Painting a cube

How many distinguishable cubes can you make by painting each face one of three colors?

How many can you make if cubes which are mirror images of each other are considered to be the same?

What about if you have n colors, instead of three?

Jonathan Dushoff

all of them.

57

56

27*Cn,2 + 30*Cn,3 + 80*Cn,4 + 60*Cn,5 + 30*Cn,6 or, if mirror images are considered the same, 27*Cn,2 + 29*Cn,3 + 52*Cn,4 + 30*Cn,5 + 15*Cn,6

i should have warned that thie first one was a problem appearing on the hoeflin power test. a definite no-no.

Yes. I also received this information by email from Andreas Gunnarsson. I am not personally impressed with the idea of the test as an intelligence test, or the idea of sitting down to do the whole thing and get a score, but it definitely has some nice puzzles. It is at http://www.eskimo.com/~miyaguch/power.html. Since I am pointing r.p. to it, I suppose I have a responsibility to request that people respect the suggested limitations on discussion of these problems.

: a definite no-no.

This presumably implies that I cribbed the puzzle from Hoeflin, which would be insulting, silly (given the simplicity of Hoeflin's puzzle, and the fact that mine has the non-obvious extension to mirror images), and - for the record - false.

: [Spoilers without spoiler space]

How come?

Jonathan Dushoff

Are you saying that my answers are wrong (which is quite possible), or asking me how I obtained them?

:> :>How come?

: Are you saying that my answers are wrong (which is quite possible), or : asking me how I obtained them?

I would guess he's asking you why you didn't leave any spoiler space, since that's the only feature of your post he brought attention to.

I thought I was fairly clear: I am asking how come you posted spoilers without spoiler space. Although I am also curious why there was no explanation: were the problems that boring?

Jonathan Dushoff

I should have used spoiler space. I usually use spoiler space, and on this occasion I forgot. I apologise for this.

I thought the problems were good ones, and will post the reasoning behind my replies, if there is any more evidence of demand for it.

Yes please, I've been trying to whittle my solutions down to 56 but I can't catch all the symmetries.

looks like i stirred up some trouble here. hey, sorry man. yes it is perfectly reasonable to imagine that someone could dream hoeflin's puzzles up. and I DO believe you. 'Compromise' has been a term applied to Hoeflin's 4 so-called 'Power Tests' for a long time. They were originally designed as take home tests with no tme limit and no help from any source other than books - except for the Power which allows only the mighty pen. It was hoped that they would measure high IQ's unmeasurable by other traditional tests. The answers began leaking however, and by 1995, the Mega was pretty much trashed. The main reason for the spread of these answers is news groups where one can do vast searches and even turn up the answers to specific problems. Usual people wanting a solution could pose a 'tough' problem to a pack of bloodthirsty puzzlists and receive the credit, while the answer remained in the archives waiting for other vultures. the problem was eventually recognized once the tests were online at Darryl's site there. so people could call these cheaters wanting a solution to: how many pieces can you cut a cube of butter into using 5 planar strokes? so the cheaters got better and started doing 3 things: 1) trivially modifying the puzzle while superficially making it appear different 2) making it harder. for example: how many pieces can you cut a cube of butter into with N planar knife strokes. now post it at sci.math and it looks like a tricky university problem that has stumped some poor kid. 3) camoflage it within a bunch of other problems. I guess my alarm bell went off for number 3. sorry.