We (Lucy and Natassia) tried tackling the 'how many 1 inch circles on a piece of paper that is 8.5"x11"' problem and discovered:

The sheet of paper is 8.5"x11" inches but half circles are not allowed. So the easiest way to organize the circles is simply in a grid like format of 8 circles x 11 circles, which gives a grand total of 88 circles in a 8.5" x 11" rectangle. However there is a way to pack more circles in this space...

...which is by staggering the rows of circles (see picture above). Instead of lining them up as if they were in a 8"x11" grid, if you move the first row up against one side, and the next row on top up against the other side and so on, therefore it is then possible to fit 12 rows instead of only 11 and a total of 96 circles in the rectangle.

We were wondering how this might work in a 3D version, which would be a 8.5"x8.5"x11" rectangular cube filled with 1 inch diameter spheres. Possibly each plane/sheet of spheres (96 in total according to above proof) stacked in a staggered fashion on top of each other would allow for a new "sheet" of spheres, like the staggering in 2D allowed for another row? Would the total amount of spheres be 9x8x11 or just 8x8x11? Please comment with ideas!

Here is a video showing our thought process: