Shortest Path Finding in Mazes by Active and Passive Particles

Part of the Emergence, Complexity and Computation book series (ECC, volume 32)


Maze solving and finding the shortest path or all possible exit paths in mazes can be interpreted as mathematical problems which can be solved algorithmically. These algorithms can be used by both living entities (such as humans, animals, cells) and non-living systems (computer programs, simulators, robots, particles). In this chapter we summarize several chemistry-based concepts for maze solving in two-dimensional standard mazes which rely on surface tension driven phenomena at the air-liquid interface. We show that maze solving can be implemented by using: (i) active (self-propelled) droplets and/or (ii) passive particles (chemical entities).



J. Č. was financially supported by the Czech Science Foundation (Grant No. 17-21696Y). Other authors acknowledge the financial support of the Hungarian Research Fund (OTKA K104666). Financial support for R. T. by the Marie Heim-Vogtlin Program under project no PMPDP2-139698 is gratefully acknowledged. D. U. and I. L. gratefully acknowledge the financial support of the National Research, Development and Innovation Office of Hungary (TÉT12JP-1-2014-0005).


  1. 1.
    A. Adamatzky, Hot ice computer. Phys. Lett. A 374, 264–271 (2009)CrossRefGoogle Scholar
  2. 2.
    A. Adamatzky, Slime mold solves maze in one pass, assisted by gradient of chemo-attractants. IEEE Trans. Nanobiosci. 11, 131–134 (2012)CrossRefGoogle Scholar
  3. 3.
    A. Adamatzky, Physical maze solvers. All twelve prototypes implement 1961 Lee algorithm, in Emergent Computation: A Festschrift for Selim G. Akl, ed. by A. Adamatzky (Cham, Springer International Publishing, 2017), pp. 489–504Google Scholar
  4. 4.
    A. Braun, R. Tóth, I. Lagzi, Künstliche Intelligenz aus dem Chemiereaktor. Nachr. Chem. 63, 445–446 (2015)CrossRefGoogle Scholar
  5. 5.
    J. Čejková, M. Novák, F. Štěpánek, M.M. Hanczyc, Dynamics of chemotactic droplets in salt concentration gradients. Langmuir 30, 11937–11944 (2014)CrossRefGoogle Scholar
  6. 6.
    J. Čejková, T. Banno, F. Štěpánek, M.M. Hanczyc, Droplets as liquid robots. Artif. Life 23, 528–549 (2017)CrossRefGoogle Scholar
  7. 7.
    J. Čejková, S. Holler, N.T. Quyen, C. Kerrigan, F. Štěpánek, M.M. Hanczyc, Chemotaxis and chemokinesis of living and non-living objects, in Advances in Unconventional Computing, ed. by A. Adamatzky (Springer, 2017), pp. 245–260Google Scholar
  8. 8.
    A.E. Dubinov, A.N. Maksimov, M.S. Mironenko, N.A. Pylayev, V.D. Selemir, Glow discharge based device for solving mazes. Phys. Plasmas 21, 093503 (2014)CrossRefGoogle Scholar
  9. 9.
    M.J. Fuerstman, P. Deschatelets, R. Kane, A. Schwartz, P.J.A. Kenis, J.M. Deutch, G.M. Whitesides, Solving mazes using microfluidic networks. Langmuir 19, 4714–4722 (2003)CrossRefGoogle Scholar
  10. 10.
    I. Lagzi, S. Soh, P.J. Wesson, K.P. Browne, B.A. Grzybowski, Maze solving by chemotactic droplets. J. Am. Chem. Soc. 132, 1198–1199 (2010)CrossRefGoogle Scholar
  11. 11.
    P. Lovass, M. Branicki, R. Tóth, A. Braun, K. Suzuno, D. Ueyama, I. Lagzi, Maze solving using temperature-induced Marangoni flow. RSC Adv. 5, 48563–48568 (2015)CrossRefGoogle Scholar
  12. 12.
    T. Nakagaki, H. Yamada, A. Tóth, Maze-solving by an amoeboid organism. Nature 407, 470–470 (2000)CrossRefGoogle Scholar
  13. 13.
    T. Nakagaki, H. Yamada, A. Tóth, Path finding by tube morphogenesis in an amoeboid organism. Biophys. Chem. 92, 47–52 (2001)CrossRefGoogle Scholar
  14. 14.
    Y.V. Pershin, M. Di Ventra, Solving mazes with memristors: a massively parallel approach. Phys. Rev. E 84, 046703 (2011)CrossRefGoogle Scholar
  15. 15.
    D.R. Reyes, M.M. Ghanem, G.M. Whitesides, A. Manz, Glow discharge in microfluidic chips for visible analog computing. Lab Chip 2, 113–116 (2002)CrossRefGoogle Scholar
  16. 16.
    O. Steinbock, A. Tóth, K. Showalter, Navigating complex labyrinths: optimal paths from chemical waves. Science 267, 868–871 (1995)CrossRefGoogle Scholar
  17. 17.
    O. Steinbock, P. Kettunen, K. Showalter, Chemical wave logic gates. J. Phys. Chem. 100, 18970–18975 (1996)CrossRefGoogle Scholar
  18. 18.
    K. Suzuno, D. Ueyama, M. Branicki, R. Tóth, A. Braun, I. Lagzi, Maze solving using fatty acid chemistry. Langmuir 30, 9251–9255 (2014)CrossRefGoogle Scholar
  19. 19.
    Y. Yu, G. Pan, Y. Gong, K. Xu, N. Zheng, W. Hua, X. Zheng, Z. Wu, Intelligence-augmented rat cyborgs in maze solving. PLoS ONE 11, e014775 (2016)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Chemistry and Technology PraguePragueCzechia
  2. 2.Laboratory for High Performance Ceramics, EmpaDübendorfSwitzerland
  3. 3.School of MathematicsUniversity of EdinburghEdinburghUK
  4. 4.Faculty of EngineeringMusashino UniversityTokyoJapan
  5. 5.Department of PhysicsBudapest University of Technology and Economics and MTA-BME Condensed Matter Research GroupBudapestHungary

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