{"id":32,"date":"2019-06-08T19:53:02","date_gmt":"2019-06-08T19:53:02","guid":{"rendered":"http:\/\/arthur-benjamin.104.42.120.246.xip.io\/?page_id=32"},"modified":"2025-05-28T22:19:49","modified_gmt":"2025-05-28T22:19:49","slug":"research-papers","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/benjamin\/research-papers\/","title":{"rendered":"Research Papers"},"content":{"rendered":"\n

Color scheme for paper themes: 
            Papers on combinatorics are RED<\/span>. 
            Papers on mathematical games are BLUE<\/span>. 
            Papers on miscellaneous topics are GREEN<\/span>. 
* Stars denote undergraduate co-authors.<\/p>\n\n\n\n

\n\n\n\n

Published Works<\/h2>\n\n\n
    \n
  1. The Bisection Method: Which Root?\u00a0<\/strong><\/span>
    The American Mathematical Monthly<\/em>, Vol. 94, No. 9, pp. 861-863, November 1987.\u00a0
    [PDF file]<\/a><\/li>\n
  2. Turnpike Structures for Optimal Maneuvers\u00a0<\/strong><\/span>
    Ph.D. Thesis, The Johns Hopkins University, Department of Mathematical Sciences, 1989.<\/em>\u00a0\u00a0\u00a0\u00a0Advisor: Alan J. Goldman.\u00a0
    [PDF file]<\/a><\/li>\n
  3. Graphs, Maneuvers, and Turnpikes\u00a0<\/strong><\/span>
    Operations Research,<\/em>\u00a0Vol. 38, No. 2, pp. 202-216, March\/April 1990.\u00a0
    This paper was awarded the\u00a0    George Nicholson Prize\u00a0<\/a>from the Operations Research Society of America in 1988.\u00a0
    [PDF file]<\/a><\/li>\n
  4. Another Way of Counting\u00a0NN<\/sup><\/em>\u00a0<\/strong><\/span>
    (with Fritz Juhnke*)
    SIAM Journal on Discrete Mathematics<\/em>, Vol. 5, No. 3, pp. 377-379, August 1992.\u00a0
    [PDF file]<\/a><\/li>\n
  5. Optimization in Chemical Kinetics\u00a0<\/strong><\/span>
    (with Gordon J. Hogenson*)
    The UMAP Journal<\/em>, Vol. 13, No. 1, pp. 17-24, Spring 1992.\u00a0
    [PDF file]<\/a><\/li>\n
  6. Optimal Leapfrogging\u00a0<\/strong><\/span>
    (with Joel Auslander* and Daniel S. Wilkerson*)
    Mathematics Magazine<\/em>\u00a0(cover article),\u00a0Vol. 66, No. 1, pp.14-19, February, 1993.\u00a0
    [PDF file]<\/a><\/li>\n
  7. Optimal Blackjack Strategy with ‘Lucky Bucks’<\/strong><\/span>
    (with Eric L. Huggins*)
    The UMAP Journal<\/em>, Vol. 14, No. 4, pp. 309-318, Spring 1993.\u00a0
    [PDF file]<\/a><\/li>\n
  8. Localization of Optimal Strategies in Certain Games\u00a0<\/strong><\/span>
    (with Alan J. Goldman)
    Naval Research Logistics<\/em>, Vol. 41, No. 5, pp. 669-676, August 1994.\u00a0
    [PDF file]<\/a><\/li>\n
  9. Optimal Klappenspiel\u00a0<\/strong><\/span>
    (with Derek L. Stanford*)
    The UMAP Journal<\/em>\u00a0(cover article),\u00a0Vol. 16, No. 1, pp. 11-20, Spring 1995.\u00a0
    [PDF file]<\/a><\/li>\n
  10. Sensible Rules for Remembering Duals–The S-O-B Method\u00a0<\/strong><\/span>
    SIAM Review<\/em>, vol. 37, No. 1, pp. 85-87, 1995.\u00a0
    [PDF file]<\/a><\/li>\n
  11. How Do I Marry Thee? Let Me Count the Ways!\u00a0<\/strong><\/span>
    (with Cherlyn Converse and Henry A. Krieger)
    Discrete Applied Mathematics<\/em>, 59, pp. 285-292, 1995.\u00a0
    [PDF file]<\/a><\/li>\n
  12. Determinants of the Tournaments\u00a0<\/strong><\/span>
    (with Clifford A. McCarthy*)
    Mathematics Magazine<\/em>, Vol. 69, No. 2, pp. 133-135, April 1996.\u00a0
    [PDF file]<\/a><\/li>\n
  13. Combinatorics and Campus Security\u00a0<\/strong><\/span>
    The UMAP Journal<\/em>\u00a0(cover article),\u00a0vol. 17.2, pp. 111-116, Summer 1996.\u00a0
    [PDF file]<\/a><\/li>\n
  14. Strong Chromatic Index of Subset Graphs\u00a0<\/strong><\/span>
    (with Jennifer J. Quinn)
    The Journal of Graph Theory,<\/em>\u00a0Vol. 24, No. 3, pp. 267-273, 1997.\u00a0
