[#P2494] Distinct nonzero triangle areas in the 9 by 9 grid

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Problem. Let \(A_9\) be the largest size of a set \(S\subseteq\{0,\ldots,8\}^2\) for which the doubled areas of all triangles with vertices in \(S\) are positive and pairwise distinct. Determine \(A_9\).

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Definitions and notation

1Problem setup

Remark 1. The doubled area of \(p,q,r\) is the absolute value of the integer determinant \((q-p)\times(r-p)\).

Definition 1. Pairwise distinct means that two different three-element subsets of S never determine the same doubled area.

2What counts as a solution

Exhibit an \(A_9\)-point set and certify by exhaustive search or a mathematical upper bound that no \((A_9+1)\)-point set exists.

1Status

Saved packet · July 24, 2026

What counts as a solution

Saved solution (The exact maximum is A_9=6). An explicit six-point set has 20 different positive doubled areas, and exhaustive search excludes every seven-point subset.^[1]

Verification

The six points \[ \{(8,7),(2,7),(3,6),(0,2),(7,1),(1,0)\} \] give the 20 doubled areas \[ 1,6,7,8,9,10,13,17,21,28,29,30,31,34,35,36,37,41,42,43. \] They are positive and pairwise distinct, proving \(A_9\geq6\).

Every doubled triangle area in the grid is an integer between 1 and 64. The alphabet count \(\binom{k}{3}\leq64\) gives the preliminary bound \(k\leq8\). The exhaustive certificate then enumerates every increasing seven-element subset of the 81 grid points. When a candidate point is appended, it forms one new triangle with each pair of selected points. The branch continues exactly when these new doubled areas are positive, mutually different, and absent from the prior area mask.

The search reaches no depth-seven leaf. Since the condition passes to subsets, any valid set of at least seven points would contain a valid seven-point subset. Thus \(A_9\leq6\), and \[ A_9=6. \]

1Packet records

Show recorded summaries

Record	Kind	Assessment
An explicit six-point set has 20 different positive doubled areas, and exhaustive search excludes every seven-point subset.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 1 An explicit six-point set has 20 different positive doubled areas, and exhaustive search excludes every seven-point subset.^[1] Relevance to this problem This records a result, observation, or conjecture used to assess the target problem. Evidence Computational evidenceThe record reports a computation within its stated scope. Reproduction requires a separate replay. Record state established Scope all subsets of the 81 lattice points {0,...,8}^2 whose triangle doubled areas are positive and pairwise distinct Sources Reference 1 Details The six points \[ \{(8,7),(2,7),(3,6),(0,2),(7,1),(1,0)\} \] give the 20 doubled areas \[ 1,6,7,8,9,10,13,17,21,28,29,30,31,34,35,36,37,41,42,43. \] They are positive and pairwise distinct, proving \(A_9\geq6\). Every doubled triangle area in the grid is an integer between 1 and 64. The alphabet count \(\binom{k}{3}\leq64\) gives the preliminary bound \(k\leq8\). The exhaustive certificate then enumerates every increasing seven-element subset of the 81 grid points. When a candidate point is appended, it forms one new triangle with each pair of selected points. The branch continues exactly when these new doubled areas are positive, mutually different, and absent from the prior area mask. The search reaches no depth-seven leaf. Since the condition passes to subsets, any valid set of at least seven points would contain a valid seven-point subset. Thus \(A_9\leq6\), and \[ A_9=6. \] Open full record →
OEIS A322740 lists 65 as the least rectangle area for seven points, one above the 8 by 8 coordinate span here.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 OEIS A322740 lists 65 as the least rectangle area for seven points, one above the 8 by 8 coordinate span here.^[1][2][3] Relevance to this problem This records a line of attack on the target problem, including what it establishes and where it stops. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope prior results for seven lattice points with all 35 triangle areas positive and different in a rectangle of coordinate area at most 65 Sources Reference 1, Reference 2, Reference 3 What happened OEIS A322740 asks for the least product \(L_1L_2\) of coordinate spans of a lattice rectangle containing \(n\) points whose \(\binom n3\) triangle areas are positive and pairwise different. It reports \[ a(6)=30,\qquad a(7)=65. \] The six-point configuration listed there fits inside the present grid. Its reported seven-point threshold also predicts the upper bound here: \(\{0,\ldots,8\}^2\) has coordinate-span area \(8\cdot8=64\). The 2018 IBM Research challenge uses the same geometric condition for 11 points and seeks a small containing rectangle. Pinchasi's 2008 theorem concerns the minimum number of different triangle areas forced by an arbitrary planar point set, a neighboring extremal question. The focused search found no peer-reviewed source stating the exact square-grid maximum \(A_9\). OEIS gives a directly relevant prior computation without a replayable search trace on the entry. The independent artifact proves the needed seven-point nonexistence for this grid. Related: Computation 1 Open full record →
The direct search visits 4,699,843 admissible partial sets and finds no valid seven-point leaf.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 The direct search visits 4,699,843 admissible partial sets and finds no valid seven-point leaf. Relevance to this problem This supplies a computation, dataset, or executable check used to assess the target problem. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every seven-element subset of the 9 by 9 point grid {0,...,8}^2 Entry point `join source_lines with newline, compile with the stated command, and run` Runtime C++20 standard library with the GCC or Clang unsigned __int128 extension Details Points are ordered first by \(x\), then by \(y\), and every subset has one increasing representation in this order. A 128-bit mask stores the positive doubled areas already used. At depth \(d\), adding a point creates \(\binom d2\) new triangles. A zero determinant causes a collinearity rejection. A repeated bit within the new values or against the stored mask causes an area-collision rejection. The loop's shortage cutoff merely leaves enough later points to reach seven. The exhaustive run visits 4,699,843 admissible nodes. It rejects 2,963,134 extensions for a zero area and 44,877,104 for a repeated positive area, out of 52,540,080 candidate extensions. Visit counts by selected-set size are \[ [1,75,2850,70770,983307,3255572,387268,0]. \] The terminal zero proves nonexistence at size seven. A running FNV-1a trace of the depth, next index, and both halves of the area mask ends at `ade14ba0a60dc0f4`. Before the search, the same executable checks all 20 triangles of the displayed six-point witness. Its stable six-line output has SHA-256 digest `ab06239e5dd989c76cf8fcf77971fe67eb844118219c1d6ab47102319d75a916`. Related: Computation 1 Open full record →
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Notes and companion material

