Which way does a cylinder on a slab roll after the slab is removed? An article by Tokieda[1] begins: 

I lay a cylinder on a sheet of paper. I pull the sheet from under the
  cylinder, briskly or gently. When the sheet is pulled out, which way will the cylinder roll?

The answer is that, although the cylinder will move and roll some while still on the paper, once the paper is completely removed from under the cylinder, it comes to a dead stop on the table beneath the paper. 
The article continues:

What happens if the sheet is thick? Explain why the outcome depends only on the
  sheet’s thickness and not on how vigorously I pull out the sheet.

It seems to me the same thing will happen as before. While the cylinder is on the sheet it will move and roll in the same way. When the sheet is removed (assuming no normal or friction force from the corner of the sheet), the cylinder simply falls down to the table. Falling a longer or shorter distance doesn't seem like it should affect the final horizontal motion of the cylinder. So what is the paper getting at here?
[1] Tokieda, Tadashi. "Roll models." The American Mathematical Monthly 120.3 (2013): 265-282.
 A: I think the article is mistaken, or else was asking a trick question which seemed to imply thickness matters, when in fact it doesn't.
Here's a gif of the experiment with a sheet of paper:

Here's one with a book:

The pen stops about equally well in each case and behaves about the same in each case.
A: I do not have access to the article right now, so this is just speculation about what the author is intending.
For a thin sheet, there is minimal energy loss as the cylinder rolls.  As the sheet is pulled to the left, the cylinder gains translational and rotational momentum.  When the sheet is removed, the table's friction opposes both and assuming no other losses, comes to a stop.
As the sheet increases in thickness, I can guess two different additional effects are possible.  
One is that at the end of the paper, the cylinder can roll off the "edge".  At that point the force from the paper is no longer normal to the table.  This will increase the momentum in the direction away from the paper.  As the paper becomes thicker, this edge effect becomes larger (because it maintains edge contact for longer).  This will cause the cylinder to roll away from the paper after rolling off the edge. 
The other is that the paper is removing some energy from the cylinder as it rolls.  This energy loss is likely greater as the thickness increases.  The "rolling friction" will decrease the rotational momentum and increase the linear momentum.  When the cylinder is released to the table, I would expect it to continue rolling toward the paper rather than come to a stop.