    [PDF file]<\/a><\/li>\n
  15. Bounds on a Bug\u00a0<\/strong><\/span>
    (with Matthew T. Fluet*)
    The UMAP Journal<\/em>, Vol 20, No. 1, pp. 5-9, 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  16. The Best Way to Knock ‘m Down\u00a0<\/strong><\/span>
    (with Matthew T. Fluet*)
    The UMAP Journal<\/em>, Vol 20, No. 1, pp. 11-20, 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  17. Magic “Squares” Indeed!\u00a0<\/strong><\/span>
    (with Kan Yasuda*)
    The American Mathematical Monthly,\u00a0<\/em>Vol. 106, No. 2, pp. 152-156, February, 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  18. Unevening the Odds of\u00a0 “Even Up”<\/strong><\/span>
    (with Jennifer J. Quinn)
    Mathematics Magazine<\/em>, Vol 72, No. 2, pp. 149-150, 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  19. Why the Player Never Wins in the Long Run at LA Blackjack\u00a0<\/strong><\/span>
    (with Michael Lauzon* and Christopher Moore*)
    The UMAP Journal<\/em>, Vol 20, No. 2, pp. 127-138, Summer 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  20. Recounting Fibonacci and Lucas Identities\u00a0<\/strong><\/span>
    (with Jennifer J. Quinn)
    The College Mathematics Journal<\/em>, Vol 30, No. 5, pp. 359-366, November, 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  21. Fibonacci and Lucas Identities through Colored Tilings<\/strong><\/span>
    (with Jennifer J. Quinn)
    Utilitas Mathematica<\/em>, Vol 56, pp. 137-142, November, 1999.\u00a0
    [PDF file]<\/a><\/li>\n
  22. A Rational Solution to Cootie<\/strong><\/span>
    (with Matthew T. Fluet*)
    The College Mathematics Journal<\/em>, Vol 31, No. 2, pp. 124-125, March, 2000.\u00a0
    [PDF file]<\/a><\/li>\n
  23. Paths to a Multinomial Inequality<\/strong><\/span>
    (with Jennifer J. Quinn)
    Ars Combinatoria<\/em>, Vol 55, pp. 117-122, 2000.\u00a0
    [PDF file]<\/a><\/li>\n
  24. Phased Tilings and Generalized Fibonacci Identities<\/strong><\/span>
    (with Jennifer J. Quinn and Francis E. Su)
    The Fibonacci Quarterly<\/em>, Vol 38, No. 3, pp. 282-288, June-July 2000.\u00a0
    [PDF file]<\/a><\/li>\n
  25. Counting on Continued Fractions<\/strong><\/span>
    (with Jennifer J. Quinn and Francis E. Su)
    Mathematics Magazine<\/em>, Vol 73, No. 2, pp. 98-104, April, 2000.\u00a0
    [PDF file]<\/a><\/li>\n
  26. Random Approaches to Fibonacci Identities<\/strong><\/span>
    (with Gregory M. Levin, Karl Mahlburg*, and Jennifer J. Quinn)
    The American Mathematical Monthly<\/em>, Vol 107, No. 6, pp. 511-516, June-July, 2000.\u00a0
    [PDF file]<\/a><\/li>\n
  27. What’s Best?\u00a0<\/strong><\/span>
    (with Matthew T. Fluet*)
    The American Mathematical Monthly<\/em>, Vol 107, No. 6, pp. 560-562, June-July, 2000.\u00a0
    [PDF file]<\/a><\/li>\n
  28. Fermion-Boson Transformation in Fractional Quantum Hall Systems<\/span>\u00a0<\/strong>
    (with John J. Quinn, Arkadius Wojs, and Jennifer J. Quinn)
    Physica E<\/em>, Vol. 9, No. 4, pp. 701-708, April 2001.\u00a0
    [PDF file]<\/a><\/li>\n
  29. Optimal Token Allocations in Solitaire Knock ‘m Down<\/strong><\/span>
    (with Matthew T. Fluet* and Mark E. Huber)
    The Electronic Journal of Combinatorics<\/em>, Vol 8, No. 2, pp. 1-8, #R2, 2001.\u00a0
    [PDF file]<\/a><\/li>\n
  30. Composite Fermions and Integer Partitions<\/strong><\/span>
    (with Jennifer J. Quinn, Arkadius Wojs, and John J. Quinn)
    Jounal of Combinatorial Theory, Series A<\/em>, Vol 95, 390-397, 2001.\u00a0
    [PDF file]<\/a><\/li>\n
  31. Transformation of Statistics in Fractional Quantum Hall Systems<\/strong><\/span>
    (with John J. Quinn, A. Wojs, and Jennifer J. Quinn)
