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Hmmm, I'm afraid I must disagree on most points.
09/03/2019, 17:38:09

    Ramillies writes:

    If whatever part of the pencil that is uppermost is the top, then it's some of its edges. The edges don't carry any numbers, so the probability is 0, definitely not 1/5.

    Or you could argue that there are two faces that are "on top", and hence the probability should be 2/5. Definitely not 1/5.

    You are right that normal pencils truly are hexagonal prisms, but there is no problem in rolling them (I just tried). Rock it in your hand and roll it on the table, so that it rolls forward a bit like a wheel (around the longest direction of the pencil as an axis). Dropping the pencil from a height, on the contrary, seems to make it hard to randomize the result properly, because in a drop, the pencil just tends to land in whatever position it was before you dropped it. (I still recall my brother "rolling" dice in a similar manner. He was always surprised that he got mostly the same number again and again!)

    All in all, if you get rid of the word "pencil", then most of your objections are solved and the problem doesn't change at all. It's just a playful trick, nothing more.

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