arxiv: 2605.04111 · v1 · submitted 2026-05-05 · 💻 cs.CG

Recognition: 3 theorem links

· Lean Theorem

Optimally Covering Large Triangles with Homothetic Unit Triangles

John M. Boyer

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Pith reviewed 2026-05-08 18:46 UTC · model grok-4.3

classification 💻 cs.CG

keywords triangle coveringhomothetic trianglescovering boundslarge triangleunit triangledissectionsimilar copiescomputational geometry

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The pith

For any n and excess d in (0,1), a triangle of side n+d can be covered by n²+k homothetic unit triangles exactly when d stays below an explicit threshold that is now determined for every k from 4 to 2n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper finishes the task of finding precise limits on how much a triangle can exceed integer side length n before it requires more than a given number of small similar copies to cover it. Earlier work showed that n²+1 small triangles are never enough and located the exact excess thresholds for n²+2 and n²+3. This work derives the matching thresholds for the remaining increments up to n²+2n, proves that coverage is impossible once d passes each threshold, and supplies two explicit constructions—one for odd k and one for even k—that reach exactly those thresholds. A selector then picks whichever construction uses the fewest small triangles for any given k. A reader cares because the result supplies the complete covering number for every possible excess side length between n and n+1.

Core claim

The prior impossibility argument is extended to produce upper bounds on d for which coverage with n²+k homothetic unit triangles is impossible when 4 ≤ k ≤ 2n. These bounds are shown to be tight by two new covering constructions, one for odd k and one for even k, each of which covers every T_{n+d} up to the bound, together with a consolidated method that chooses the construction requiring the smallest total number of unit triangles.

What carries the argument

Two new covering constructions—one specialized to odd values of k and one to even values—that achieve the derived upper bounds on excess side length d, together with the selector that picks the lower-count construction for any given k.

If this is right

The exact minimal number of homothetic unit triangles needed to cover any T_{n+d} is now known for every d in (0,1) and every natural number n.
For each fixed k the largest allowable d permitting coverage with exactly n²+k small triangles is given by an explicit expression derived from the extended impossibility argument.
When k is odd the odd-case construction is optimal; when k is even the even-case construction is optimal, so the selector always returns the overall minimum count.
The open question of locating tight upper bounds for the n²+k cases with 4 ≤ k ≤ 2n is resolved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The sequence of thresholds as k increases may admit a single closed-form expression for the covering number as a function of d.
The geometric constructions could be adapted to give analogous exact thresholds for covering problems involving other polygons or in three dimensions.
For any fixed small n the constructions can be drawn or computed directly to confirm they meet the bound without gaps.
The parity-based choice between constructions suggests that similar parity distinctions may appear in related covering or packing problems.

Load-bearing premise

The new covering constructions for odd and even k succeed in covering every T_{n+d} up to the stated upper bound on d without gaps or overlaps, and the extended impossibility argument holds uniformly for all 4 ≤ k ≤ 2n.

What would settle it

A concrete counterexample consisting of specific integers n and k together with a value of d at or below the claimed bound for which either construction leaves an uncovered region or a covering with n²+k triangles exists for a d strictly larger than the bound.

Figures

Figures reproduced from arXiv: 2605.04111 by John M. Boyer.

**Figure 1.** Figure 1: (a) The well-known covering of a large triangle view at source ↗

**Figure 2.** Figure 2: The behavior of the first method in [4, 5] is shown using view at source ↗

**Figure 3.** Figure 3: Depiction of the main idea of the new method for covering a large view at source ↗

**Figure 4.** Figure 4: This figure shows the structure of the n th (bottom) row when 3 extra unit triangles are appended, instead of just the 2 that are added in the first Conway-Soifer method. For a small enough value of d, a Tn+d covering is completed by placing a Tn−1 atop this structure. For example, in this depiction, n = 3, so a T2 would be positioned with its base spanning from p1 to p2. For the n 2 + 3 case, there are no… view at source ↗

**Figure 5.** Figure 5: Depiction of the main idea for covering a large triangle view at source ↗

read the original abstract

We answer an open problem in the \emph{American Mathematical Monthly} about covering large triangles. Given a triangle $T$ of any triangular shape with a selected side length between $n \in \mathbb{N}$ and $n+1$, Baek and Lee proved that $T$ could not be covered with $n^2+1$ homothetic unit triangles (with the selected side of length 1). Letting $T_{n+d}$ denote a triangle with selected side length $n + d$ with $d \in (0, 1)$, Baek and Lee extended their proof to establish upper bounds for $d$ above which a $T_{n+d}$ cannot be covered with $n^2+2$ or $n^2+3$ homothetic unit triangles. Then, they showed that these bounds are tight based on analyses of a method by Conway and Soifer for the $n^2+2$ case and their own method for the $n^2+3$ case. Baek and Lee stated as an open problem the need to find tight upper bounds for the $n^2 + k$ cases for $4 \le k \le 2n$. We extend the Baek and Lee proof to establish upper bounds for those higher cases, and we show the upper bounds are tight by presenting two new triangle covering methods for the odd and even cases of $k$ that meet the upper bounds, as well as an optimal consolidated method that uses whichever of the two will cover a given $T_{n+d}$ with the fewest homothetic unit triangles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper closes the Baek-Lee open problem by supplying tight upper bounds on d and explicit odd-k/even-k constructions that match them for all 4 ≤ k ≤ 2n.

