ProofAtlas

Decimal reverse-and-add dynamics

The 196 problem and Lychrel numbers

Collaboration beta

Does repeatedly reversing the decimal digits of 196 and adding ever produce a palindrome? The source reports exact quotient dynamics, a partial seed invariant and exclusions of fourteen infinite families, while leaving all-time palindrome avoidance unresolved.

N0=196,Nt+1=Nt+rev(Nt),∃t≥0:Nt=rev(Nt)?
Known results and sources
Illustration: Vertical decimal tiles show 196+691=887 next to an unresolved palindrome question.
Illustration: Reverse the digits of 196 and add to obtain 887; whether the orbit ever reaches a palindrome remains open.

Research problem

Exact mathematical statement

Starting from the positive decimal integer 196, repeatedly add the integer obtained by reversing its ordinary decimal digits. Reversal discards leading zeros. Determine whether any iterate is palindromic:

N0=196,Nt+1=Nt+rev(Nt),∃t≥0:Nt=rev(Nt)?N_0=196,\qquad N_{t+1}=N_t+\operatorname{rev}(N_t),\qquad \exists t\ge0: N_t=\operatorname{rev}(N_t)?

A positive answer requires a connected finite trajectory ending at a palindrome. A negative answer requires exclusion for every nonnegative iteration index. The source supplies neither a terminating seed trajectory nor a complete permanent-safety invariant.

Problem infographic

Problem at a glance

Illustration: The finite prefix 196→887→1675→7436 accompanies the reverse-and-add recurrence and a symbolic ABCCBA palindrome pattern.
Illustration: A finite nonpalindromic prefix does not settle whether some later iterate from 196 is a palindrome.

Current mathematical picture

Where work on The 196 problem and Lychrel numbers stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureUsing the closed prefix domain K as a complete safety invariant

The source gives 1000000001 in A0 inside K and a separate nonseed chain with fifteen safe normalized blocks before reaching A0; neither witness is claimed to start at 196. A strictly stronger seed-tail class may still work if it preserves exact quotient legality, forward closure and complete danger avoidance.

Route status · Narrowed route
Main reduction196-T24: sufficient tail certificate

The source reports T(B1 union B3) subset A0. Therefore a forward-invariant, exactly connected tail class avoiding A0–A3 would exclude all six danger components. Avoidance without closure does not satisfy this implication.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeFind an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components.Task status · Ready to work on

Work mapped so far

The 196 problem and Lychrel numbers in numbers

1.3kretained lines of mathematical investigation1,250 in the current working snapshot
Argument development
828 · 66%
Explored or eliminated routes
80 · 6%
Computational analysis
93 · 7%
Open obligations
73 · 6%
Definitions and setup
176 · 14%
9selected mapped statements2routes investigated3open questions3contribution-ready tasks
How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedSep 7, 2026
Source recordedSep 7, 2026

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

14 selected steps

Scroll horizontally to explore the route

Working route overview for The 196 problem and Lychrel numbersA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Does reversing and adding, starting from 196, ever reach a palindrome? — Depends on missing premiseDoes reversing and adding,starting from 196, everreach…196-T24: sufficient tail certificate — Depends on missing premise196-T24: sufficient tailcertificateCurrent reduction — Depends on missing premiseCurrent reduction103-state prefix invariant K — Depends on missing premise103-state prefix invariant K196-O16: five-prefix phase obstruction — Depends on missing premise196-O16: five-prefix phaseobstruction196-O17: corrected threshold grammar — Depends on missing premise196-O17: corrected thresholdgrammar196-T23: first B1 exclusion — Depends on missing premise196-T23: first B1 exclusion196-T33: fourteen infinite families — Depends on missing premise196-T33: fourteen infinitefamiliesClosing target — Depends on missing premiseClosing targetUsing the closed prefix domain K as a complete safety invariant — stoppedUsing the closed prefixdomain K as a completesafety…Fixed five-prefix/phase or conjunctive local-feature safety models — stoppedFixed five-prefix/phase orconjunctive local-featuresafety…Find an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components. — OpenFind an exactly connectedseed-tail class closed underT…Control surviving A0–A3 configurations beyond the fourteen infinite monotone families already excluded from the actual seed orbit. — OpenControl surviving A0–A3configurations beyond thefourteen…Develop an invariant language with information outside the precise five-prefix/phase and fixed conjunctive grammar obstructions. — OpenDevelop an invariantlanguage with informationoutside…
Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded
Narrowed routeUsing the closed prefix domain K as a complete safety invariant

