The article discusses improving the lower bound for s(17), the side of the smallest square into which 17 unit squares can be packed. John Bidwell found in 1998 that s(17)≤4.6756…, while Trevor Green proved in 2000 that 4.4452…≤s(17) using 16 unavoidable points in a 4×6 grid. Sam Burns recently improved the lower bound to 4.4811(?) using 268 weighted points with ChatGPT assistance. The current work further improves this to 4.5058(?)≤s(17) using 168 points in a 29×29 grid with linear programming optimization. The new solution has an empty margin of 0.77565 and internal grid side of 3.9545. Each unit square must include at least one weighted point, ensuring that fitting 17 squares forces overlap. This automatically improves lower bounds for s(18), s(19), and s(20). Future work includes exploring larger grids and finding non-symmetrical versions.

The method involves placing weighted points in a square of side 4.5058 and proving that any unit square placed in this region must cover points with total weight at least 1. Since 17 unit squares would require 17 total weight but only 16.9166 is available, a contradiction arises. The weights were optimized using linear programming and rounded to nice fractions for clarity.

The distributions show discrete clustering in corners with strange bars near the center, suggesting potential for further improvement through larger grids or additional rotation directions.