+104 Three squares of a 5 \times 5 grid are colored. Two colorings are conisdered equivalent if one coloring can be rotated to form the other coloring, such as the two colorings below, and the grid can be reflected as well. We also want the three squares to be in a row. Find the number of distinct colorings.
+100 1. Center bar:
Middle row, cols 2-4 = positions (3,2)-(3,3)-(3,4)
This can be rotated to vertical or reflected, but always stays at the center.
2. Off-center:
Top row, cols 1-3: (1,1)-(1,2)-(1,3)
Top row, cols 2-4: (1,2)-(1,3)-(1,4)
Top row, cols 3-5: (1,3)-(1,4)-(1,5)
But these are all equally distant from their accompanying edge; by symmetry, place (1,1)-(1,2)-(1,3) is same as (1,3)-(1,4)-(1,5) under reflection.
So for each "run of 3" not at the center, there are only two types by distance from the edge: edge and one-step-in.
So per orientation, horizontal and vertical, that gives:
Center
Edge
One in from edge
But rotations mix horizontals and verticals. So these three types exhaust the possibility for straight bars:
Case 1: Centered three-in-a-row (middle row & col)
Case 2: Edge three-in-a-row (either on top/left or bottom/right edge)
Case 3: Offset three-in-a-row (one away from edge, but not center)
Centered bar: ((3,2)-(3,3)-(3,4)) (also ((2,3)-(3,3)-(4,3)), rotated)
Edge bar: ((1,1)-(1,2)-(1,3)) (can be rotated/reflected to any edge)
Offset bar: ((1,2)-(1,3)-(1,4))
So 3 distinct orbits for straight bars.
B. Diagonal Bars:
Now, consider the diagonal three-in-a-row:
((1,1)-(2,2)-(3,3))
((2,2)-(3,3)-(4,4))
((3,3)-(4,4)-(5,5))
Similar situation on other main diagonal.
Only possibilities for "bar on a diagonal":
Main diagonal, centered
Central run: ((2,2)-(3,3)-(4,4))
Main diagonal, edge
Edge run: ((1,1)-(2,2)-(3,3))
((3,3)-(4,4)-(5,5))
The two runs at the ends of the diagonal are symmetric by 180° rotation.
Similarly, diagonals above/below the main diagonal, but only have length 3.
Also, shorter diagonals: length 3, only one way to pick three in a row.
But are these different up to symmetry? For a 5x5, all "sloping" bars at the edge are the same under symmetry.
Diagonal (main): Centered and edge.
And diagonal bars not on the true main diagonal or true anti-diagonal can be rotated to each other.
So, just as before:
Diagonal centered
Diagonal edge
So, in total, for "three-in-a-row" on a diagonal, we only get two types: centered and edge.
In conclusion, there are 5 distinct types of three-in-a-row colorings (orbits under symmetry):
Final Answer;
\(\boxed{5}\)
There are 5 distinct colorings of three squares in a row (of any direction) in a 5 by 5 grid, up to rotation and reflection.
