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bonanova
Daedalian Member

 Posted: Wed Aug 15, 2012 3:22 am    Post subject: 1 An old chestnut proves that it takes six saw cuts to transform a 3x3x3 cube into 27 1x1x1 cubes, How many saw cuts are required to similarly transform a 4x4x4 cube into 64 1x1x1 cubes?_________________ Vidi, vici, veni. Last edited by bonanova on Wed Aug 15, 2012 4:02 am; edited 2 times in total
MNOWAX
0.999... of a Troll

 Posted: Wed Aug 15, 2012 3:59 am    Post subject: 2 I'm thinking - 8 Cut the cube in half, take the two halves and put them on top of each other and cut them horizontally, getting 1X4X4 stack them up and make three cuts to make 1X1X4, then stack them up and cut three more times to make your 1X1X1s. - gotta be a better solution, right?_________________The Man The Myth The Legend MNOWAX
Courk
Daedalian Member

 Posted: Wed Aug 15, 2012 4:25 am    Post subject: 3 6? Do what mnowax said at first: cut the cube in half along one face and then stack it against itself (so it looks like an 8x4x2), and cut that in half again, yielding four 4x4x1 blocks. Take those, stack them together so they look like the 4x4x4, and cut down the center of a different face. Restack it to look like the 8x4x2 again, cut down that center, you now have a bunch of 1x1x4's. Again, arrange them to look like the 4x4x4, cut down the middle. Place in 8x4x2 configuration, make final cut.
novice
No harm. Pun intended!

 Posted: Wed Aug 15, 2012 7:54 am    Post subject: 4 I think Courk's answer is as good as it gets, because each cut is straight, and all pieces are "convex" (maybe that requires a separate proof?), so a cut can only cut any given piece into two new pieces. Hence each cut can at best double the total amount of pieces. You start with one piece, so to get 64 = 2^6 pieces you need at least 6 cuts.
Elethiomel
Daedalian Member

 Posted: Wed Aug 15, 2012 8:57 am    Post subject: 5 And without looking at the specifics of Courk's method, you could just build on Novice's observation, and note that in any optimal six-cut strategy, every intermediate piece must have an even number of blocks. So just line up all your pieces and cut all of them in half, with a single long cut. 6 times.
Thok
Oh, foe, the cursed teeth!

 Posted: Wed Aug 15, 2012 9:44 am    Post subject: 6 A different argument for why Courk's 6 is as good as it gets: there's an interior block that must have all 6 sides cuts, and each side must be cut be done on a different pass.
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