    Physica E<\/em>, Vol 11, pp. 182-185, 2001.\u00a0
    [PDF file]<\/a><\/li>\n
  32. A Stirling EnCOUNTer with Harmonic Numbers<\/strong><\/span>
    (with Greg O. Preston* and Jennifer J. Quinn)
    Mathematics Magazine<\/em>, Vol. 75, No. 2, pp. 95-103, April 2002.\u00a0
    [PDF file]<\/a><\/li>\n
  33. Analysis of the N-Card Game Le Her<\/strong><\/span>
    (with Alan J. Goldman)
    Journal of Optimization Theory and Applications<\/em>, Vol 114, No. 3, pp. 695-704, September, 2002.\u00a0
    [PDF file]<\/a><\/li>\n
  34. Two Quick Combinatorial Proofs of “the sum of the cubes”\u00a0<\/strong><\/span>
    (with Michael E. Orrison)
    The College Mathematics Journal<\/em>, Vol 33, No. 5, pp. 406-408, November, 2002.\u00a0
    [PDF file]<\/a><\/li>\n
  35. The Fibonacci Numbers — Exposed More Discretely<\/strong><\/span>
    (with Jennifer J. Quinn)
    Mathematics Magazine<\/em>,\u00a0(cover article),\u00a0Vol 76, No. 3, pp. 182-192, June 2003.\u00a0
    [PDF file]<\/a><\/li>\n
  36. A Probabilistic View of Certain Weighted Fibonacci Sums<\/strong><\/span>
    (with Judson D. Neer*, Daniel E. Otero, and James A. Sellers)
    The Fibonacci Quarterly<\/em>, Vol 41, No. 4, pp. 360-364, August, 2003.\u00a0
    [PDF file]<\/a><\/li>\n
  37. A Combinatorial Approach to Hyperharmonic Numbers<\/strong><\/span>
    (with David Gaebler* and Robert Gaebler*)
    INTEGERS: The Electronic Journal of Combinatorial Number Theory<\/em>, Vol 3, pp. 1–9, #A15, 2003.\u00a0
    [PDF file]<\/a><\/li>\n
  38. Linear Recurrences Through Tilings and Markov Chains<\/strong><\/span>
    (with Christopher R.H. Hanusa* and Francis E. Su)
    Utilitas Mathematica<\/em>, Vol 64, pp. 3-17, November 2003.\u00a0
    [PDF file]<\/a><\/li>\n
  39. Fibinomial Identities<\/strong><\/span>
    (with Jennifer J. Quinn and Jeremy A. Rouse*)
    Applications of Fibonacci Numbers, Volume 9<\/em>, (Frederic T. Howard, ed.), Kluwer Academic Publishers, pp. 19-24, 2004.\u00a0
    [PDF file]<\/a><\/li>\n
  40. Recounting Binomial Fibonacci Identities<\/strong><\/span>
    (with Jeremy A. Rouse*)
    Applications of Fibonacci Numbers, Volume 9<\/em>, (Frederic T. Howard, ed.), Kluwer Academic Publishers, pp. 25-28, 2004.\u00a0
    [PDF file]<\/a><\/li>\n
  41. Proofs that Really Count: The Magic of Fibonacci Numbers and More<\/strong><\/span>
    (with Jennifer J. Quinn)
    Mathematical Adventures for Students and Amateurs,\u00a0<\/em>(David F. Hayes and Tatiana Shubin, editors), Spectrum Series of MAA, pp. 83-98, 2004.\u00a0
    [PDF file]<\/a><\/li>\n
  42. A (Not So) Complex Solution to a^2 + b^2 = c^n<\/strong><\/span>
    (with Arnold M. Adelberg and David I. Rudel*)
    Mathematics Magazine,\u00a0<\/em>Vol 77, No. 4, pp. 300-302, October 2004.\u00a0
    [PDF file]<\/a><\/li>\n
  43. Combinatorial Proofs of Fermat’s, Lucas’s, and Wilson’s Theorems<\/strong><\/span>
    (with Peter G. Anderson and Jeremy A. Rouse*)
    The American Mathematical Monthly,\u00a0<\/em>Vol 112, No. 3, pp. 266-268, March 2005.\u00a0
    [PDF file]<\/a><\/li>\n
  44. Counting on Determinants\u00a0<\/strong><\/span>
    (with Naiomi T. Cameron)
    The American Mathematical Monthly,<\/em>\u00a0Vol 112, No. 6, pp. 481-492, June-July 2005.\u00a0
    [PDF file]<\/a><\/li>\n
  45. Recounting the Odds of an Even Derangement<\/strong><\/span>
    (with Curtis D. Bennett and Florence Newberger)
    Mathematics Magazine,\u00a0<\/em>Vol 78, No. 5, pp. 387-390, December 2005\u00a0
    [PDF file]<\/a><\/li>\n
  46. Pythagorean Primes and Palindromic Continued Fractions<\/strong><\/span>
    (with Doron Zeilberger)
