27 Cubes

Probability Level 1

You can make a cube out of 27 cubes of the same size like this:

But can you also make a cube out of 27 cubes that aren't all the same size?

Assumption: Each cube has a finite size.

No Yes

2 solutions

Geoff Pilling
Nov 19, 2016

Take a 3x3x3 cube. Combine 8 of them that are in a 2x2x2 cube leaving 20 cubes. Then divide one of the original ones up to make a new 2x2x2 cube with smaller cubes. The cube is now made up of 27 cubes of 3 different sizes.

So, $\boxed{\text{yes}}$ you can make a cube using 27 cubes without all of them being the same size.

I like this solution. Very interesting.

Nelson M. Martinez - 4 years, 6 months ago

Hmmm, this is nice. Do you have an accompanying image that goes with your solution? I think it's much easier to visualize your explanation with an image next to it ;)

Pi Han Goh - 4 years, 6 months ago

Sure... Lemme see if I can come up with one...

Geoff Pilling - 4 years, 6 months ago

I'm a tad confused and think I'm missing something obvious! Some help would be appreciated. If you take a 3x3 cube, made up of 27 mini cubes, and combine 8 of them to make a 2x2 cube, are you not left with 27 - 8 = 19 mini cubes? Thanks

Tom Dean - 4 years, 6 months ago

Yes. 19 mini cubes and 1 larger cube for a total of 20 cubes.

Geoff Pilling - 4 years, 6 months ago

so in this case you are assuming that the cube was already made out of one/ or you just reversed the thing?(Since it can be made up into different cubes, the reverse must also be true)

Diyah Muhammed - 4 years, 6 months ago

Hrithik Singla
Dec 3, 2016

well i have a simple solution let dimention and volume of each cube be one so if we decrease the dimention by 2 times of anyone cube we can increase the dimension of any other cube by 2 times so that it occupies the space left by the cube we decreased.
we can expess it in form of an algabric equation : n = vol of 1 small cube and larger cube be 27 n so ; 27 n = 10n/2 + 22n || there can be a combination for each 3 cubes whose sum is balanced as 3n

27 Cubes

2 solutions

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