The source gives 1000000001 in A0 inside K and a separate nonseed chain with fifteen safe normalized blocks before reaching A0; neither witness is claimed to start at 196. A strictly stronger seed-tail class may still work if it preserves exact quotient legality, forward closure and complete danger avoidance.

Route status · Narrowed route
Narrowed routeFixed five-prefix/phase or conjunctive local-feature safety models

The source's phase-generation argument forces reachable A0 words under O16. O17's exact feature forcing fills its thresholded domain, with the below-threshold quantifier explicitly corrected. Additional ancestry conditions, arbitrary Boolean grammar modes, phase-conditioned interior rules or global ordering information are outside those stated obstructions.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Find an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components.Suggested move: Suggested next step: Specify a candidate tail class with its exact quotient domain and seed connection, then separate the inclusion, closure and four avoidance obligations before proposing any certificate check.
Ready to work on
02
Control surviving A0–A3 configurations beyond the fourteen infinite monotone families already excluded from the actual seed orbit.Suggested move: Suggested next step: Classify one remaining entry family with explicit run parameters, legal ancestors and seed-index constraints, recording which cases are outside the fourteen-family theorem.
Ready to work on
03
Develop an invariant language with information outside the precise five-prefix/phase and fixed conjunctive grammar obstructions.Suggested move: Suggested next step: State the additional ancestry or global-order information a proposed recognizer retains, and check explicitly that its definition lies outside O16 and O17 before pursuing it.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen problem

It remains conjectural that repeatedly reversing the decimal digits of 196 and adding never reaches a palindrome. Finite trajectories without a palindrome do not establish nontermination, and 196 remains a candidate rather than a proved decimal Lychrel number.

[1]
External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

  1. Computational resultWalker reports 2,415,836 iterations from 196, reaching one million digits without a palindrome. His report explicitly says he did not independently rerun the calculation.[2]
  2. Authoritative summaryOEIS identifies Lehmer's 1938 Sphinx item as its earliest known reference to the 196 problem; this collection did not inspect the original item.[1]
2 cited sources2 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focus196 problem
Related problemDecimal reverse-and-add trajectories

OEIS supplies the recurrence and initial trajectory beginning 196, 887, 1675, 7436.

[1]
Stronger or generalized formDoes every decimal starting integer eventually reach a palindrome?

Walker's account poses this universal question. Proving nontermination for 196 would give a counterexample; finding a palindrome from 196 would settle only that trajectory.

[2]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

  • dataset · source linked; not reproduced by ProofAtlasOEIS A006960 initial trajectory

    The sequence entry supplies initial iterates and links a longer table. No linked table or program was executed.

    [1]
  • computation · not independently reproducedWalker's historical million-digit experiment

    The primary report links its program and checkpoint. The resource is a reported finite computation, not evidence of permanent avoidance; neither download was opened here.