    INTEGERS: The Electronic Journal of Combinatorial Number Theory<\/em>, 5(1), #A30, 2005.\u00a0
    [PDF file]<\/a><\/li>\n
  47. Revisiting Fibonacci and Related Sequences<\/strong><\/span>
    (with Jennifer J. Quinn)
    Mathematics Teacher<\/em>, Vol 99, No. 5, pp. 357-361, December 2005\/January 2006.\u00a0
    [PDF file]<\/a><\/li>\n
  48. Combinatorial Interpretations of Spanning Tree Identities<\/strong><\/span>
    (with Carl R. Yerger*)
    Bulletin of the Institute for Combinatorics and its Applications,<\/em>\u00a0Vol 47, pp. 37-42, 2006.\u00a0
    [PDF file]<\/a><\/li>\n
  49. Self-Avoiding Walks and Fibonacci Numbers<\/strong><\/span>
    The Fibonacci Quarterly<\/em>, Vol. 44, No. 4, pp. 330-334, November 2006.\u00a0
    [PDF file]<\/a><\/li>\n
  50. Summing Cubes by Counting Rectangles\u00a0<\/strong><\/span>
    (with Jennifer J. Quinn and Calyssa Wurtz*)
    The College Mathematics Journal<\/em>, November 2006.\u00a0
    [PDF file]<\/a><\/li>\n
  51. A Combinatorial Solution to Intertwined Recurrences<\/strong><\/span>
    (with Michael D. Hirschhorn)
    The Australasian Journal of Combinatorics<\/em>, Vol 37, pp. 101-116, 2007.\u00a0
    [PDF file]<\/a><\/li>\n
  52. Fibonacci Deteminants — A Combinatorial Approach<\/strong><\/span>
    (with Naiomi T. Cameron and Jennifer J. Quinn)
    The Fibonacci Quarterly<\/em>, Vol. 45, No. 1, pp. 39-55, February 2007.\u00a0
    [PDF file]<\/a><\/li>\n
  53. A Combinatorial Proof of Vandermonde’s Determinant<\/strong><\/span>
    (with Gregory P. Dresden)
    The American Mathematical Monthly<\/em>, Vol. 114, No. 4, pp. 338-341, April 2007.\u00a0
    [PDF file]<\/a><\/li>\n
  54. The Probability of Relatively Prime Polynomials<\/strong><\/span>
    (with Curtis D. Bennett)
    Mathematics Magazine<\/em>, (cover article), Vol. 80, No. 3, pp. 196-202, June 2007.\u00a0
    [PDF file]<\/a><\/li>\n
  55. Recounting Determinants for a Class of Hessenberg Matrices\u00a0<\/strong><\/span>
    (with Mark A. Shattuck)
    INTEGERS: The Electronic Journal of Combinatorial Number Theory<\/em>, Vol. 7, #A55, pp. 1–7, 2007.\u00a0
    [PDF file]<\/a><\/li>\n
  56. Estimating Winning Probabilities in Backgammon Races\u00a0<\/strong><\/span>
    (with Andrew M. Ross and Michael Munson)
    Optimal Play: Mathematical Studies of Games and Gambling<\/em>, (Stewart N. Ethier and William R. Eadington, editors), Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, pp. 269–291, 2007.\u00a0
    [PDF file]<\/a><\/li>\n
  57. Paint it Black — A Combinatorial Yawp<\/strong><\/span>
    (with Jennifer J. Quinn, James A. Sellers, and Mark A. Shattuck)
    Mathematics Magazine<\/em>, Vol. 81, No. 1. pp. 45-50, February 2008.\u00a0
    [PDF file]<\/a><\/li>\n
  58. The 99th Fibonacci Identity<\/strong><\/span>
    (with Alex K. Eustis* and Sean S. Plott*)
    The Electronic Journal of Combinatorics<\/em>, R34 (A Fibonacci number!), February 25, 2008.\u00a0
    [PDF file]<\/a><\/li>\n
  59. An Alternate Approach to Alternating Sums: A Method to Die For\u00a0<\/strong><\/span>
    (with Jennifer J. Quinn)
    The College Mathematics Journal<\/em>, Vol. 39, No. 3. pp. 191-201, September 2008.\u00a0
    [PDF file]<\/a><\/li>\n
  60. Tiling Proofs of Recent Sum Identities Involving Pell Numbers<\/strong><\/span>
    (with Sean S. Plott* and James A. Sellers)
    The Annals of Combinatorics<\/em>, Vol. 12, No. 3, pp. 271–278, October 2008.\u00a0
    [PDF file]<\/a><\/li>\n
  61. Recounting the Sums of Cubes of Fibonacci Numbers<\/strong><\/span>
    (with Timothy A. Carnes* and Benoit Cloitre)
    Congressus Numerantium, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications<\/em>, (William Webb, ed.), Vol 194, pp. 45-51, 2009.\u00a0
