squares

Prompt

You are advising on an active optimization competition entry. I need **creative, novel, and mathematically testable** ideas I can implement and verify **locally** (Python). You do **not** have access to my workspace, so I’m giving you full context below. Please propose concrete, falsifiable approaches (with enough mathematical/algorithmic detail to implement), prioritized by likely impact on the total score. ## Problem: “Squares in Squares” (Kaggle-style packing) - For each integer `n = 1..1000`, place `n` **unit squares** (side 1) with **continuous** center coordinates `(x,y)` and rotation `angle` (radians) inside the smallest **axis-aligned** enclosing square of side `L`. - **No interior overlap** between any two unit squares. **Exact contacts are rejected** by the official checker (we must enforce a strictly positive clearance, e.g. `~1e-12`). - Submission: `id,n,square_id,x,y,angle` (one row per square). `L` is computed from placed squares. ## Metric (lower is better) - Per puzzle: `score(n) = L / sqrt(n) - 1` - **Leaderboard/total = Σ_{n=1}^{1000} score(n)** - Grid/trivial for n: `L0 = ceil(sqrt(n))`, `score0(n)=L0/sqrt(n)-1`. **Full grid total ≈ 28.93766991655**. - Missing or **invalid/rejected** puzzle is charged the grid for that n. ## Current status (locally verified with official-checker logic) - **Best verified total: 22.1750063410** (0 invalid). **555/1000** puzzles beat grid, **445 at grid**. - **Target: < 22.0544214515**. **Gap: +0.1205848895** (need total reduction > 0.120585). - Decomposition (approx.): 445 at-grid n contribute ~7.14 to total; 554 improved n contribute ~15.00. - Structural headroom: of 445 at-grid, 173 are n≤324 (treated as known grid-optimal in frontier) and **272 are n>324 contributing ~2.221 score** (upper band of each k-band: improved packings exist for small j but j above a threshold is grid). Closing that 2.22 requires a genuinely new upper-band construction for those cases. **No single n can close the gap** (max single-target headroom observed < gap); we need many verified improvements or a few big ones. ## What has been tried (and is largely exhausted/negative) — do not re-propose as-is I need ideas that are **different in kind** from these, or a concretely justified variant that escapes why they failed. **Data/public imports (exhausted):** extensive sweeps of public kernels/datasets/catalogs/certificates and live frontier/witness sources for covered ranges → ~0 importable wins into the current store (frontier n≤324 absorbed; many 0-win scans incl. large cert sets, Karimi two-strip, etc.). Treat public data as saturated. **Local optimality / jammed basins (strong evidence):** - Contact-graph LP on store packings: for 199/200 puzzles (n=1..200) the optimal global shrink `t*=0` ⇒ **strict local minima under continuous moves** (no small feasible improvement even with free rotations, in that linearized sense). J4: some cases have 2nd-order/null directions but predictor-corrector/null-space/basin-hop moves led to infeasible overlap regions (curved constraint manifold). - Store packings broadly **jammed/tight** (tested): uniform/single-axis compression 0, row-relaxation 0, angle perturb+repair 0, strip/frame re-propagation 0, void-insertion audit 0 hits, monotone/transfer/rotation audits 0 (or only float-epsilon). Many improved n>324 are numerically relaxed grids yet still local minima. **Bounded local search / polish (many negatives, ~hundreds of target-approach combos, 0 per-n gains):** - SA/annealing variants (temp schedules, basin hopping, swap/grid permutations), compression-from-store, pressure/compaction ladders, perturb-and-repair, destroy-rebuild, relax-and-project, LNS variants, cluster moves, evolutionary/pop recombination/crossover (worse in tested cases), slip/dislocation, seam carving, insertion/add-one/insert-with-repair ladders, remove-and-compress, NPT/hard-particle MC / parallel-tempering attempts (tested on samples), SLSQP/penalty continuation, etc. - 2D rigid-body physics compaction (pressure/shrink, contact dynamics, diverse seeds/modes, long runs, plateau-guarded) on 