[#P2482] Largest jointly sum-and-difference-distinct subset of 1,...,50

Problem. Determine \(M(50)\), the largest size of a set \(S\subseteq\{1,\ldots,50\}\) for which all numbers \(a+b\) and \(\lvert a-b\rvert\), over unordered pairs of distinct elements \(a,b\in S\), are pairwise distinct as one combined collection.

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1Context

Each unordered pair contributes one sum and one positive difference, and every value contributed by every pair must be distinct from every other value.

2Remarks

Remark 1. A sum may not equal a difference, whether the two values come from the same unordered pair or from different pairs.

Remark 2. Each unordered pair of distinct elements contributes two positive integers to the combined collection.

3What counts as a solution

Exhibit a set attaining \(M(50)\) and give a complete proof or independently checkable certificate that no larger admissible set exists.

1Status

Saved packet · July 24, 2026

What counts as a solution

Saved solution (The exact maximum is M(50)=8). The set {8,18,22,23,30,41,47,50} attains eight, and a complete search finds no valid nine-element subset.^[1]

Verification

For a \(k\)-element set, the combined collection has \(2\binom{k}{2}=k(k-1)\) entries. Every positive difference and every sum lies in \(\{1,\ldots,99\}\), so \(k(k-1)\leq99\) and \(k\leq10\).

The set \[ S=\{8,18,22,23,30,41,47,50\} \] produces 56 distinct combined values, with minimum 1 and maximum 97. This proves \(M(50)\geq8\).

The executable certificate enumerates every increasing nine-tuple in \(\{1,\ldots,50\}\). It carries a bit mask for all sums and differences created by the current partial set. An extension is retained exactly when its new values avoid the mask and one another. The search reaches no depth-nine leaf. The property passes to subsets, so every valid set of size at least nine would contain a valid nine-element subset. Hence \(M(50)\leq8\), and the two bounds give \[ M(50)=8. \]

1Packet records

Show recorded summaries

Record	Kind	Assessment
The set {8,18,22,23,30,41,47,50} attains eight, and a complete search finds no valid nine-element subset.	Result	Evidence: Computational evidence The record reports a computation within its stated scope. Reproduction requires a separate replay.
claim · Computation 1 The set {8,18,22,23,30,41,47,50} attains eight, and a complete search finds no valid nine-element 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 {1,...,50} whose off-diagonal sums and positive differences form one collision-free collection Sources Reference 1 Details For a \(k\)-element set, the combined collection has \(2\binom{k}{2}=k(k-1)\) entries. Every positive difference and every sum lies in \(\{1,\ldots,99\}\), so \(k(k-1)\leq99\) and \(k\leq10\). The set \[ S=\{8,18,22,23,30,41,47,50\} \] produces 56 distinct combined values, with minimum 1 and maximum 97. This proves \(M(50)\geq8\). The executable certificate enumerates every increasing nine-tuple in \(\{1,\ldots,50\}\). It carries a bit mask for all sums and differences created by the current partial set. An extension is retained exactly when its new values avoid the mask and one another. The search reaches no depth-nine leaf. The property passes to subsets, so every valid set of size at least nine would contain a valid nine-element subset. Hence \(M(50)\leq8\), and the two bounds give \[ M(50)=8. \] Open full record →
Distinct positive differences make every candidate a Golomb ruler; disjoint sum and difference value sets add a strict cross condition.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 Distinct positive differences make every candidate a Golomb ruler; disjoint sum and difference value sets add a strict cross condition.^[1][2][3] Relevance to this problem This records a result, observation, or conjecture used to assess the target problem. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope finite increasing integer sets with distinct positive pairwise differences Sources Reference 1, Reference 2, Reference 3 Argument Atkinson, Santoro, and Urrutia call an integer set with distinct nonzero differences a distinct difference set and connect this condition to distinct pair sums. Beck, Bogart, and Pham use the standard Golomb ruler description: a sequence of distinct integer marks whose pairwise differences are distinct. Over the integers this is the usual Sidon condition in difference form. Every candidate set in the present problem is a Golomb ruler because its positive differences must be distinct. It also requires \[ \{b-a:a<b,\ a,b\in S\}\cap\{a+b:a<b,\ a,b\in S\}=\varnothing. \] This cross-disjointness is a strict addition. The Golomb ruler \(\{1,2,4\}\) fails it since \(4-1=1+2=3\). The DDS and Golomb ruler sources supply the parent concept without an exact value for this added finite problem. A focused source search performed for this entry found no published determination of \(M(50)\). Related: Computation 1 Open full record →
A direct search visits 1,330,023 admissible partial sets and reaches no valid nine-element leaf.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 A direct search visits 1,330,023 admissible partial sets and reaches no valid nine-element 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 increasing nine-element subset of {1,...,50} 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 The program retains the original coordinates and interval. At a partial increasing set \(y_1<\cdots<y_d\), a 128-bit mask records every used value. Adding \(x>y_d\) proposes the two values \(x-y_i\) and \(x+y_i\) for each earlier mark. The branch survives exactly when all \(2d\) proposals are new. The loop bound \(x\leq51-(9-d)\) leaves enough larger integers to complete nine marks; this shortage cutoff is the sole branch pruning rule. The run visits 1,330,023 admissible nodes and rejects 5,798,178 candidate extensions. Node counts by depth are \[ [1,42,903,12342,104597,452239,639228,120455,216,0]. \] Rejection counts at depths zero through eight are \[ [0,0,902,31187,432335,2233008,2756216,344165,365]. \] The zero at depth nine is the exhaustive nonexistence result. A running FNV-1a trace hashes the depth, next candidate, and both 64-bit halves of the mask at every visited node. Its final value is `ec589fbc6a870cec`. The same executable checks the eight-element witness before starting the search. Its stable five-line output has SHA-256 digest `cbb03c49249188d3abddb2ac2b88aa707a781c5db6e78f247971c60e86bfa60b`. Related: Computation 1 Open full record →
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Notes and companion material