    [PDF file]<\/a><\/li>\n
  62. When does (F_m)^L divide F_n? A Combinatorial Solution<\/strong><\/span>
    (with Jeremy A. Rouse*)
    Congressus Numerantium, Proceedings of the Eleventh International Conference on Fibonacci Numbers and their Applications<\/em>, (William Webb, ed.), Vol 194, pp. 53-58, 2009.\u00a0
    [PDF file]<\/a><\/li>\n
  63. A Combinatorial Approach to Fibonomial Coefficients\u00a0<\/strong><\/span>
    (with Sean S. Plott*)
    The Fibonacci Quarterly<\/em>, Vol. 46\/47, No. 1, pp. 7-9, February 2008\/2009.\u00a0
    See also Errata: Vol 48, No. 3, pp. 276, August 2010.\u00a0
    [PDF file]<\/a>\u00a0[Errata]<\/a><\/li>\n
  64. Counting on Chebyshev Polynomials\u00a0<\/strong><\/span>
    (with Daniel Walton*)
    Mathematics Magazine,\u00a0<\/em>Vol. 82, No. 2, pp. 117-126, April 2009.\u00a0
    [PDF file]<\/a><\/li>\n
  65. Compression Theorems for Periodic Tilings and Consequences\u00a0<\/strong><\/span>
    (with Alex Eustis* and Mark Shattuck)
    The Journal of Integer Sequences<\/em>, Vol. 12, Article 09.6.3, 2009.\u00a0
    [PDF file]<\/a><\/li>\n
  66. Combinatorial Trigonometry with Chebyshev Polynomials\u00a0<\/strong><\/span>
    (with Larry Ericksen, Pallavi Jayawant, and Mark Shattuck)
    Journal of Statistical Planning and Inference, Proceedings of the 6th International Conference on Lattice Path Combinatorics,<\/em>, Vol. 140, pp. 2157-2160, 2010.\u00a0
    doi:10.1016\/j.jspi.2010.01.011\u00a0
    [PDF file]<\/a><\/li>\n
  67. Combinatorially Composing Chebyshev Polynomials\u00a0<\/strong><\/span>
    (with Daniel Walton*)
    Journal of Statistical Planning and Inference, Proceedings of the 6th International Conference on Lattice Path Combinatorics,<\/em>\u00a0Vol 140, pp. 2161-2167, 2010.\u00a0
    doi:10.1016\/j.jspi.2010.01.011, 2010.\u00a0
    [PDF file]<\/a><\/li>\n
  68. Catalan Determinants — A Combinatorial Approach<\/strong><\/span>
    (with Naiomi T. Cameron, Jennifer J. Quinn and Carl R. Yerger*)
    Congressus Numerantum, Proceedings of the Twelfth Conference on Fibonacci Numbers and their Applications,<\/em>\u00a0(William Webb, ed.), Vol. 200, pp. 27-34, 2010.\u00a0
    [PDF file]<\/a><\/li>\n
  69. The Lucas Triangle Recounted<\/strong><\/span>
    Congressus Numerantum, Proceedings of the Twelfth Conference on Fibonacci Numbers and their Applications,<\/em>\u00a0(William Webb, ed.), Vol. 200, pp. 169-177, 2010.\u00a0
    [PDF file]<\/a><\/li>\n
  70. Fibonacci Identities Derived from Path Counting in Automata\u00a0<\/strong><\/span>
    (with Dale Gerdemann)
    Congressus Numerantum, Proceedings of the Thirteenth Conference on Fibonacci Numbers and their Applications,<\/em>\u00a0(Florian Luca and Pante Stanica, eds.), Vol. 201, pp. 237-256, 2010.\u00a0
    [PDF file]<\/a><\/li>\n
  71. An Amazing Mathematical Card Trick\u00a0<\/strong><\/span>
    The Mathematical Intelligencer<\/em>, Vol. 32, Number 2, pp. 37-41, Summer 2010.\u00a0
    doi:10.1007\/s00283-009-9123-1
    [PDF file]<\/a><\/li>\n
  72. Sums of Evenly Spaced Binomial Coefficients\u00a0<\/strong><\/span>
    (with Bob Chen* and Kimberly Tucker)
    Mathematics Magazine\u00a0<\/em>, Vol. 83, No. 5, pp. 370-373, December 2010.
    [PDF file]<\/a><\/li>\n
  73. Third and Fourth Binomial Coefficients\u00a0<\/strong><\/span>
    (with Jacob N. Scott*)
    The Fibonacci Quarterly<\/em>, Vol. 49, No. 2, pp. 99-101, May 2011.\u00a0
    [PDF file]<\/a><\/li>\n
  74. The Combinatorialization of Linear Recurrences\u00a0<\/strong><\/span>
    (with Halcyon Derks* and Jennifer J. Quinn)\u00a0
    The Electronic Journal of Combinatorics,<\/em>\u00a0Vol. 18(2), P12, Zeilberger Issue, June 11, 2011.
