orders fully countedEvery solution is indexed for N = 2–20.
A deceptively simple finite-geometry problem
No three in a line.
Each grid size N defines a separate finite-geometry challenge: choose an N × N lattice and place the theoretical maximum of 2N points—without any three sharing a line in any direction. Horizontal and vertical lines force exactly two points into every row and column; every diagonal and rational slope adds another global constraint. The search space explodes while valid boards become extraordinarily rare.
One representative per square-symmetry orbit.
known 2N witnesses through N = 76Every N from 2–74, plus N = 76, has a recorded construction.
first recorded gapThe maintained table’s first order without a known 2N board.
Solution growth
Symmetries excludedQuick board
N =
This mirrors the solution currently selected in the full browser below.
Open the full browserNumber of no-three-in-line configurations by grid size N.
Pattern lab
Where solutions thin out—and where dots gather.
The rarity curve compares valid boards with the much larger universe of boards that merely put two dots in every row and column. Select an N to inspect its aggregate geometry below.
The hidden breakdown
How rare is a valid solution?
Higher means rarer. Both sides count oriented boards; the main symmetry checkbox intentionally does not alter this comparison.
Valid placements
—
Row/column feasible
—
Solution yield
—
Solution cloud · all orientations
Where dots are most likely to land
Each background tile measures that lattice point across every distinct placement. Warm points are used more than the uniform expectation; cool points are used less.
Building the N = 20 cloud…
—
Click a heat tile to ask: “when this point is occupied, where do the other dots go?”
Permutation A
Permutation B
Solution 1
Structural fingerprints
How the dots connect and separate
Rows and columns form a degree-two bipartite graph, so every board decomposes into cycles. The gap profile measures the separation of each row and column pair.
Tracing cycles…
Most common cycle patterns
dots per cycleRow + column pair gaps
—Gap in lattice steps
Constraint geometry
Which directions survive—and how locked is a board?
Bubble size is how often a primitive slope occurs; color is raw enrichment against all lattice-cell pairs. The pressure board counts the non-axis selected pairs that would forbid each empty point beyond the already-saturated rows and columns.
Interpretation note
Axis suppression is forced by row/column balance. Treat the bubble colors as a raw diagnostic—not a cap-specific effect—and focus future tests on balance-corrected nonaxis residuals.
Primitive-slope fingerprint
—
Raw suppressedRaw enriched
Selected board’s line pressure
——
Across-N fingerprints
Signals that persist as the grid grows
These summaries use every oriented placement. The highlighted point follows the Pattern Lab’s selected N.
Mean pair gap ÷ board width
—
Largest cycle’s share
—
Asymmetric orbit representatives
—
Corner and noncorner edge use
—
The boards
Move a slider—or use its arrow buttons—to inspect each configuration. Chart points open a quick copy above.