\times4$ rectangle\n $P_1=(0,0),\\;P_2=(3,0),\\;P_3=(0,4),\\;P_4=(3,4).$\nThe six pairwise distances are\n $P_1P_2=3,\\;P_1P_3=4,\\;P_1P_4=5,\\;P_2P_3=5,\\;P_2P_4=4,\\;P_3P_4=3$,\nall integers.  No three of the four vertices are collinear, and the largest distance equals $5$.  Hence $\\operatorname{diam}\\{P_1,P_2,P_3,P_4\\}=5$.\n\nStep 6.  Conclusion.  \nA diameter of $5$ is attainable, and a diameter $\\le4$ is impossible.  Therefore\n\n  least possible diameter $=\\boxed{5}$.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.625943",
        "was_fixed": false,
        "difficulty_analysis": "Compared with the original triangle problem, the enhanced variant adds several layers of complexity.\n\n1. More objects and constraints.  \n   The configuration involves four points instead of three, which increases the number of distance conditions from three to six and introduces the additional “no three collinear’’ constraint.\n\n2. Global-versus-local reasoning.  \n   One must consider all pairwise distances simultaneously and understand how short lattice distances constrain the \\emph{entire} layout, not just a single side.\n\n3. Number-theoretic classification.  \n   The key lemma classifies sums of two squares up to 16, invoking modular arithmetic (work modulo 4) and elementary facts about representations as sums of squares.\n\n4. Geometric synthesis.  \n   Translating the numerical lemma into geometric language (all very short integral vectors are axis-parallel) and then using it to rule out a whole class of configurations requires a blend of combinatorial geometry and arithmetic.\n\n5. Exhaustion of small cases without brute force.  \n   Instead of enumerating hundreds of point sets, the argument shows abstractly that diameter ≤ 4 cannot occur, a more delicate task than the original “length 1 or 2 is impossible’’ proof.\n\nAltogether these elements demand a broader toolkit—modular arithmetic, properties of sums of squares, and global geometric reasoning—making the new kernel variant substantially harder than the original."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}