# How to Solve the Torus Puzzle

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Definitions and Notation

#### 2.2. A Basic Tool: Three Elements Rotations

- 1.
- ${W}^{q}\left({\mathit{r}}_{i}\right)$, ${S}^{p}\left({\mathit{c}}_{j}\right)$, ${E}^{q}\left({\mathit{r}}_{i}\right)$, ${N}^{p}\left({\mathit{c}}_{j}\right)$→ clockwise on A,
- 2.
- ${S}^{p}\left({\mathit{c}}_{j}\right)$, ${W}^{q}\left({\mathit{r}}_{i}\right)$, ${N}^{p}\left({\mathit{c}}_{j}\right)$, ${E}^{q}\left({\mathit{r}}_{i}\right)$→ counterclockwise on A,
- 3.
- ${N}^{p}\left({\mathit{c}}_{j}\right)$, ${W}^{q}\left({\mathit{r}}_{i}\right)$, ${S}^{p}\left({\mathit{c}}_{j}\right)$, ${E}^{q}\left({\mathit{r}}_{i}\right)$→ clockwise on B,
- 4.
- ${W}^{q}\left({\mathit{r}}_{i}\right)$, ${N}^{p}\left({\mathit{c}}_{j}\right)$, ${E}^{q}\left({\mathit{r}}_{i}\right)$, ${S}^{p}\left({\mathit{c}}_{j}\right)$→ counterclockwise on B,
- 5.
- ${E}^{q}\left({\mathit{r}}_{i}\right)$, ${N}^{p}\left({\mathit{c}}_{j}\right)$, ${W}^{q}\left({\mathit{r}}_{i}\right)$, ${S}^{p}\left({\mathit{c}}_{j}\right)$→ clockwise on C,
- 6.
- ${N}^{p}\left({\mathit{c}}_{j}\right)$, ${E}^{q}\left({\mathit{r}}_{i}\right)$, ${S}^{p}\left({\mathit{c}}_{j}\right)$, ${W}^{q}\left({\mathit{r}}_{i}\right)$→ counterclockwise on C,
- 7.
- ${S}^{p}\left({\mathit{c}}_{j}\right)$, ${E}^{q}\left({\mathit{r}}_{i}\right)$, ${N}^{p}\left({\mathit{c}}_{j}\right)$, ${W}^{q}\left({\mathit{r}}_{i}\right)$→ clockwise on D,
- 8.
- ${E}^{q}\left({\mathit{r}}_{i}\right)$, ${S}^{p}\left({\mathit{c}}_{j}\right)$, ${W}^{q}\left({\mathit{r}}_{i}\right)$, ${N}^{p}\left({\mathit{c}}_{j}\right)$→ counterclockwise on D.

#### 2.3. Solution

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

## 3. Upper and Lower Bounds of diam$\left({\mathcal{M}}_{\mathit{m},\mathit{n}}\right)$

**Theorem**

**1.**

- 1.
- For any configurations $\mathit{X}$, $\mathit{Y}\in {\mathcal{M}}_{m,n}$ with $\mathit{X}\equiv \mathit{Y}$, $dn(\mathit{X},\mathit{Y})\le 4mn+c\xb7max\{m,n\}$ for some constant c.
- 2.
- For a given $0<\epsilon <1$, there are integers m and n such that for any configuration $\mathit{X}\in {\mathcal{M}}_{m,n}$ there is a configurations $\mathit{Y}\in {\mathcal{M}}_{m,n}$ for which $dn(\mathit{X},\mathit{Y})\ge \epsilon mn$.

**Proof.**

## 4. Hardness

- INSTANCE:
- A bound $B\in {Z}^{+}$, a set $A={\ell}_{1}$, $\dots \phantom{\rule{0.166667em}{0ex}}$, ${\ell}_{3m}$ of $3m$ positive integers such that $B/4<{\ell}_{i}<B/2$ for each $1\le i\le 3m$, and ${\sum}_{i=1}^{3m}{\ell}_{i}=mB$.
- QUESTION:
- Can A be partitioned into m disjoint sets ${A}_{1}$, $\dots \phantom{\rule{0.166667em}{0ex}}$, ${A}_{m}$ such that ${\sum}_{\ell \in {A}_{i}}\ell =B$ for each $1\le i\le m$?

**Observation**

**1.**

**Theorem**

**2.**

**Proof.**

**Claim**

**1.**

**Proof.**

**Claim**

**2.**

**Proof.**

## 5. Discussion

## Acknowledgements

## References

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## Share and Cite

**MDPI and ACS Style**

Amano, K.; Kojima, Y.; Kurabayashi, T.; Kurihara, K.; Nakamura, M.; Omi, A.; Tanaka, T.; Yamazaki, K.
How to Solve the Torus Puzzle. *Algorithms* **2012**, *5*, 18-29.
https://doi.org/10.3390/a5010018

**AMA Style**

Amano K, Kojima Y, Kurabayashi T, Kurihara K, Nakamura M, Omi A, Tanaka T, Yamazaki K.
How to Solve the Torus Puzzle. *Algorithms*. 2012; 5(1):18-29.
https://doi.org/10.3390/a5010018

**Chicago/Turabian Style**

Amano, Kazuyuki, Yuta Kojima, Toshiya Kurabayashi, Keita Kurihara, Masahiro Nakamura, Ayaka Omi, Toshiyuki Tanaka, and Koichi Yamazaki.
2012. "How to Solve the Torus Puzzle" *Algorithms* 5, no. 1: 18-29.
https://doi.org/10.3390/a5010018