    [2]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

  • Formalization targetA proposed formalization must fix ordinary decimal reversal, leading-zero handling, iteration indexing, and palindrome equality.
  • Formalization targetAn infinite negative answer needs an invariant covering the connected trajectory for every iteration; a positive answer needs an exact connected finite trajectory. These are planning requirements.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes
Statements by mathematical role9 selected mapped statements
  • theorem candidate1 of 91
  • reduction2 of 92
  • lemma3 of 93
  • special case1 of 91
  • negative result2 of 92
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included
  • retained route statementDoes reversing and adding, starting from 196, ever reach a palindrome?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statement103-state prefix invariant Kintermediate
  • retained route statement196-T23: first B1 exclusionintermediate
  • retained route statement196-T24: sufficient tail certificateintermediate
  • retained route statement196-T33: fourteen infinite familiesintermediate
  • retained route statement196-O16: five-prefix phase obstructionintermediate
  • retained route statement196-O17: corrected threshold grammarintermediate
  • Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
  • Useful failureUsing the closed prefix domain K as a complete safety invariantreported failure
  • Useful failureFixed five-prefix/phase or conjunctive local-feature safety modelsreported failure
  • Research targetFind an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components.open
  • Research targetControl surviving A0–A3 configurations beyond the fourteen infinite monotone families already excluded from the actual seed orbit.open
  • Research targetDevelop an invariant language with information outside the precise five-prefix/phase and fixed conjunctive grammar obstructions.open
  • ComputationThe source describes exact automata preimages, independent carry-stack comparisons and finite pumping checks for selected families and feature languages.Reported M14 and M9 language comparisons and the 707-state output NFA support their named finite or language claims. Interrupted reconstruction is inconclusive; these executable checks are not proof-assistant verification or an all-time safety certificate. · reported unreproduced
  • Narrowed routeUsing the closed prefix domain K as a complete safety invariantThe source gives 1000000001 in A0 inside K and a separate nonseed chain with fifteen safe normalized blocks before reaching A0; neither witness is claimed to start at 196. A strictly stronger seed-tail class may still work if it preserves exact quotient legality, forward closure and complete danger avoidance.
  • Narrowed routeFixed five-prefix/phase or conjunctive local-feature safety modelsThe source's phase-generation argument forces reachable A0 words under O16. O17's exact feature forcing fills its thresholded domain, with the below-threshold quantifier explicitly corrected. Additional ancestry conditions, arbitrary Boolean grammar modes, phase-conditioned interior rules or global ordering information are outside those stated obstructions.
How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeFind an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

  • Supply a complete argument with every imported premise identified.
  • Survive an independent attempt to falsify the proposed step.

Continue the mathematics

Contribute

ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

Read-only beta · actions unavailable
Prepared starting pointFind an exactly connected seed-tail class closed under T and disjoint from all A0–A3 components.

The 196 problem and Lychrel numbers · ready to start

Mathematical updatesFollow this problem

Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Does repeatedly reversing the decimal digits of 196 and adding ever produce a palindrome? The source reports exact quotient dynamics, a partial seed invariant and exclusions of fourteen infinite families, while leaving all-time palindrome avoidance unresolved.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
Return mathematical workReturn what you or your agent found

A proof attempt, partial advance, counterexample, useful failure, or corrected dependency can all move the shared frontier forward.

Proof attempt or partial resultSupporting notes or data
Hosted agentRun this task with a hosted agent

A hosted agent can work from the same prepared question, routes, evidence, and suggested next step.

Your own AI agentConnect an outside research agent

Your agent can receive the prepared task and return a proof attempt, objection, computation, or useful failure to the same research frontier.

Sources and references2 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    A006960: Reverse and Add! sequence starting with 196software or dataset · OEIS Foundation · OEIS A006960 · accessed Sep 7, 2026
  2. 2
    Three Years Of Computing: Final Report On The Palindrome Questauthoritative webpage · John Walker · Fourmilab · 1990-05-25 · accessed Sep 7, 2026

Important qualifications

  • This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
  • No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
  • No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
  • Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
  • Earliest-reference information is catalog testimony, not a priority finding; no proposer or exact proposed year is inferred.
  • The historical computation is not represented as the latest search record. Some informal source comments use definite Lychrel language; the conjectural posture controls this metadata.
  • No source download, checkpoint, program, or exact formalization was inspected or executed.

Continue exploring

Compare another research frontier

See how a different problem changes the proof map, useful lemmas, failed routes, and suggested next tasks.

Explore all research workspaces

Expanded visual

Open original image