    [ PDF file]<\/a><\/li>\n
  75. Combinatorics of Two Toned Tilings<\/strong><\/span>
    (with Phyllis Chinn, Jacob N. Scott*, and Greg Simay)
    The Fibonacci Quarterly<\/em>, Vol. 49, No. 4, pp. 290-297, November 2011.\u00a0
    [PDF file]<\/a><\/li>\n
  76. Squaring, Cubing, and Cube Rooting<\/strong><\/span>
    The College Mathematics Journal<\/em>, Vol. 43, Number 1, pp. 58-63, January, 2012.\u00a0
    [PDF file]<\/a><\/li>\n
  77. Challenging Magic Squares for Magicians\u00a0<\/strong><\/span>
    (with Ethan J. Brown*)
    The College Mathematics Journal<\/em>, Vol. 45, Number 2, pp. 92-100, March 2014. (This paper was reprinted in\u00a0The Best Writing in Mathematics 2015,\u00a0<\/em>edited by Mircea Pitici, Princeton University Press, pp. 26-37, 2015.
    [PDF file]<\/a><\/li>\n
  78. Counting on r-Fibonacci Numbers<\/strong><\/span>
    (with Curtis R. Heberle*)
    The Fibonacci Quarterly<\/em>, Vol. 52, Number 2, pp. 121-128, May 2014.
    [PDF file]<\/a><\/li>\n
  79. Combinatorial Proofs of Fibonomial Identities<\/strong><\/span>
    (with Elizabeth Reiland*)
    The Fibonacci Quarterly<\/em>, Vol. 52, Number 5, pp. 28-34, December 2014.
    [PDF file]<\/a><\/li>\n
  80. A Bijective Proof of a Derangement Recurrence<\/strong><\/span>
    (with Joel Ornstein*)
    Proceedings of the 17th International Conference on Fibonacci Numbers and Their Applications,\u00a0<\/em>pp. 28-29, December 2017.\u00a0
    [PDF file]<\/a><\/li>\n
  81. The BINGO Paradox<\/strong><\/span>
    (with Joseph Kisenwether and Ben Weiss)
    Math Horizons<\/em>, Vol. 25, Number 1, pp. 18-21, September 2017.\u00a0
    Also appeared in\u00a0The Best Writing on Mathematics, 2018,\u00a0<\/em>Mircea Pitici, Editor, Princeton University Press, pp. 108-116, 2019.\u00a0
    [PDF file]<\/a><\/li>\n
  82. Challenging Knight’s Tours<\/span>\u00a0<\/strong>
    (with Sam K. Miller*)
    Math Horizons (cover article),\u00a0<\/em>Number 3, Vol. 25, pp. 18-21, February 2018.
    [PDF file]<\/a><\/li>\n
  83. Factoring Numbers with Conway’s 150 Method<\/strong><\/span>
    The College Mathematics Journal<\/em>, Vol. 49, No. 2, pp. 122-125, March 2018.
    [PDF file]
    <\/a><\/li>\n
  84. BINGO Paradoxes\u00a0<\/strong><\/span>
    (with Barry A. Balof, Jay Cordes, and Joseph Kisenwether)
    to appear in\u00a0Mathematics of Various Entertaining Subjects (MOVES),\u00a0<\/em>Princeton University Press, 2019.<\/li>\n
  85. The Long and Short of Benford’s Law<\/strong><\/span>
    Math Horizons<\/em>, Number 4, Vol. 25, pp. 8-9, April 2019.
    [PDF file]<\/a><\/li>\n
  86. Unified Tiling Proofs of a Family of Fibonacci Identities<\/strong><\/span>
    (with Joshua Crouch* and James Sellers)
    The Fibonacci Quarterly<\/em>, Vol. 57, Number 1, pp. 29-31, February 2019.
    [PDF file]<\/a><\/li>\n
  87. A Tiling Interpretation of q-Binomial Coefficients<\/span><\/strong>
    (with Jonathan J. Azose*)
    Fibonacci Quarterly,<\/em> Number 2, Vol. 58, pp. 99-125, May 2020.
    [PDF file]<\/a><\/li>\n
  88. Counting on Euler and Bernoulli Number Identities<\/strong><\/span>
    (with John Lentfer* and Thomas C. Martinez*)
    Fibonacci Quarterly,<\/em> Number 5 (Special Open Access Issue), Vol. 58, pp. 30-33, December 2020.
    [PDF file]<\/a><\/li>\n
  89. A Tiling Interpretation of a Generalized Zeckendorf Theorem<\/span><\/strong>
    (with John Lentfer)
    Integers: The Electronic Journal of Combinatorial Number Theory,<\/em> Volume 22, #A104, November 2022.
    [PDF file]<\/a><\/li>\n
  90. Counting on Hosoya’s Triangle<\/strong><\/span>
    (with Daniela Elizondo)
    The Fibonacci Quarterly,<\/em> Vol. 60, Number 5, pp. 47-55, December 2022.
    [PDF file]<\/a><\/li>\n
  91. A Proof of the Optimal Leapfrogging Conjecture
    <\/strong><\/span>(with Sam K. Miller)
    Involve: A Journal of Mathematics,<\/em> Vol. 18, Number 1, pp. 105-122, 2025.
    [PDF file]<\/a>
    <\/strong><\/span><\/li>\n
  92. Counting on Padovan Numbers<\/strong><\/span>
    (with Delaney Cohn)
    to appear in The Fibonacci Quarterly,<\/em> Vol. 63, Number 2, June, 2025.

