Prove or disprove that a cube can be cut in 27 smaller cubes in less than 6 cuts

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M

Mauri

#3
I agree with Pancho, unless.....

Is the cube made of solid material or is it bendable? Is bending the pieces allowed?
 

smryan

Perspective.
#4
I can get 27 cubes in 6 cuts... slice each face in 3rds.

I can also get 64 cubes in 6 cuts - that requires rearranging the cubes as they get cut.

But in less then 6 cuts the pieces are not cubes. They are shoe-box shaped.
 
M

MIREGMGR

#5
Here's a technique that is sort of cheating:

Start with a cube, sized as 3x3x3 of the final-cube size. Make two parallel cuts. Now you have three pieces, each 3x3x1 of the final-cube size.

Line those three pieces up and make two more cuts. Now you have nine pieces, each 3x1x1 of the final-cube size.

Now line up those nine pieces in a flat row. This is easiest to envision as occurring on the table of a bandsaw. The nine pieces will be cut sequentially, not three at a time. As each piece is cut, what results is one 2x1x1 piece, and one 1x1x1 piece. As it's generated, immediately move each 2x1x1 piece to the end of the feed row, aligned so that its midpoint will be passed through by the blade.

After the nine 3x1x1 pieces are done being cut, the blade begins to cut through the nine 2x1x1 pieces...all in one feed process, sort of.

When you're done, you've made five cuts, and you have 27 1x1x1 cubes.
 
M

Mauri

#6
That sounds like what I had in mind.
I was thinking if the material is flexible, take your 9 3x1x1 pieces and fold them so you have 9 1.5x2x1 and then make one cut through two layers 1" from the open side.
 
T

True Position

#7
Here's a technique that is sort of cheating:

Start with a cube, sized as 3x3x3 of the final-cube size. Make two parallel cuts. Now you have three pieces, each 3x3x1 of the final-cube size.

Line those three pieces up and make two more cuts. Now you have nine pieces, each 3x1x1 of the final-cube size.

Now line up those nine pieces in a flat row. This is easiest to envision as occurring on the table of a bandsaw. The nine pieces will be cut sequentially, not three at a time. As each piece is cut, what results is one 2x1x1 piece, and one 1x1x1 piece. As it's generated, immediately move each 2x1x1 piece to the end of the feed row, aligned so that its midpoint will be passed through by the blade.

After the nine 3x1x1 pieces are done being cut, the blade begins to cut through the nine 2x1x1 pieces...all in one feed process, sort of.

When you're done, you've made five cuts, and you have 27 1x1x1 cubes.
That's not sort of cheating, that's very much cheating. If you extend the problem to including 'painting each face a different color based on each cut' there's a problem. Using your method you'd have multiple faces with the same color. This is the equivalent of cuts turning at 90 degrees.
 
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G

George Weiss

#8
I must be confused?
I see a square being cut into 3 slabs with 2 cuts
I see 3 slabs being cut into 9 smaller retangles with 2 cuts
I see 9 rectangles being cut into 27 smaller squares with 2 cuts
I see 27 smaller cubes
 
M

MIREGMGR

#9
What's a cut?

Is it when a tool enters a volume of material, and proceeds through the mass--ignoring intermediate seams at right angles to the cut line, where one piece of material is butted up against another piece--until it comes out the other side/end of the volume?

Or is it a certain length or area of cutting? Or does it have to be a certain shape of cut?
 
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