Answer by Douglas Zare for Minimal blocking objects with shadows like a cube
Lower bound:Let there be $c$ cubes in a blocking configuration. Consider the graph whose vertices are cubes so that cubes are connected if they share a face. Any connected graph has at least $c-1$...
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Answer by Zack Wolske for Minimal blocking objects with shadows like a cube
Edited image to correct an error, as per Michael Biro's commentHere are a few small cases which can each be done with $\frac{3}{2}(n^2+n) - 5$ blocks. They're built by placing $n^2$ "X" blocks to give...
View ArticleAnswer by Michael Biro for Minimal blocking objects with shadows like a cube
Here is a way to get $26$ for the $C_3(4)$ case, and $2(n^2-n) + 2$ in general. In the bottom level, add $4n-5$ cubes around the outside, leaving one out adjacent to a corner. For the second level,...
View ArticleAnswer by Joel David Hamkins for Minimal blocking objects with shadows like a...
For the main case ($3\times 3\times 3$), here is a solution using 13 blocks: 1 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0Update: For the $4\times 4\times 4$ case, here is a solution using only...
View ArticleAnswer by ARupinski for Minimal blocking objects with shadows like a cube
For $C_3(n)$ an absolute upper bound is $3(n-1)^2+3$ which can always be attained by taking the cubes on three faces adjacent to a given corner, removing the corner itself, and removing all but one...
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Minimal blocking objects with shadows like a cube
This is a more geometric version of the previous question,"Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$.View an $n \times n \times n$ cube $C_3(n)$ as formed of...
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