ProofAtlas

Discrete geometry and square packing

Packing 11 unit squares in a square

Collaboration beta

What is the smallest square that can contain eleven unit squares when each may rotate independently? The handoff gives an exact construction and local bounds, but still needs a universal argument that no smaller packing exists.

L11=L*=2(3τ+2)1+2τ-τ2,p(τ)=0on the stated isolated branch.
Known results and sources
Illustration: Eleven unit squares fit in an elementary four-by-four container, with unused space making clear that this is only a feasible example.
Illustration: This elementary L = 4 packing demonstrates feasibility and object count. It is not the conjectured optimal arrangement.

Research problem

Exact mathematical statement

Let L₁₁ be the infimum of L>0 for which eleven closed unit squares with pairwise disjoint interiors fit in a closed square of side L. Each unit square may translate and rotate independently; touching at edges or vertices is allowed.

Define

p(t)=5t8-10t7-2t6+14t5+12t4-6t3+2t2+2t-1.p(t)=5t^8-10t^7-2t^6+14t^5+12t^4-6t^3+2t^2+2t-1.

Let τ be the unique root in the exact rational interval

0.36576930760467729338854501814331131<τ<0.36576930760467729338854501814331132,0.36576930760467729338854501814331131 <\tau< 0.36576930760467729338854501814331132,

and set

L*=2(3τ+2)1+2τ-τ2.L_*=\frac{2(3\tau+2)}{1+2\tau-\tau^2}.

The target is

L11=L*.L_{11}=L_*.

Problem infographic

Problem at a glance

Illustration: An elementary eleven-square packing in a square of side four is separated from the exact algebraic candidate L* and the unresolved global optimality question.
Illustration: Arbitrary independent rotations and touching boundaries are allowed. The displayed L = 4 example is not the candidate; the open target is L₁₁ = L*.

Current mathematical picture

Where work on Packing 11 unit squares in a square stands

Open problem

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

Useful failureReplacing the actual I02 boundary square by its maximal common core

T040/T062 reports ten separated unit squares at side 3.877 avoiding the live-grid maximal B core. Two fixed neighbors force ξ>49/50 and η>99/100, contradicting the common-center condition ξ+η≤1. Position conditioning or explicit shared-center constraints can strengthen the model. The fixed-neighbor contradiction is not a uniform exclusion of moving I02 configurations.

Route status · Narrowed route
Main reductionCurrent reduction

After strictifying a hypothetical improvement and centering its true smaller container, the source reports an exhaustive necessary incidence cover by I02, I19, and I20, with twenty globally labelled descriptors. Each family retains continuous centers, independent angles, complete probe roles, and one shared center for each square.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeControl the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused.Task status · Ready to work on

Work mapped so far

Packing 11 unit squares in a square in numbers

1.2kretained lines of mathematical investigation1,195 in the current working snapshot
Argument development
735 · 62%
Explored or eliminated routes
48 · 4%
Computational analysis
100 · 8%
Open obligations
172 · 14%
Definitions and setup
140 · 12%
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 Packing 11 unit squares in a squareA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Is the exact eleven-square construction globally optimal? — Depends on missing premiseIs the exact eleven-squareconstruction globallyoptimal?Current reduction — Depends on missing premiseCurrent reductionThree continuous necessary families — Depends on missing premiseThree continuous necessaryfamiliesA completed local neighborhood — Depends on missing premiseA completed localneighborhoodA sharp bound under thirteen projection orders — Depends on missing premiseA sharp bound under thirteenprojection ordersClosing target — Depends on missing premiseClosing targetExact maximal core for one avoider — Depends on missing premiseExact maximal core for oneavoiderExact upper construction — Depends on missing premiseExact upper constructionShared-owner displacement coupling — Depends on missing premiseShared-owner displacementcouplingReplacing the actual I02 boundary square by its maximal common core — stoppedReplacing the actual I02boundary square by itsmaximal…Using the old NE octagon as a complete I19 obstacle — stoppedUsing the old NE octagon asa complete I19 obstacleControl the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused. — OpenControl the full I02 familywith boundary avoider B,interior…Control I19 with actual B and NE avoiders and nine singleton owners; the fixed old-octagon and fixed-neighbor examples do not exclude the full moving family. — OpenControl I19 with actual Band NE avoiders and ninesingleton…Obtain a sharp bound for the entire I20 family with independently oriented SW and NW avoiders and nine singleton owners. — OpenObtain a sharp bound for theentire I20 family withindependently…
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 routeReplacing the actual I02 boundary square by its maximal common core

T040/T062 reports ten separated unit squares at side 3.877 avoiding the live-grid maximal B core. Two fixed neighbors force ξ>49/50 and η>99/100, contradicting the common-center condition ξ+η≤1. Position conditioning or explicit shared-center constraints can strengthen the model. The fixed-neighbor contradiction is not a uniform exclusion of moving I02 configurations.