    <\/li>\n<\/ol>\n\n\n

    Book Reviews, Problem Solutions, and Letters<\/h2>\n\n\n
      \n
    1. Aspects of Combinatorics (by Victor Bryant) and Excursions in Calculus. (by Robert M. Young)\u00a0<\/strong><\/span>
      SIAM Newsletter<\/em>, (Discrete Mathematics Activity Group), Winter, 1994.<\/li>\n
    2. Discrete Mathematics: Elementary and Beyond. (by L. Lovasz, J. Pelikan, and K. Vesztergombi)\u00a0<\/strong><\/span>
      The American Mathematical Monthly<\/em>, Vol 111, No. 7, pp. 636-638, August-September, 2004.\u00a0
      [PDF file]<\/a><\/li>\n
    3. Across the Board: The Mathematics of Chessboard Problems. (by John J. Watkins)\u00a0<\/strong><\/span>
      The Mathematical Intelligencer<\/em>, Vol. 27, No. 2, 2005.\u00a0
      [PDF file]<\/a><\/li>\n
    4. Letter to the Editor: Sury’s Parent of Binet’s Formula<\/strong><\/span>
      Mathematics Magazine<\/em>, Vol 78, No. 2, p. 97, April 2005.\u00a0
      [PDF file]<\/a><\/li>\n
    5. Q.954 and A.954, Quickie Problem and Solution.\u00a0
      (with Michel Bataille)\u00a0<\/strong><\/span>
      Mathematics Magazine<\/em>, Vol 78, No. 4, October 2005.\u00a0
      [PDF file]<\/a><\/li>\n
    6. Proof Without Words: Alternating Sums of Odd Numbers<\/strong><\/span>
      Mathematics Magazine<\/em>, Vol 78, No. 5, p. 385, December 2005.\u00a0
      [PDF file]<\/a><\/li>\n
    7. Solution to Problem 1751, A Combinatorial Identity
      (with Andrew Carman*)\u00a0<\/strong><\/span>
      Mathematics Magazine<\/em>, Vol 80, No. 4, pp.309-310, October 2007.\u00a0
      [PDF file]<\/a><\/li>\n
    8. Reader Reflections (Division by 1001)
      (with Patrick Vennebush)\u00a0<\/strong><\/span>
      Mathematics Teacher<\/em>, Vol 104, No. 5, p. 332, December 2010\/January 2011.\u00a0
      [Word file]<\/a>\u00a0[PDF file]<\/a><\/li>\n
    9. Solution to Problem 952, A Binomial Coefficient Identity
      (with Lizard Reiland*)\u00a0<\/strong><\/span>
      The College Mathematics Journal<\/em>, Vol 43, No. 3, pp.260-261, May 2012.
      [PDF file]<\/a><\/li>\n
    10. Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks (by Persi Diaconis and Ron Graham)\u00a0<\/strong><\/span>
      SIAM Review<\/em>, Vol. 54, No. 3, pp. 609-612, 2012. (Featured Review)
      [PDF file]
      <\/a><\/li>\n
    11. Math Bite: Multiplying by 9
      (with Rohan Chandra*)\u00a0<\/strong><\/span>
      The College Mathematics Journal<\/em>, Vol 47, No. 4, pp.281, September 2016.
      [PDF file]<\/a><\/li>\n<\/ol>\n\n\n

      Other Papers<\/h2>\n\n\n
        \n
      1. Mathematical Magic for Skeptics<\/strong><\/span>
        (with Michael B. Shermer)
        Skeptical Inquirer<\/em>, Vol. 18, No. 1, pp. 52-62, Fall 1993.\u00a0\u00a0<\/li>\n
      2. Enumerating\u00a0Backgammon Positions: The Perfect Hash<\/strong><\/span>
        (with Andrew M. Ross*)
        Interface<\/em>, Vol. 16, No. 1, pp. 3-10, Spring 1997.\u00a0
        [PDF file]<\/a>\u00a0<\/li>\n
      3. Math Major Wins College Jeopardy<\/strong><\/span>
        Math Horizons<\/em>, pp. 24-26, April 1999.\u00a0\u00a0<\/li>\n
      4. A Better Way to Memorize Pi: The Phonetic Code<\/strong><\/span>
        Math Horizons<\/em>, p. 17, February 2000.\u00a0\u00a0<\/li>\n
      5. The Mathematical Web<\/strong><\/span>
        (with Jennifer J. Quinn)
        Math Horizons<\/em>, pp. 27-29, February, 2001.\u00a0\u00a0<\/li>\n
      6. Proof With Words: 2 + 11 \u2013 1 = 12<\/strong><\/span>
        Mathematics Magazine<\/em>, Vol. 74, No. 2, p. 98, April 2001.\u00a0
        [Word file]<\/a>\u00a0\u00a0<\/li>\n
      7. Mathematical Constance (a poem dedicated to Constance Reid)<\/strong><\/span>
        The American Mathematical Monthly<\/em>, Vol. 108, No. 5, p. 423, May 2001.\u00a0
        [Word file]<\/a>\u00a0\u00a0<\/li>\n
      8. Math Major, Math Major (a song)<\/strong><\/span>
        Math Horizons<\/em>, p. 28, September 2001.\u00a0
