{"paper":{"title":"Parallel packing a square with isosceles right triangles and equilateral triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Any collection of equilateral triangles homothetic to a given one with total area at most √3/4 packs parallel into a unit square.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chen-Yang Su","submitted_at":"2026-05-10T08:09:21Z","abstract_excerpt":"Suppose that $I$ is a unit square. Let $T$ (resp. $\\Delta$) be an isosceles right triangle (resp. an equilateral triangle). We prove that any collection of triangles homothetic to $T$ (resp. $\\Delta$), whose total area does not exceed $\\frac{1}{2}$ (resp. $\\frac{\\sqrt{3}}{4}$), can be parallel packed into $I$. These upper bounds are tight."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"any collection of triangles homothetic to Δ, whose total area does not exceed √3/4, can be parallel packed into I. The upper bound of √3/4 is tight.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"All triangles are homothetic to the given equilateral triangle Δ with a side parallel to a side of the unit square, restricting them to parallel packing without individual rotations.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Any collection of same-orientation equilateral triangles with total area ≤ √3/4 packs into a unit square, and the bound is tight.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Any collection of equilateral triangles homothetic to a given one with total area at most √3/4 packs parallel into a unit square.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"6c5689716a1f66e93c233b2ea7f224934a10ce19a4f1892f2a9fbe816b5839de"},"source":{"id":"2605.09406","kind":"arxiv","version":2},"verdict":{"id":"014caa38-301e-4241-a5dd-dbaab4d72c28","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T02:47:36.342218Z","strongest_claim":"any collection of triangles homothetic to Δ, whose total area does not exceed √3/4, can be parallel packed into I. The upper bound of √3/4 is tight.","one_line_summary":"Any collection of same-orientation equilateral triangles with total area ≤ √3/4 packs into a unit square, and the bound is tight.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"All triangles are homothetic to the given equilateral triangle Δ with a side parallel to a side of the unit square, restricting them to parallel packing without individual rotations.","pith_extraction_headline":"Any collection of equilateral triangles homothetic to a given one with total area at most √3/4 packs parallel into a unit square."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.09406/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T07:42:01.475602Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T19:36:00.880236Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T13:01:18.289192Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:18:14.789323Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"18fbe75dfbf51c41e9c22ee0ca8f368cfbc8bcc54b628b4b3e2513a178a98c69"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"2b050376f3b928c40053c28bda2d456132bf4cf83ddaaeddd767abe42b00060b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}