How do you arrange 6 pencils so that each one touches the other five? And what about 7 or 8?
You need 2 layers. First arrange 2 pencils that spread outwards touching by the erasers, like a V, then slide one with the pencil tip in between. Then you can repeat the process with 3 more, sideways, laying on the original, this achieves 6:
You can also achieve an arrangement of 7 by making 3 V's of 2, an arranging them like a ninja star (not really sure how else to explain it) you may need to tie them with the V's because 1 pencil has to be on the other, it could be tough to balance. This is 6, you can add one more, upright to achieve a 7th:
I believe 8 may be impossible.
Note: The matchstick version, and images, can be found at BrainDen.
I did not use it for the first solution, I found that myself (really proud) but it's where the second answer came from.
The solution for 7 pencils - without using the ends - was only recently discovered:
Seven mutually touching infinite cylinders - Sándor Bozóki, Tsung-Lin Lee, Lajos Rónyai. It was presented at "Gathering 4 Gardner" in May 2014, because the original 7-cigarette version, allowing the ends to be used, was popularised by a Martin Gardner column in Scientific American 50 years ago.
If the pencils are cylinders, and do not have to have equal radii, nine mutually touching cylingers is possible. See page 15 of this paper.