2–300 sweep + targeted sets: **0 improvements** observed. **Constructive / structural families (many negatives):** - Axis/45° hybrids, sheared/tilted lattice blocks (AABB worse or not helpful; 45° blocks act as interlocking devices), block/row swaps & crossovers, tilted-row/row-based band constructions for upper-band at-grid (row-count/height conflicts shown in tested forms), Ostashev V-band attempts, pinwheel DP, periodic strips, two-strip (Karimi/Ryu) cannot reach our at-grid band, J2 MIS on fixed grids, chevron/stair bands, tapered strips, lattice clipping, vacancy buckling, rectangle/guillotine DP mis-estimates, etc. Analytic capacity models often overcount; only real-coordinate packings validated by checker count. - At-grid n>324 upper band (272 n, ~2.221 score): add-one from improved n-1 negative (no free slot); one-sided strip extensions 0 vs store; composition inequalities hold. **Audits/targeting (mostly 0 real gains):** J8 rotation/slack sweeps, waste/headroom ranking, small-n rotation audits, monotone closure attempts — no verified improvements (beware float-epsilon). **Inspiration already tested (0):** quick double-group rigid-motion prototype and angle-class constrained compaction on sample n → 0. Scaling/tiling small-n packings to n=m*k^2 gives `score(n)=score(m)` but translates **much worse overall** than current store on applicable targets (except negligible ~1e-10 perfect-square clearance artifacts) — not helpful. ## What might still be worth exploring (open directions — need *concrete* realizations) - **Parameterized constructive generators (build-from-scratch)** producing *different* contact structures: e.g. double-hex/two-group “wall-eater” families (variable splits/angles/offsets), block+tilted/wedge/other motifs with count/L prescreen + legalization/repair, especially for n>324 and the 272 upper-band at-grid n and high-ΔTOTAL targets. - **Motif/quadrant seeding + repair:** use proven small motifs/blocks as seeds with freedom/optimization in the remainder (not simple equality-composition stacking, which is negative). - **Strongly structured low-dimensional or symmetry-constrained search / basin escape** that changes contact structure (sequential perturb+interleaved quench done adaptively, hard-feasible/NPT or event-driven compression variants, informed ruin-and-recreate on defect/stress regions, nonlinear contact-manifold moves) — must differ materially from exhausted variants. - **Impact-targeted recombination/transfer** across n with repair, or fixed-L count-max/insertion in windows with robust legalization. ## Environment & constraints for testing your ideas - Local: **Python 3.11** (numpy/scipy/pandas available), **no local NVIDIA GPU** (CPU-bound). I can run scripts and have a local `checker.py` implementing: SAT separation + `1e-12` clearance, bbox `L`, `score=L/sqrt(n)-1`, total over 1..1000 (invalid/missing→grid), merge/score/build. - A result counts only if **checker-verified** (valid, no overlaps, no exact contacts). Fast, falsifiable prototypes preferred. - Current packings for any n can be treated as known inputs (I can feed best-known coords/angles to a method). I can try ideas on selected n (especially high-impact / upper-band at-grid n>324). ## What I want from you 1. **3–6 distinct, creative ideas** most likely to beat the current store on some n and reduce the total (explain the mechanism: why it can reach a *different* basin/structure than jammed local optima). 2. For each: **mathematical formulation / construction or search procedure**, key parameters, how to **prescreen** (count/L bounds) before expensive validation, and a **concrete local test plan** (which n ranges to try first, success criteria, how to verify with the checker). 3. Flag any idea that is essentially equivalent to an exhausted category above (and skip it) vs. what makes it genuinely different. 4. If relevant, suggest how to target the **272 at-grid n>324 upper-band** (~2.221 potential) or other high-ΔTOTAL opportunities. Be bold but rigorous: prioritize ideas I can actually implement and falsify locally with the checker.