Square Packing n = 11

The n = 11 square packing puzzle is one of those elegant, unsolved problems in mathematics that captivates both enthusiasts and professionals around the world. On the surface, the challenge seems simple: how do you fit 11 congruent squares into the smallest possible  larger square? However, this task remains unresolved, and its complexity highlights the interesting interplay between geometry, optimization, and mathematical theory.

When you first start you may try to put them in normally in a square like configuration but you will quickly realize you can only fit 9 doing this, after you decide to rotate one or more of them you will fit 10 in one of many possible ways. You will see there is still a lot of empty space so you just need to find a way to fit the 11th block. 

There is many other interesting packing problems like n = 5 which requires one in each corner and one rotated 45 degrees in the middle. Also n = 17 is the first one that requires blocks to be at 3 different angles and looks very random. looking through others types of polygons and shapes like trying to fit squares in a circle is extremely interesting because the solutions are sometimes very bizzare and it is not known if there is a better one yet to be discovered.

For the n = 11 problem the tightest known packing of 11 squares is inside a square of side length approximately 3.877084 found by Walter Trump or exactly  √(n⁸ - 20n⁷ + 178n⁶ - 842n⁵ + 1923n ⁴  - 496n³ - 6754n² + 12420n - 6865) = 0.

If you give up and cannot find the way to make them fit here is a link to the solution https://shorturl.at/GPg8S 

Please note the blocks are made 0.99in x 0.99in to ensure it will fit well keeping the big box at the correct 3.877084