n=11 is smallest number for which no optimal square packing 

https://web.archive.org/web/20180505055454/http://www.combinatorics.org/files/Surveys/ds7/ds7v5-2009/ds7.html#previousresults
https://en.wikipedia.org/wiki/Square_packing#In_a_square

is known. The best known packing as shown in photo has outer square size of 3.877084 times the size of a unit square. I tried with bigger values before, because I was not sure whether 3D print would be exact enough to fit. The final value I used is 3.8771, and the 11 squares fit inside. It is a tight fit, so if a factor lass than 3.877084 is possible, then with a different arrangement.

I created the model with JSCAD, here is share link you can use to open the model in your browser:

"use strict" const jscad = require('@jscad/modeling') const { subtract } = jscad.booleans const { cuboid } = jscad.primitives const { translate } = jscad.transforms const l=15 const s=8 const t=3 const f=3.8771 function main() { var ret = [] for(var j=0; j<4; ++j) for(var i=0; i<6; ++i) { if (j==0 || i==0 || (i==5 && j<3)) ret.push(translate([-70 + i*(l+s),j*(l+s),0],cuboid({size: [l,l,t]}))) } ret.push(subtract( translate([-10 , 55, 0],cuboid({size: [f*l+2*l,f*l+2*l,t]})), translate([-10 , 55, 0],cuboid({size: [f*l,f*l,t]})), )) return ret } module.exports = { main }