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is following step right proving?

it is first time I prove something so I am wonder whether it is right or wrong.

---given problem---

there is a cube whose one side is 3 inch long.

a person cuts 6 times for making 27 cubes whose one side is 1 inch long.

is there a way to cut less than 6 times for making 27 cubes?

if not, prove it.

---proving---

for any shape of object,

for making one or more cube, I should cut 6 times

because only one plane would be made for each cut.

and cube have 6 plane.

making 27 cubes is more than making a cube.

so making 27 cubes require at least cutting 6 times which time is times of cutting for making a cube.

I tried to prove "for making one or more cube, it require cutting at least 6 times"

does it same as proving "there is no way to make 27 cube with cutting less than 6 times"?

thanks!

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3  
If you are referring to the little cube inside, then this sounds good to me. That cube doesn't share any of its faces with the big 3x3x3 cube, and a single cut cannot produce more than one face, as you said (I think?) –  Jyrki Lahtonen Jul 22 '11 at 8:58
1  
If I want to make 8 cubes, I only need to cut 3 times. But your argument would still suggest cutting 6 times because "making 8 cubes is more than making a cube". So there is something missing in your argument. –  Rahul Jul 22 '11 at 9:01
1  
(And the missing thing is in Jyrki's comment, which I didn't see before posting...) –  Rahul Jul 22 '11 at 9:02
1  
The way I saw Jyrki's comment expressed originally was to think of painting the exterior of the original cube. The inner cube has no paint, so needs 6 cuts, no matter how you rearrange the pieces between cuts. This distinguishes the 3x3x3 case from the 2x2x2 case. –  Ross Millikan Jul 22 '11 at 15:30
    
I quite like Ross's use of paint here. Admittedly my choice of phrase "shares a face" is not quite as precise as we might like it to be, but at least you understood. –  Jyrki Lahtonen Jul 23 '11 at 7:43

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