# Any Monotone Function Is Realized by Interlocked Polygons

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem 1.**

## 2. Construction

## 3. Exploding Sliding Block Puzzle

**Theorem 2.**

**Corollary 3.**

**Theorem 4.**

- (1)
- When all blocks are rectangles, every instance of the exploding sliding block puzzle has a solution, that is, any collection of rectangles is free.
- (2)
- When all blocks are rectangles except one block, the exploding sliding block puzzle in Theorem 2 and the decision problem in Corollary 3 are PSPACE-complete.

- (1)
- Any convex shapes (in any dimension) can be separated by simultaneous explosion, as proved by de Bruijn (see [5, page 2] for a comprehensive references). When we only consider orthogonal arrangements of rectangles, we have a simpler proof. We take any rectangle that touches the bounding box (which exists). Slide it out to infinity, and repeat. (Update the bounding box each time.)
- (2)
- We use the frame in Figure 11 instead of one in Figure 3. Suppose that the trigger block moves to right. After moving the block A to up in the frame, we can remove the block B from inside of C. Then we can remove all dummy blocks. Using this room, each of the blocks of size 1 × 2 in the gadgets can be rotated and slid out from C one by one. Finally, the block A can be slid out after a rotation. Therefore, we have the claim. □

## 4. Concluding Remarks

## Acknowledgements

## References

- Berlekamp, E.R.; Conway, J.H.; Guy, R.K. Winning Ways for Your Mathematical Plays; A K Peters Ltd.: Natick, MA, USA, 2001–2003; Volume 1–4. [Google Scholar]
- Hearn, R.A.; Demaine, E.D. Games, Puzzles, and Computation; A K Peters Ltd.: Natick, MA, USA, 2009. [Google Scholar]
- Leeuwen, J. Handbook of Theoretical Computer Science; Elsevier Science Publishers: Amsterdam, The Netherlands, 1990. [Google Scholar]
- Rectangular Jam. Available online: http://home.r01.itscom.net/iwahiro/main/eng_contents/eng_intro.html (accessed on 17 March 2012).
- Demaine, E.D.; Langerman, S.; O’Rourke, J.; Snoeyink, J. Interlocked open and closed linkages with few joints. Comput. Geom. Theory Appl.
**2003**, 26, 37–45. [Google Scholar] [CrossRef]

**Figure 2.**Interlocked polygons freed by removing any one, two, or three of the polygons (from left to right).

(a) | (b) |

**Figure 8.**A construction for $f({x}_{1},{x}_{2},{x}_{3})=(({x}_{1}\wedge {x}_{2})\vee {x}_{3})\wedge ({x}_{1}\vee {x}_{3})$.

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**MDPI and ACS Style**

Demaine, E.D.; Demaine, M.L.; Uehara, R.
Any Monotone Function Is Realized by Interlocked Polygons. *Algorithms* **2012**, *5*, 148-157.
https://doi.org/10.3390/a5010148

**AMA Style**

Demaine ED, Demaine ML, Uehara R.
Any Monotone Function Is Realized by Interlocked Polygons. *Algorithms*. 2012; 5(1):148-157.
https://doi.org/10.3390/a5010148

**Chicago/Turabian Style**

Demaine, Erik D., Martin L. Demaine, and Ryuhei Uehara.
2012. "Any Monotone Function Is Realized by Interlocked Polygons" *Algorithms* 5, no. 1: 148-157.
https://doi.org/10.3390/a5010148