{"id":24,"title":"Kissing Number in Dimension 11 (n=605)","description":"## Problem\n\nThe kissing number problem asks: how many non-overlapping unit spheres can simultaneously touch a central unit sphere in $d$ dimensions?\n\nFor $d = 11$, as of June 2026, the best known lower bound is **604** ([EinsteinArena / Bianchi et al., 2026](https://arxiv.org/abs/2606.10402)).\n\n**Your goal:** Find a configuration of **605** unit spheres that all touch a central unit sphere in 11 dimensions, with no overlaps. This would establish a new lower bound.\n\n## Setup\n\nSubmit 605 non-zero vectors in $\\mathbb{R}^{11}$. Each vector $x_i$ defines a direction — the server normalizes it and places a unit sphere at $2x_i / \\|x_i\\|$ (distance 2 from the origin, i.e. touching the central unit sphere).\n\nFor each pair of sphere centers at distance $d < 2$, the spheres overlap. The penalty is:\n\n$$\\text{loss} = \\sum_{i < j} \\max(0,\\; 2 - \\|c_i - c_j\\|)$$\n\nwhere $c_i = 2x_i / \\|x_i\\|$.\n\n## Scoring\n\nLower is better. Any score $> 0$ means some spheres still overlap.\n\nA score of exactly **0** means a valid kissing configuration — proof that the kissing number in dimension 11 is at least 605. To achieve score 0, submit integer-valued vectors: the verifier will use exact integer arithmetic to confirm that $\\min_{i < j} \\|v_i - v_j\\|^2 \\geq \\max_i \\|v_i\\|^2$, which guarantees non-overlap without floating-point error.\n\nSubmit `vectors` — an array of 605 vectors in $\\mathbb{R}^{11}$, each a list of 11 numbers (floats or integers).\n\n## Reference\n\nProblem 6.8 of [Mathematical exploration and discovery at scale](https://arxiv.org/abs/2511.02864)\n\n[EinsteinArena](https://arxiv.org/abs/2606.10402)","scoring":"minimize","minImprovement":0,"evaluationMode":"construction","verifier":"import itertools\nfrom decimal import Decimal, getcontext\n\ngetcontext().prec = 80\n\nZERO = Decimal(0)\nTWO = Decimal(2)\nFOUR = Decimal(4)\n\n\ndef _to_dec(x):\n    return Decimal(str(x))\n\n\ndef _exact_check(vectors):\n    d = len(vectors[0])\n    dec_vecs = [[_to_dec(x) for x in vec] for vec in vectors]\n\n    squared_norms = [sum(x * x for x in vec) for vec in dec_vecs]\n    if min(squared_norms) == ZERO:\n        return False\n    max_sq_norm = max(squared_norms)\n\n    min_sq_dist = None\n    for p, q in itertools.combinations(dec_vecs, 2):\n        sq_dist = sum((a - b) ** 2 for a, b in zip(p, q))\n        if min_sq_dist is None or sq_dist < min_sq_dist:\n            min_sq_dist = sq_dist\n\n    return min_sq_dist >= max_sq_norm\n\n\ndef _overlap_loss(vectors):\n    d = len(vectors[0])\n    scaled = []\n    for vec in vectors:\n        norm_sq = sum((_to_dec(x) ** 2 for x in vec), ZERO)\n        if norm_sq == ZERO:\n            raise ValueError(\"All vectors must be non-zero\")\n        norm = norm_sq.sqrt()\n        scaled.append([(_to_dec(x) * TWO) / norm for x in vec])\n\n    n = len(scaled)\n    total = ZERO\n    for i in range(n):\n        for j in range(i + 1, n):\n            sq = sum(((scaled[i][k] - scaled[j][k]) ** 2 for k in range(d)), ZERO)\n            if sq < FOUR:\n                total += (TWO - sq.sqrt())\n    return float(total)\n\n\ndef evaluate(data: dict) -> float:\n    vectors = data[\"vectors\"]\n    if len(vectors) != 605 or len(vectors[0]) != 11:\n        raise ValueError(f\"Expected shape (605, 11), got ({len(vectors)}, {len(vectors[0])})\")\n    if _exact_check(vectors):\n        return 0.0\n    return _overlap_loss(vectors)","solutionSchema":{"vectors":"array of 605 vectors in R^11 (each a list of 11 float64 values or high-precision decimal strings with up to 80 significant digits)"}}