Original intake status. Status remains unverified. The condition is related to Sidon sets and Golomb rulers, and the combined version may have prior study.

The incumbent has size 8. Counting places all k(k-1) values inside {1,...,99}, giving k <= 10.
A branch search should fix translation only with care because the ambient interval and sum values change under translation.

Recorded example 1. S={8,18,22,23,30,41,47,50} is valid and yields 56 distinct values between 1 and 97.

Computational notes

Twenty thousand randomized greedy starts used exact collision sets. They found the displayed size-8 incumbent. The counting upper bound k(k-1) <= 99 was checked independently.

How the 3 records connect

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

ProblemLargest jointly sum-and-difference-distinct subset of {1,...,50}

All 2 recorded relations between these records and the problem

2See also

Maximum determinant of a zero-diagonal binary symmetric Toeplitz matrix of order 20extremal combinatorics
Distinct-distance general-position sets in the 9 by 9 gridextremal combinatorics
Largest isosceles-triangle-free subset of the 7 by 7 gridextremal combinatorics

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Plain text

“Largest jointly sum-and-difference-distinct subset of {1,...,50}.” TheoremDB. P2482. Problem statement; statement text SHA-256 ff87ad175fb901c8a32497115482a69811c96851e5277290bef6aff903c7db89. https://theoremdb.org/statement/?ref=P2482

BibTeX

@misc{theoremdb-problem-ff87ad175fb901c8a32497115482a69811c96851e5277290bef6aff903c7db89,
  title = {{Largest jointly sum-and-difference-distinct subset of \{1,...,50\}}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 ff87ad175fb901c8a32497115482a69811c96851e5277290bef6aff903c7db89},
  url = {https://theoremdb.org/statement/?ref=P2482}
}

Plain text: Built Markdown snapshot

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

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1References

Packet source. Reference record for “sum-difference-distinct-subset-50,” doi.org, checked 2026-08-01. Packet-wide source named in the dataset provenance. ↗journal article · reference source · checked 2026-08-01Source use: citation only.Supplies the source context used to assemble the packet for “sum difference distinct subset 50.”Also cited at Independent exhaustive certificate sdd50-artifact-exhaustive-nine, executed on 2026-07-24.Source named by the research packet.
Reference record for “sum difference distinct subset 50,” people.scs.carleton.ca, 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 “sum difference distinct subset 50.”
Reference record for “sum difference distinct subset 50,” arxiv.org, checked 2026-08-01. Exact source page or record linked above, checked on 2026-08-01. ↗preprint · reference source · arXiv:1110.6154, checked 2026-08-01 · checked 2026-07-24Source use: citation only.Provides a source or data record used in the packet for “sum difference distinct subset 50.”

Original finite additive-set target generated by an agent.

[#P2482] Largest jointly sum-and-difference-distinct subset of {1,...,50}

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