The Mathematical Intelligencer

, Volume 32, Issue 2, pp 41–48 | Cite as

The Early History of the Brick Factory Problem

  • Lowell Beineke
  • Robin Wilson
Years Ago David E. Rowe, Editor


Complete Graph Mathematical Intelligencer Complete Bipartite Graph Plane Drawing Labor Camp 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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We wish to express our thanks to Vera Sós, Paul Turán’s widow, for supplying the photograph of him, and to Anthony Hill for many helpful conversations and access to his geometrical notebooks. We should also like to thank Richard Guy, Bruce Richter, Marjorie Senechal, and David Rowe for their helpful comments.


  1. [1]
    D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993), 333–348.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Blažek and M. Koman, A minimal problem concerning complete plane graphs, In: Theory of Graphs and Its Applications (ed. M. Fiedler), Czechoslovak Academy of Sciences (1964), 113–117.Google Scholar
  3. [3]
    Alex Brodsky, Stephane Durocher, and Ellen Gethner, The rectilinear crossing number of K 10 is 62, Electron. J. Combin. 8, No. 1 (2001), Research Paper 23, 1–30.Google Scholar
  4. [4]
    H. E. Dudeney, Perplexities, Strand Magazine 46, No. 271 (July 1913), 110; solution in Strand Magazine 46, No. 272 (August 1913), 221.Google Scholar
  5. [5]
    H. E. Dudeney, Amusements in Mathematics, Thomas Nelson and Sons (1917), Problem 251 and solution, 73, 200.Google Scholar
  6. [6]
    Yona Friedman, Towards a Scientific Architecture (transl. Cynthia Lang), Cambridge, MA: MIT Press (1975), 63.Google Scholar
  7. [7]
    M. R. Garey and D. S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Discrete Methods 4 (1983), 312–316.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    L. Glebskeii and G. Salazar, The crossing number of C m × C n is (m – 2)n for all but finitely many n for each m, J. Graph Theory 47 (2004), 53–72.CrossRefMathSciNetGoogle Scholar
  9. [9]
    H. P. Goodman, The complete n-point graph, Letter to Nature, 190, No. 4778 (27 May 1961), 840.Google Scholar
  10. [10]
    M. Grohe, Computing crossing numbers in quadratic time, J. Comput. System Sci. 68 (2004), 285–302.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Richard K. Guy, A combinatorial problem, Nabla (Bull. Malayan Math. Soc.) 7 (1960), 68–72.Google Scholar
  12. [12]
    Richard K. Guy, The decline and fall of Zarankiewicz’s theorem, In: Proof Techniques in Graph Theory (ed. F. Harary), New York: Academic Press (1969), 63–69.Google Scholar
  13. [13]
    Frank Harary and Anthony Hill, On the number of crossings in a complete graph, Proc. Edinb. Math. Soc. (II) 13 (1962–63), 333–338.Google Scholar
  14. [14]
    F. Harary, P. C. Kainen, and A. J. Schwenk, Toroidal graphs with arbitrarily high crossing numbers, Nanta Math. 6 (1973), 58–69.zbMATHMathSciNetGoogle Scholar
  15. [15]
    Anthony Hill, Catalogue of the Retrospective Exhibition, Arts Council of Great Britain, Hayward Gallery, 1983.Google Scholar
  16. [16]
    D. J. Kleitman, The crossing number of K 5,n, J. Combin. Theory 9 (1970), 315–323.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    T. Kövari, V. Sós, and P. Turán: On a problem of K. Zarankiewicz, Colloq. Math. 3 (1954), 50–57.zbMATHGoogle Scholar
  18. [18]
    David E. Kullman, The utilities problem, Math. Magazine 52 (1979), 299–302.CrossRefGoogle Scholar
  19. [19]
    F. T. Leighton, New lower bound techniques for VSLI, In: Proceedings of the 22nd Annual Symposium on Foundations of Computer Science, Washington, D. C.: IEEE Computer Society (1981), 1–12.Google Scholar
  20. [20]
    Sam Loyd, Jr., Sam Loyd and His Puzzles, New York, Barse (1928), 6, 87–88.Google Scholar
  21. [21]
    Nadine C. Myers, The crossing number of C m × C n: a reluctant induction, Math. Magazine 71 (1998), 350–359.zbMATHGoogle Scholar
  22. [22]
    T. H. O’Beirne, Christmas puzzles and paradoxes, 51: For boys, men and heroes, New Scientist 12, No. 266 (21 December 1961), 751–753.Google Scholar
  23. [23]
    Shengjun Pan and R. Bruce Richter, The crossing number of K 11 is 100, J. Graph Theory 56 (2007), 128–134.Google Scholar
  24. [24]
    R. B. Richter and G. Salazar, Crossing numbers, Topics in Topological Graph Theory (eds. L. W. Beineke and R. J. Wilson), Cambridge University Press (2009), 133–150.Google Scholar
  25. [25]
    Paul Turán, A note of welcome, J. Graph Theory 1 (1977), 7–9.CrossRefGoogle Scholar
  26. [26]
    K. Urbanik, Solution du problème posé par P. Turán, Colloq. Math. 3 (1955), 200–201.MathSciNetGoogle Scholar
  27. [27]
    D. R. Woodall, Cyclic-order graphs and Zarankiewicz’s crossing-number conjecture, J. Graph Theory 17 (1993), 657–671.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    K. Zarankiewicz, The solution of a certain problem on graphs of P. Turan, Bull. Acad. Polon. Sci. Cl. III. 1 (1953), 167–168.zbMATHMathSciNetGoogle Scholar
  29. [29]
    K. Zarankiewicz, On a problem of P. Turan concerning graphs, Fund. Math. 41 (1954), 137–145.zbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIndiana University — Purdue University Fort WayneFort WayneUSA
  2. 2.Department of Mathematics and StatisticsThe Open UniversityMilton KeynesUK

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