The constraint couples every triple, so failed branches can be indexed by the first repeated determinant value. That makes negative search results easy to compare across solvers.

Original intake status. Status not established. No literature search was performed. Related distinct-area and no-three-in-line questions have a substantial literature.

When a point is added, every newly formed triangle area must be positive, absent from the old area set, and distinct among the new triangles.
The finite area alphabet gives an additional upper bound: a k-point solution needs at least binomial(k,3) distinct integers among the attainable doubled areas.

Recorded example 1. Six valid points in the 9 by 9 grid are \((8,7),(2,7),(3,6),(0,2),(7,1),(1,0)\).

Computational notes

Exact backtracking with integer determinants gave \(A_n=3,4,4,5,5,6,6\) for \(2\le n\le8\). The n=8 search visited 1003458 recursive nodes. A seeded greedy search of 100000 random point orders found the stated 6-point n=9 construction and found no 7-point construction; the greedy failure is not an upper bound.

How the 3 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemDistinct nonzero triangle areas in the 9 by 9 grid

All 2 recorded relations between these records and the problem

Complete seven-point branch certificate is evidence for The exact maximum is A_9=6
OEIS reports a matching rectangle threshold supports The exact maximum is A_9=6

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Distinct-distance general-position sets in the 9 by 9 griddiscrete geometry
Largest isosceles-triangle-free subset of the 7 by 7 griddiscrete geometry

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Plain text

“Distinct nonzero triangle areas in the 9 by 9 grid.” TheoremDB. P2494. Problem statement; statement text SHA-256 0180e5b83aad9e8f738248cfa04d6cba4b90b17fe996296a20050abd3f1e4281. https://theoremdb.org/statement/?ref=P2494

BibTeX

@misc{theoremdb-problem-0180e5b83aad9e8f738248cfa04d6cba4b90b17fe996296a20050abd3f1e4281,
  title = {{Distinct nonzero triangle areas in the 9 by 9 grid}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 0180e5b83aad9e8f738248cfa04d6cba4b90b17fe996296a20050abd3f1e4281},
  url = {https://theoremdb.org/statement/?ref=P2494}
}

Plain text: Built Markdown snapshot

This problem includes 3 records joined by 2 typed links, sourced from oeis.org^[1], current as of July 24, 2026.

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1References

Packet source. Reference record for “distinct-triangle-areas-grid-9,” oeis.org, checked 2026-08-01. Packet-wide source named in the dataset provenance. ↗reference database · reference source · checked 2026-08-01Source use: citation only.Supplies the source context used to assemble the packet for “distinct triangle areas grid 9.”Also cited at Independent exhaustive certificate dta9-artifact-complete-seven-search, executed on 2026-07-24.Source named by the research packet.
Reference record for “distinct triangle areas grid 9,” research.ibm.com, checked 2026-08-01. Exact source page or record linked above, checked on 2026-08-01. ↗website · reference source · checked 2026-07-24Source use: citation only.Provides a source or data record used in the packet for “distinct triangle areas grid 9.”
Reference record for “distinct triangle areas grid 9,” doi.org, checked 2026-08-01. Exact source page or record linked above, checked on 2026-08-01. ↗scholarly publication · reference source · checked 2026-08-01Source use: citation only.Provides a source or data record used in the packet for “distinct triangle areas grid 9.”

Original finite lattice problem with exact determinant tests.

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[#P2494] Distinct nonzero triangle areas in the 9 by 9 grid

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