P89 n=20: exact-grid refinement of the previous public certificate. The normalized triangle leg grows from 0.306059864 to 0.306059869; the minimized, upward-rounded container side falls from 3.267334654504061337 to 3.267334601126683486. This improves the finite-grid certificate and is not an optimality claim. Method: keep all turns and quarter rotations fixed. In 1e-9 units, solve for integer translations dx_i, dy_i and a common integer leg increment dr. Keep every transformed vertex inside the unit square. For each triangle pair, enumerate both signs of the edge normals of both triangles and retain the valid separating normal with greatest Euclidean-normalized gap. With fixed orientations these constraints are linear. Maximize dr with |dx_i|,|dy_i| <= L and 1 <= dr <= L using scipy.optimize.milp/HiGHS (5 seconds, relative MIP gap 0). Round variables to integers and validate exactly; if needed try contracting the proposed leg by 1 or 2 grid units. Apply L=2,10,100,1000 in order, retaining only exact valid improvements. Accepted increments at L=2,10,100 are respectively 1,3,1 grid units. A further sweep L=2,10,100,1000,10000,100000 found no improvement. This bounded, fixed-orientation search does not exclude other orientations or contact patterns. Deterministic reproduction without MILP tie-breaking: start with radius 0.306059864 and the following rows [x,y,turn,quarter] in order: ["0.836419346","0.141695898","0.035798063",2] ["0.484597641","0.161371937","0.028065246",0] ["0.179019530","0.144206158","0.028065274",2] ["0.809383420","0.161371955","0.028065283",0] ["0.503805224","0.144207546","0.028065238",2] ["0.189764257","0.451352128","0.028065274",0] ["0.501288508","0.450608561","0.028065255",0] ["0.814534609","0.446972387","0.035798250",0] ["0.806059915","0.761106049","-0.000000010",0] ["0.193940119","0.761106091","0.000000006",1] ["0.505657603","0.446704229","0.028065249",2] ["0.499999987","0.761106097","0.000000014",0] ["0.817394032","0.444495515","0.035798250",2] ["0.783582986","0.999999995","-0.414213560",1] ["0.194965443","0.446704217","0.028065270",2] ["0.500000004","0.761106093","-0.000000004",2] ["0.216417007","0.999999994","-0.414213560",1] ["0.999999993","0.783582987","-0.414213554",0] ["0.000000001","0.783582980","0.414213562",1] ["0.153029980","0.153029963","0.000000086",0] Add these integer [dx,dy] rows times 1e-9 to the corresponding centers, preserve turn and quarter, and set radius=0.306059869: [[105,-5],[-21,3],[-26,2],[9,-5],[6,-22],[11,10],[31,-16],[13,-7],[-47,49],[-1,3],[12,0],[4,3],[-45,43],[-3,4],[24,-3],[-7,13],[3,5],[3,3],[0,8],[-19,-1]] Validation: parse all decimals on the 1e-9 grid. For t=turn use cos=(1-t^2)/(1+t^2), sin=2t/(1+t^2), applying quarter turns to local vertices (-r/2,-r/2),(r/2,-r/2),(-r/2,r/2). Check all 60 vertices inside [0,1]^2 and all 190 pairs with exact integer separating axes after clearing denominators. Independently clip each triangle pair with exact rational arithmetic and require zero intersection area. Both checks pass. With R=306059869, stored score=ceil(10^27/R)=3267334601126683486.
P89 · 等腰直角三角形装入正方形 · 讨论