Route status · Narrowed route
Narrowed routeUsing the old NE octagon as a complete I19 obstacle

T060 reports ten actual units at side 3.877 with the old-octagon clearance. A new tip in the exact maximal NE core lies inside the P11 owner, preventing completion by the actual NE square. The maximal interior core removes this specific witness, but its strengthened relaxation and the moving I19 family still require a global analysis.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Control the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused.Suggested move: Suggested next step: Keep one actual B center in its triangle coordinates, combine its moving-neighbor constraints with the NW/NE fan inequality, and cover every parameter branch before claiming a uniform incompatibility.
Ready to work on
02
Control I19 with actual B and NE avoiders and nine singleton owners; the fixed old-octagon and fixed-neighbor examples do not exclude the full moving family.Suggested move: Suggested next step: Use the exact maximal NE core or its new sixteen-gon while retaining actual B and all owners, then prove a moving-neighbor version of the shared-center obstruction and cover its complement.
Ready to work on
03
Obtain a sharp bound for the entire I20 family with independently oriented SW and NW avoiders and nine singleton owners.Suggested move: Suggested next step: Outside the certified candidate neighborhoods, force all thirteen full-projection inequalities or prove equally sharp alternatives; inside a correctly mapped ball, apply the centered radius 1/60000 and growth bound ρ_c/2400 while retaining ℓ<L*.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026
Current statusOpen problem

The unrestricted optimum for eleven unit squares is not established by the cited results. A published lower bound is 2 + 4/√5; the current catalog lists a construction with side approximately 3.87708359002281 without marking the eleven-square row proved optimal.

[1][3]
External progress

What the literature has established

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

  1. Peer reviewedStromquist establishes the lower bound s11 ≥ 2 + 4/√5 for arbitrary orientations.[1]
  2. Historical sourceThe survey and catalog attribute the retained eleven-square construction to Walter Trump in 1979.[2][3]
3 cited sources1 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusPacking eleven unit squares in a square
Related problemEleven squares restricted to 0° and 45° orientations

Stromquist determines the optimum 2 + 2√(8/9) when all squares use those two orientations. That additional restriction changes the feasible set.

[1]

Formal and computational footholds

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

  • dataset · source linked; not reproduced by ProofAtlasCurrent squares-in-squares packing catalog

    The catalog lists the eleven-square construction and its numerical side length. Its diagram and coordinates were not imported, rendered, or independently verified.

    [3]

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
  • lemma5 of 95
  • special case1 of 91
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 statementIs the exact eleven-square construction globally optimal?
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExact upper constructionintermediate
  • retained route statementA sharp bound under thirteen projection ordersintermediate
  • retained route statementA completed local neighborhoodintermediate
  • retained route statementThree continuous necessary familiesintermediate
  • retained route statementShared-owner displacement couplingintermediate
  • retained route statementExact maximal core for one avoiderintermediate
  • 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 failureReplacing the actual I02 boundary square by its maximal common corereported failure
  • Useful failureUsing the old NE octagon as a complete I19 obstaclereported failure
  • Research targetControl the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused.open
  • Research targetControl I19 with actual B and NE avoiders and nine singleton owners; the fixed old-octagon and fixed-neighbor examples do not exclude the full moving family.open
  • Research targetObtain a sharp bound for the entire I20 family with independently oriented SW and NW avoiders and nine singleton owners.open
  • ComputationThe source reports exact continuum covers for local owner obstructions, including a third T055 implementation retaining all signed obstacle and owner-pair axes.The new T055 cover is reported to have 6,021 nodes and 3,011 leaves, followed by a separate complete leaf source report. It proves the same scoped local obstruction and adds no new parent exclusion. No cover or analytic proof was independently checked for this draft. · reported unreproduced
  • Narrowed routeReplacing the actual I02 boundary square by its maximal common coreT040/T062 reports ten separated unit squares at side 3.877 avoiding the live-grid maximal B core. Two fixed neighbors force ξ>49/50 and η>99/100, contradicting the common-center condition ξ+η≤1. Position conditioning or explicit shared-center constraints can strengthen the model. The fixed-neighbor contradiction is not a uniform exclusion of moving I02 configurations.
  • Narrowed routeUsing the old NE octagon as a complete I19 obstacleT060 reports ten actual units at side 3.877 with the old-octagon clearance. A new tip in the exact maximal NE core lies inside the P11 owner, preventing completion by the actual NE square. The maximal interior core removes this specific witness, but its strengthened relaxation and the moving I19 family still require a global analysis.
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 bridgeControl the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused.

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

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Prepared starting pointControl the full I02 family with boundary avoider B, interior avoiders NW and NE, eight singleton owners, and only P01 unused.

Packing 11 unit squares in a square · ready to start

Mathematical updatesFollow this problem

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Research contextPrepared context for any AI agent

What is the smallest square that can contain eleven unit squares when each may rotate independently? The current work gives an exact construction and local bounds, but still needs a universal argument that no smaller packing exists.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 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
    Packing 10 or 11 Unit Squares in a Squarepeer reviewed result · Walter Stromquist · The Electronic Journal of Combinatorics · 2003-03-18 · DOI 10.37236/1701 · accessed Sep 7, 2026
  2. 2
    Packing Unit Squares in Squares: A Survey and New Resultssurvey or monograph · Erich Friedman · The Electronic Journal of Combinatorics, Dynamic Survey 7 · 2009 · accessed Sep 7, 2026
  3. 3
    Squares in Squaressoftware or dataset · David Ellsworth · accessed Sep 7, 2026

Important qualifications

  • A construction's discovery date or constructor does not establish the optimization problem's proposer or proposal date; those fields remain unknown.
  • The current packing catalog reports a smallest-known configuration. Its rounded side length is not asserted to be the exact optimum or an independently checked geometric certificate.
  • No linked geometry, numerical computation, program, or formal proof was reproduced. Empty formalization fields are not evidence of absence.
  • This private draft grants no mathematical acceptance, rights, authorship, contributor credit, or publication authority.

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