Joerg Arndt looked at connected polycubes with no three cubes centers in a row, in any direction. The largest is a unique 26-cube. The specific cube coordinates are as follows:
cubes={{0,1,2},{0,1,3},{0,2,1},{0,2,2},{1,0,3},{1,1,3},{1,1,4},{1,2,1},{1,2,4},{1,3,1},{1,3,2},{1,4,2},{2,0,2},{2,0,3},{2,1,0},{2,1,1},{2,2,4},{2,2,5},{2,3,3},{2,3,4},{3,0,2},{3,1,1},{3,1,2},{3,2,3},{3,3,3},{4,2,3}};
The graphical representation of this structure is displayed below:
Graphics3D[{Opacity[.7],Cuboid/@cubes},Boxed->False,SphericalRegion->True]
Out[]=
This arrangement is notably unconventional. To verify the absence of a three-in-a-row configuration, the following Wolfram Language code is employed, which yields empty set:
In[]:=
[◼]
FindExtraordinaryLines
[cubes]
Out[]=
{}
The Wolfram Language function FindExtraordinaryLines identifies lines passing through three or more points, enabling exploration of the given set of points in any direction. A related problem is described in a Wolfram Demonstration No-Three-in-Line Problem, involving the placement of 2N points on an NxN grid such that no three points are collinear. A known solution is a 52x52 square with 104 points, and the possibility of a larger solution remains uncertain.
For a no-four-in-a-row 2D polyomino (in any direction), the largest solution is 15 squares. Rodolfo Kurchan’s book Mesmerizing Math Puzzles seems to be the first publication of this puzzle.
In[]:=
squares={{0,0},{1,0},{1,3},{1,4},{2,0},{2,1},{2,4},{3,1},{3,3},{3,4},{4,1},{4,2},{4,3},{5,2},{6,2}};
The visual representation is given by:
In[]:=
Graphics[{EdgeForm[Red],Rectangle[#]}&/@squares]
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Quoting from the math-fun mailing list (it also might wind up in OEIS):
Forbidding that any three cubes lie on a line leads to the following counts (for n >= 1):
​
{1, 1, 1, 4, 5, 18, 33, 67, 82, 129, 190, 278, 365, 450, 483, 479, 455, 422, 356, 284, 208, 138, 72, 28, 8, 1, 0, 0, 0}
​
So, there are no such things with more than 26 cubes. (Counts are modulo the octahedral group.) For the two-dimensional version one can verify using pencil and paper that one does not get far at all. Would we expect that there is some dimension D such that the sequence does not terminate?
​
BTW, using rhombic dodecahedra (same as edge-connected cubes) or truncated octahedra (same as corner-connected cubes) the sequences do not seem to terminate already in three dimensions.
Jeorg did more explorations. In one, he looked at No-4-in-a-line, with a further restriction that no cube was at the midpoint of two others. That led to reasonable growth:
{1,1,1,2,3,7,11,20,32,59,96,155,234,334,464,643,890,1204,1486,1903,2330,2899,3437,4102,4654,5494,6121,6982,7645,8725,9562,10597,11137,11895,12200,12674,12602,12733,12339,12191,11756,11476,10774,10373,9860,9670,9073,8603,7830,7300,6725,6077,5292,4524,3672,3016,2382,1843,1375,1077,781,555,390,275,182,111,71,42,25,14,8,7,5,3,1,0,0,0}
Ending with 75 cubes . “For this glorious object ...”
In[]:=
gloriousobject={{0,0,4},{0,1,3},{0,1,4},{0,2,3},{0,4,4},{0,4,5},{0,5,5},{0,5,6},{1,0,4},{1,0,5},{1,1,5},{1,2,3},{1,3,3},{1,3,4},{1,4,4},{1,4,7},{1,5,6},{1,5,7},{1,9,5},{1,9,6},{1,10,6},{2,1,4},{2,1,5},{2,4,7},{2,4,8},{2,10,5},{2,10,6},{3,0,3},{3,0,4},{3,1,1},{3,1,4},{3,2,0},{3,2,1},{3,3,0},{3,4,8},{3,5,8},{3,6,6},{3,7,6},{3,7,7},{3,8,7},{3,9,5},{3,10,5},{4,0,2},{4,0,3},{4,1,1},{4,1,2},{4,3,0},{4,4,0},{4,4,1},{4,5,7},{4,5,8},{4,6,6},{4,6,7},{4,8,6},{4,8,7},{4,9,5},{4,9,6},{5,3,2},{5,4,1},{5,4,2},{6,3,1},{6,3,2},{7,1,3},{7,3,1},{7,4,1},{7,4,2},{8,1,2},{8,1,3},{8,3,3},{8,4,2},{8,4,3},{9,1,2},{9,2,2},{9,2,3},{9,3,3}};
In[]:=
Graphics3D[Cuboid/@gloriousobject]
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CITE THIS NOTEBOOK

The largest no-3-in-a-row polycube is a unique 26-cube​
by Ed Pegg​
Wolfram Community, STAFF PICKS, December 10, 2023
​https://community.wolfram.com/groups/-/m/t/3080500