Smallest square in which other squares fit

Question

What is the sidelength of the smallest square in which one can fit $n$ non-overlapping squares of sidelengths $1,2,3,4,...,n$ ?

And what is the sidelength of the smallest cube in which one can fit $n$ non-intersecting cubes of sidelengths $1,2,3,4,...,n$ ?

All squares/cubes have all sides paralell, no rotation

Answer 1

For 180 dollars, you can get a 37 squares solution and there is some discussion of cubes and here's a discussion of 70 squares.