Sages and Their Anonymous Slips — Draconitic Research

Abstract

Before the game, the sages agree on a finite set of symbols and on a rule for choosing one of them. They are then seated in an unknown order, either in a row or around a circle with a common clockwise direction. Each sage sees only the adjacent sage or sages and secretly writes one symbol. The slips are then mixed, so nobody learns who wrote which symbol; everyone learns only how many times each symbol occurred. The aim is that every sage should be able to independently reconstruct the entire seating. We ask for the smallest number of distinct symbols that makes this possible. We also consider a version in which the seats have fixed public labels, and each sage is told the label of their own seat. For large numbers of sages, the minimum number of symbols is proportional to the square of the number of sages, up to a constant factor. Several small cases can be settled exactly. Four sages in a row need exactly two symbols, and so do five sages on a circle. Six sages on a circle need exactly three symbols, both with and without labeled seats; we give a short explicit rule. Five sages in a row with labeled seats also need exactly three symbols. Without seat labels, two symbols are impossible, and four are sufficient; whether three symbols suffice remains open.

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BibTeX RIS

BibTeX

@misc{khovanova2026sages,
  title = {{Sages and Their Anonymous Slips}},
  author = {Tanya Khovanova and Konstantin Knop},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33032},
  eprint = {2610.33032},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33032},
}

RIS

TY  - UNPB
TI  - Sages and Their Anonymous Slips
AU  - Tanya Khovanova
AU  - Konstantin Knop
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.33032
UR  - https://arxiv.org/abs/2610.33032
AB  - Before the game, the sages agree on a finite set of symbols and on a rule for choosing one of them. They are then seated in an unknown order, either in a row or around a circle with a common clockwise direction. Each sage sees only the adjacent sage or sages and secretly writes one symbol. The slips are then mixed, so nobody learns who wrote which symbol; everyone learns only how many times each symbol occurred. The aim is that every sage should be able to independently reconstruct the entire seating. We ask for the smallest number of distinct symbols that makes this possible. We also consider a version in which the seats have fixed public labels, and each sage is told the label of their own seat. For large numbers of sages, the minimum number of symbols is proportional to the square of the number of sages, up to a constant factor. Several small cases can be settled exactly. Four sages in a row need exactly two symbols, and so do five sages on a circle. Six sages on a circle need exactly three symbols, both with and without labeled seats; we give a short explicit rule. Five sages in a row with labeled seats also need exactly three symbols. Without seat labels, two symbols are impossible, and four are sufficient; whether three symbols suffice remains open.
ER  -

Abstract

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