Dna Computation of Solutions to Edge‐matching Puzzles Abstract Dna Computation of Solutions to Edge‐matching Puzzles

complies with the regulations of the University and meets the accepted standards with respect to originality and quality. The resilient, ancient, and fine‐tuned DNA (deoxyribonucleic acid) has inspired many researchers to harness its power for material purposes. In this work, we use synthesized DNA strands to compute the solution to an instance of edge‐matching puzzles (EMP), where the challenge is to pack a collection of edge‐coloured square tiles on a square board such that all adjacent edges match in colour. We encode tiles with DNA strands and make use of structural, chemical and enzymatic properties of DNA to effectively carry out a brute‐force search of the solution to the puzzle. The solution ultimately results as a 2‐dimensional DNA lattice encoding the position and orientation of each tile on the solution board. Our approach has been to represent a tile as the union of two half‐tiles. This conceptual representation allows for the use of a supremely powerful heuristic: polymerase chain reaction (PCR), which can be inserted at any step of the protocol to selectively amplify certain strands to exponential quantities. Our abstract formalization of half‐tiles and the DNA protocol we use to manipulate them have relevance in three ways. First, by solving an instance of the (NP‐Complete) EMP problem we make precise characterizations of the processing power of DNA Computing. Second, the 2‐ dimensional self‐assembly of half‐tiles is Turing‐complete. Thirdly, the 2‐dimensional self‐ assembly of half‐tiles can serve as a PCR‐powered model for massive nano‐scale fabrication of 2‐ dimensional DNA nano‐shapes. iv Acknowledgements