November 1st, 2018, 12:16 PM  #1 
Newbie Joined: Nov 2018 From: Poland Posts: 1 Thanks: 0  Cube, induction, block
The cube S with side 2 consists of 8 unit cubes. Define block a figure that is formed from a cube S as a result of the removal of one unit cube. Decide if the cube with dimensions 2^n⋅2^n⋅2^n from which one cube has been removed can be built by blocks. Since a block is formed by 7 unit cubes, we just need to prove that 72^(3n)−1=8^n−1 So, this is true but my teacher told me "78n−1 is a necessary condition for a solution, but I don't think on its own it is a sufficient condition." So... What's next? I don't know... Help me please Last edited by skipjack; November 2nd, 2018 at 06:51 AM. 
November 2nd, 2018, 06:53 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,098 Thanks: 1905 
For n > 1, I suspect it can't be built from undivided blocks.

November 2nd, 2018, 08:20 AM  #3  
Senior Member Joined: May 2016 From: USA Posts: 1,250 Thanks: 516  Quote:
It is of course trivially true that such a defective cube with a sidelength of 2^1 can be formed from 1 block. So that gives your base case for an induction. But you have to show how the 2^(k+1) case can be deduced from the 2^k case. And that is not just a matter of number of unit blocks. It involves a description of how to create a particular shape. When I consider trying a proof by induction, I like to start by considering the cases for n = 2 and n = 3. So consider the case where n = 2. That would be a cube (missing one unit cube somewhere) with dimensions 4x4x4. That will contain 63 unit cubes. And 9 "blocks" contain 63 unit cubes. But how do I fit those 9 "blocks" together to form the defective cube? The issue is how you fit the 9 blocks together so that 8 of the "holes" disappear while keeping the dimensions 4x4x4. Figure that out, and then figure out how to generalize that process. If you cannot figure it out for n = 2, then try to prove it is impossible. After all, you were not asked to prove that it is possible. EDIT: Of course, if skipjack's intuition is correct, it might be easier to start by trying to show why it is impossible. This might have something to do with what are the prime factors of the divisors you found. Last edited by JeffM1; November 2nd, 2018 at 08:25 AM.  

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analyis, block, cube, geometry, induction, proof 
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