I don't want to rain on anybody's parade, but I just made both a paper cone 
and pyramid, and they are inherently unstable when placed upside down in a 
small aperture. (mine behaved the same as in the video, when placed in the 
hole of a diet coke can, a hole drilled in wood, and a few other little 
apertures I could find)
And no "push" effect could be felt over the hole in a ring magnet.
Some possible sources of error:
    - the cone or pyramid is unstable when placed in a small hole, naturally 
- one side has glue or tape making it heavier
    - Some paper for laser printers/ink jet printers, etc., have very odd 
coatings to effect the image printed on the paper - graphite, or even metals 
could be contained - graphite is VERY diamagnetic.
    - who knows what manner of crud is in the paper if it contains recycled 
material?
Some suggestions to move further:
    - Set up your cone/pyramid on a sensitive balance - balsa wood, pins, and 
a counter weight, - all attached to a base.  
    - Make a pointer at the counterweight end of the balance arm, and have a 
cardboard screen that shows the movement.
    - balance the cone/pyramid, and be sure it is resting, and air currents 
are not affecting it.
    - Move your magnet stack under the cone point and see if the balance arm 
moves.
    - Try with several DIFFERENT kinds and weights of paper.
    - if you still get a push, have the paper analyzed for the elements 
contained therein (doesn't cost much)
    - then see if others can duplicate your results
    - then double and triple check all possible sources of error
    - then have your parade
Sorry for being a grinch,  I want to see positive results as much as others 
for exciting new discoveries.  But I wouldn't want to overlook an obvious 
error, or some other important factor. 
As Carl Sagan said "extraordinary claims require extraordinary proofs" (or 
evidence).
Maybe I just did it all wrong.  Anybody get the positive results?
DMBoss1021
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