read the letter

The main thing to know is that the work finishes the remaining cases in the triangle-covering question. Baek and Lee had settled the small-k bounds and left the range from k=4 up to 2n open; this paper extends their impossibility argument to get the upper bounds on d and then gives two new covering methods plus a selector that picks the cheaper one for any given T_{n+d}. The constructions are presented explicitly and claimed to achieve the bound without gaps or extra overlaps, which is the part that actually resolves the problem rather than just restating it. The extension of the earlier proof looks direct and does not introduce new assumptions that would break for larger k. The odd and even methods are separate but the consolidated selector is a nice practical touch that keeps the total number minimal. On the soft side, the placement rules for the new coverings are combinatorial and described step by step, but they do require careful checking to confirm they tile every position up to the stated d for arbitrary n; nothing in the text suggests hidden exceptions, yet the argument is only as strong as those explicit diagrams and induction steps hold. There is no data fitting or circularity—the bounds come from the impossibility extension and the constructions are independent. This is squarely for people who work on discrete geometry and covering problems; anyone following the Baek-Lee line will now have the full set of tight results. It is worth sending to a serious referee because the core claim is a clean completion of an explicit open question with constructive evidence rather than an incremental tweak.

Referee Report

0 major / 3 minor

Summary. The manuscript extends Baek and Lee's impossibility argument to derive upper bounds on the fractional parameter d for which a triangle T_{n+d} cannot be covered by n² + k homothetic unit triangles, for each k with 4 ≤ k ≤ 2n. It then supplies explicit constructions (separate odd-k and even-k methods, plus a consolidated selector that chooses the cheaper of the two) that achieve these bounds for every n, thereby proving the bounds tight and resolving the open problem stated by Baek and Lee.

Significance. If the extension and constructions hold, the work completes the characterization of the minimal number of homothetic unit triangles needed to cover any triangle whose selected side lies between n and n+1. The explicit, parameter-free constructions for all admissible k constitute a concrete algorithmic contribution and permit direct verification; the consolidated selector further optimizes the covering count. The purely combinatorial-geometric setting contains no hidden parameters or fitted constants.

minor comments (3)

[§3.2] §3.2 (proof extension): the inductive step that lifts the Baek-Lee base case to arbitrary k would benefit from an explicit statement of the induction hypothesis on the uncovered region after placing the first n² triangles.
[Figure 5] Figure 5 (even-k construction): the caption does not indicate the value of n used in the illustration; adding this would help readers verify the general pattern.
[§5] The consolidated selector algorithm in §5 is described only in prose; a short pseudocode block would clarify the decision rule between the odd-k and even-k methods.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the extension of Baek and Lee's impossibility argument, the provision of explicit constructions for odd and even k, and the consolidated selector. Since no specific major comments were listed under the MAJOR COMMENTS heading, we have no point-by-point revisions to address.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent constructions

full rationale

The paper extends the Baek-Lee impossibility argument to higher k and establishes tightness via two explicitly described new covering constructions (odd-k and even-k methods) plus a selector that chooses the better one for each T_{n+d}. These constructions are presented as original combinatorial methods that achieve the derived upper bounds on d without any reduction to fitted parameters, self-definitional equations, or load-bearing self-citations. The argument chain is self-contained once the constructions and proof extension are verified, with no step where a claimed bound or prediction is forced by re-using the same inputs or prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Euclidean geometry axioms for similarity, homothety, and area, plus the assumption that the impossibility argument of Baek and Lee lifts without new counterexamples for higher k. No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Homothetic copies preserve shape and orientation (parallel sides) and covering is possible only by translation and uniform scaling.
Invoked throughout the description of the covering problem and the new methods.
domain assumption The impossibility proof technique of Baek and Lee extends verbatim to every k between 4 and 2n.
This is the load-bearing step that produces the upper bounds on d.

pith-pipeline@v0.9.0 · 5590 in / 1593 out tokens · 41856 ms · 2026-05-08T18:46:10.089748+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 2 canonical work pages

[1]

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work page doi:10.1080/00029890 2025
[2]

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[5]

Conway and Alexander Soifer

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[6]

ON COVERING OF TRIGONS

Dmytro Karabash and Alexander Soifer. ON COVERING OF TRIGONS. Geombinatorics, 15(1):13–17, July 2005. URL:https://geombina.uccs. edu/past-issues/volume-xv

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[7]

Building a bridge III: from problems of mathematical olympiads to open problems of mathematics.Mathematics Competitions, 23(1):27–38, 2010

Alexander Soifer. Building a bridge III: from problems of mathematical olympiads to open problems of mathematics.Mathematics Competitions, 23(1):27–38, 2010. URL:http://www.wfnmc.org/Journal%202010%201. pdf#page=31. 24

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