Show HN: I made a puzzle game that gently introduces my favorite math mysteries

Overall reception

• Many readers found the game fun, engaging, and surprisingly absorbing; several played with kids or partners.
• People liked how it built intuition for the four-color theorem and then “revealed” zero-knowledge proofs (ZKPs), saying they wished more math were taught this way.
• Some suggested this format for classrooms, museums, and math-exposition contests.

Four-color theorem and nature of proofs

• Discussion on wording: “mathematicians believe” vs “know” a proof is correct.
  • Some argue the proof is fully formalized (e.g., via proof assistants) and should be stated as fact.
  • Others stress that acceptance of very long or computer-assisted proofs is partly a social process, and that “belief” is unavoidable in practice.
• Debate over whether proofs are simply correct/incorrect vs. “convincing but unchecked arguments.”
• Related examples: classification of finite simple groups, Freedman’s work, Fermat’s Last Theorem.

Zero-knowledge proofs: understanding and skepticism

• Several commenters didn’t initially see why the coloring protocol is a ZKP or a “proof” at all.
• Clarifications:
  • Prover commits to a full coloring before the verifier’s choice (post-its / prime encodings as a commitment scheme).
  • If no valid 3-coloring exists, each round has at least some chance of exposing a conflict; repeated rounds make cheating probability arbitrarily small, though never literally zero.
  • This probabilistic nature leads some to dislike calling it a “proof”; others note many cryptographic “proofs” (signatures, proof-of-work) are also probabilistic in practice.
• Discussion of digital version:
  • Use of large composite numbers as “covered colors” assumes factoring is hard.
  • Some note tension: factoring may be easier than 3-coloring in principle, so the commitment mechanism and the theorem’s hardness live in different complexity worlds.
• Various analogies are mentioned (Alibaba cave, Where’s Waldo) and external resources, but several readers still report partial confusion.

Map puzzles, graph theory, and generalizations

• People enjoyed trying to construct 3-color and (impossible) 5-color maps; the “very difficult” 5-color challenge both motivated and frustrated.
• Intuition attempts:
  • Idea that a 5-color requirement would need a planar 5-clique (or something reducible to it); others point out this is nontrivial and connects to deep conjectures (e.g., about minors).
  • Noted that the western US example needs 4 colors without containing a 4-clique.
• Readers connect maps to planar graphs: regions ↔ vertices, borders ↔ edges; only planar graphs correspond to valid maps.
• Discussion of dimensions and surfaces:
  • 0D → 1 color, 1D → 2 colors, 2D → 4 colors, 3D → unbounded colors.
  • On other surfaces (sphere, torus, Klein bottle), the required number of colors changes; specific numbers are cited from secondary sources linked in the thread.
• Some note that real-world political maps with exclaves/enclaves can violate the simple planar setting, so “4 colors suffice” for regions doesn’t directly apply if you insist all pieces of a country share a color.

UX, bugs, and suggestions

• Reported issues:
  • Next-button not working on Android Firefox.
  • Occasional coloring/backtracking bugs and performance problems with very complex user-drawn maps.
  • Confusion about whether box borders count as map borders.
• UX requests:
  • Ability to move/delete points; auto-closing or zooming when drawing near edges.
  • Prevent text selection/context menu when clicking the canvas.
  • Clearer signaling when a 3-color solution is found or impossible.
  • Progress indicator and clearer framing that ZKPs are “the main topic.”
  • More explicit wording in the ZKP steps (e.g., that the three allowed colors are fixed and known; that verification checks both “different” and “within the chosen set”).
  • FAQ entry explaining why the computer isn’t choosing colors adaptively, and stronger emphasis that all prime-encoded values are fixed upfront.
• Ideas for extensions: more continents/maps, graph-theory or more rigorous follow-up explanations, easter egg for coloring the larger graph, alternative commitment schemes (e.g., hashes).

Geopolitical naming and real-world maps

• Multiple comments criticize labeling a “map of the UK” that includes the Republic of Ireland.
• Alternatives debated: “UK & Ireland,” “Britain and Ireland,” “British Isles,” “Atlantic Archipelago,” with detailed discussion of historical and political sensitivities.
• Some argue “British Isles” is standard geographic terminology; others object due to colonial connotations and current political realities.
• One suggestion: simply remove Ireland from that particular demo map to avoid the issue.
• Separately, readers note that enclaves/exclaves and non-contiguous territories complicate applying the four-color theorem to actual country maps.