The puzzle

In his October 1976 Scientific American column, Martin Gardner flipped the classic no-three-in-line problem on its head: instead of asking for the most counters you can place on an n×n board with no three in a line, ask for the fewest counters such that every vacant cell already lies on a line through two counters. Place one more counter anywhere and you create three-in-a-line.

Two flavors exist:

• I_n — the counters themselves must be in general position (no three collinear). These are the saturated no-three-in-line sets.
• D_n — collinear counters are allowed; you only need every cell of the grid to be on a line through two of them.

What was known

Aichholzer, Eppstein and Hainzl (Computational Geometry, 2023 · arXiv:2203.13170) computed I_n exactly for n = 2…12: 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 10. For the relaxed version they noted, in prose, that n = 7 is the single board size up to 12 where allowing collinear counters helps — seven counters instead of eight — and pictured three such boards.

Then in June 2026, a preprint on Monte-Carlo tree search for grid problems (arXiv:2606.26399) reported, in its results table, an improvement for the minimum dominating set at n = 7.

Both can't be right. Nobody had published a proof either way.

Tonight's result

D₇ = 7. There is no 6-point dominating set on the 7×7 grid.

The certificate chain, end to end:

1. The "≤ 6 points" instance is UNSAT by two independent solvers (CaDiCaL 1.9.5 in 0.5 s, Kissat 4.0.4), with a DRAT unsatisfiability proof emitted by Kissat and verified by drat-trim (s VERIFIED).
2. A 7-point witness exists and passes an independent brute-force checker (all 49 cells dominated).
3. So whatever the June preprint's n = 7 table entry is, 7 is a hard floor.

And a census nobody had: there are exactly 56 optimal 7-point boards, falling into 9 classes up to rotation and reflection (the 2023 paper displayed three of them). All nine are in the interactive board above — my favorite is the tight 2×3 block with a chin, which somehow sees the entire board.

The certified table so far:

n	2	3	4	5	6	7	8
I_n (no 3 in line)	4	4	4	6	6	8	8
D_n (lines allowed)	4	4	4	6	6	7	8

n = 7 really is the anomaly, and now it's a theorem rather than a remark.

Method, briefly

Boolean encoding: a variable per cell, a variable per grid line asserting "this line holds ≥ 2 chosen points", coverage clauses per cell, cardinality via sequential counters, and lex-leader symmetry breaking over the dihedral group of the square (worth ~6× on the UNSAT side). The model reproduces every published exact value and every published solution count for n ≤ 6 (1, 5, 2, 152, 8 boards) before I trusted it with anything new. Three engines cross-check each other: OR-Tools CP-SAT for optimization and enumeration, CaDiCaL and Kissat for the proofs.

Live hunts

Two record attempts are running right now on the same encoding, SAT-direction:

• I₁₇ ≤ 14? The current record is 15 — the only odd entry in the known table, flagged by the 2023 authors as a possible counterexample to their even-cardinality conjecture. A 14-point set would both improve the bound and restore evenness.
• I₂₃ ≤ 18? The published bound is 20, sitting oddly high between its neighbors.

And the big one: I₁₃ is the first unknown exact value (the 2026 preprint confirms the exact table still stops at n = 12). The proof obligation — no saturated set of size ≤ 11 exists on 13×13 — is packaged as a CNF with calibrated runtime estimates, headed for an overnight run on my Mac mini. If it lands: I₁₃ = 12, a new entry in a fifty-year-old table.

This post updates as results land.