A carpenter's problem

[Greg Ward]

A carpenter working with a circular saw, wishes to cut a wooden cube, three inches on a side, into 27 one inch cubes.
He can do this easily by making 6 cuts through the cube keeping the pieces together in the cube shape.
Can he reduce the number of necessary cuts by rearranging the pieces after each cut?


Greg

[Ironwood]

I dont think it can be done in less than 6 cuts.

I might not be thinking outside the square, but I think 6 is it.

[tea lady]

:rolleyes: There is gonna be lots of one inch cube block sets available on ebay tomorrow. Maybe we should link this question to how to make a tumbler to make nice sanded smooth shapes. :cool:

[mic-d]

I don't think it can be done at all unless the carpenter uses a laser circular saw with a 0mm kerf. :?:?

[Greg Ward]

Got your rosewood and camphor slices and a couple of jarrah handle pieces in the car, hope to post tomorrow.
Don't want a 3" square piece of jarrah as well to cut into squares do you?

This was posed by a mathematician, they have all manner of tools available, including laser cutters and always cut perfectly straight...... in theory only of course, I mean, they never really cut anything..... but if they did.... what is the minimum number of of theoretical cuts?
Greg

[tea lady]

Got your rosewood and camphor slices and a couple of jarrah handle pieces in the car, hope to post tomorrow.
Don't want a 3" square piece of jarrah as well to cut into squares do you?

:p Might just use all 's pine off cuts.:D

[_fly_]

If we use a laser cutter can I set up mirrors to redirect the laser and do it in 1 cut or should I ship the job offshore and then who cares how many people or cuts it takes. working on it.

[joe greiner]

No.

The central cube must be cut on 6 faces. Doesn't matter how the rest of the cuts are made.:D

Cheers,
Joe

[Greg Ward]

Joe old mate..... you're correct. (although to be fair, others also correctly stated the answer as '6', but your explanation as to why it is '6' is a beautiful example of mathematical logic)
Conceptually easy to understand but a nice tidy problem.
Now what about cutting 4" cube into inch blocks?

Greg

[Ironwood]

9

[SiJ]

I reckon 6 cuts:
Cut 1 makes 2 x 4x2x4. Move end to end (so effectively 8x2x4)
Cut 2 makes 4 x 4x1x4. Rotate stacks 90deg
Cut 3 makes 8 x 2x1x4. Move stacks end to end
Cut 4 makes 16 x 1x1x4. Put stacks on side
Cut 5 makes 32 x 1x1x2. Move stacks end to end.
Cut 6 makes 64 x 1x1x1.
:2tsup:

[Wongo]

Hmm interesting...

[Greg Ward]

Cubes of 2 x 2 x 2 and 3 x 3 x 3 are unique in the sense that no matter how the pieces are arranged before each cut (provided each piece is cut somewhere) the former will always need 3 cuts and the latter six to slice into unit cubes.

Greg

[Wongo]

I was going to tell you how to do it with 5 cuts but I won't now. :U

[andrewr79]

What's this circular saw business? Give me a decent handsaw and I'll do it in 3!

Veritas® Variable Gang Saw - Lee Valley Tools (http://www.leevalley.com/en/wood/page.aspx?p=62708&c=)

:D

[Greg Ward]

The 4x4x4 cube needs nine cuts if the pieces are kept together as a cube.
But by changing the piles..... the number can be reduced to 6.....
Well done Ironwood and Sij!

There a formula for a cube of 'n' sides but I can't work out how to post it. It reads:

For a nxnxn cube, the minimum number of cuts is 3k where k s defined as:

2 (to the power of k) is greater or equal to n which is greater than 2 (to the power of k-1)

They also considered block of n-dimensions with integral sides, which are to be sliced by a minimum number of planer cuts into unit hypercubes.....so there....

Useful they considered in the cheese and sugar loaf industries

Greg