        [Word file]<\/a>\u00a0\u00a0<\/li>\n
      9. Mathematical Magic<\/strong><\/span>
        Bridges: Mathematical Connections in Art, Music, and Science; Conference Proceedings 2004<\/em>, (Reza. Sarhangi and Carlo Sequin, Eds.), pp. 259\u2013263, 2004.\u00a0
        [Word file]<\/a>\u00a0\u00a0<\/li>\n
      10. Double Birthday Magic Square\u00a0<\/strong><\/span>
        M-U-M (The Magazine for the Society of American Magicians)<\/em>, pp. 68-69, August 2006.\u00a0
        [PDF file]<\/a>\u00a0\u00a0<\/li>\n
      11. Ta Da! (A magical math Pop Quiz)\u00a0<\/strong><\/span>
        The New York Times (Education Life Section)<\/em>, pp. 32-33, Sunday, January 3, 2010\u00a0
        Online version on December 30, 2009\u00a0
        [web link]<\/a><\/li>\n
      12. Backgammon: Game On!<\/strong><\/span>
        The Best Kind of College: An Insiders\u2019 Guide to America\u2019s Small Liberal Arts Colleges,\u00a0<\/em>edited by Susan McWilliams and John E. Seery, SUNY Press, pp. 65-67, 2015.
        [PDF file]<\/a>\u00a0\u00a0<\/li>\n
      13. Magic in the Classroom<\/strong><\/span>
        Bridges: Mathematical Connections in Art, Music, and Science; Conference Proceedings 2004<\/em>, (Reza. Sarhangi and Carlo Sequin, Eds.), pp. 259\u2013263, 2015.<\/li>\n
      14. Reflections on Math and Magic<\/strong><\/span>
        Math Horizons<\/em>, Volume 30, Issue 3, p. 34, February 2023.\u00a0\u00a0
        [PDF File]<\/a>\u00a0\u00a0<\/li>\n
      15. 7-Card Stud Magic and the Gilbreath Principle<\/strong><\/span>
        Math Horizons<\/em>, Volume 30, Issue 4, pp. 26-28, April 2024.\u00a0\u00a0
        [PDF File]<\/a>\u00a0<\/li>\n<\/ol>\n\n\n

        Books<\/h2>\n\n\n
          \n
        1. Teach Your Child Math<\/em><\/strong><\/span>
          (with Michael B. Shermer)
          McGraw Hill, New York, 177 pages, 1991. (Out of Print.) \u00a0<\/li>\n
        2. Mathemagics: How to Look like a Genius Without Really Trying<\/em><\/strong><\/span>
          (with Michael B. Shermer)
          McGraw Hill (Book-of-the-Month-Club Selection), New York, 218 pages, 1993, 1998. (Out of Print. Updated as\u00a0Secrets of Mental Math.<\/em>)<\/li>\n
        3. Proofs That Really Count: The Art of Combinatorial Proof<\/em><\/strong><\/span>
          (with Jennifer J. Quinn)
          Mathematical Association of America, Dolciani Series, Washington DC, 208 pages, 2003.\u00a0
          (Order here!) <\/a><\/li>\n
        4. Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks<\/em><\/strong><\/span>
          (with Michael B. Shermer; translated into more than 10 languages!)\u00a0
          Random House, New York, 279 pages, 2006. (This is an updated version of\u00a0Mathemagics<\/em>.)<\/li>\n
        5. Biscuits of Number Theory<\/em><\/strong><\/span>
          (co-edited with Ezra B. Brown)
          Mathematical Association of America, Dolciani Series, Washington DC, 311 pages, 2009.\u00a0
          (Order here!)
          <\/a><\/li>\n
        6. The Art of Mental Calculation: Addition & Subtraction<\/em>\u00a0<\/strong><\/span>
          (with Natalya St. Clair)
          self-published with createspace.com, 60 pages, 2014.\u00a0
          (Order here!)<\/a><\/li>\n
        7. The Fascinating World of Graph Theory<\/em><\/strong><\/span>
          (with Gary Chartrand and Ping Zhang)
          Princeton University Press, Princeton NJ, 322 pages, 2015.\u00a0
          (Order here!)<\/a><\/li>\n
        8. The Magic of Math: Solving for\u00a0x\u00a0and Figuring Out Why<\/em><\/strong><\/span>
          Basic Books, New York, NY, 336 pages, 2015. (A\u00a0New York Times Bestseller!)<\/em><\/li>\n<\/ol>\n\n\n

          Video Courses (from The Great Courses<\/em>)<\/h2>\n\n\n
            \n
          1. The Joy of Mathematics<\/em><\/strong><\/span>
            The Great Courses, Chantilly, VA (24 thirty minute video), 2007.\u00a0
            (Order here!)<\/a><\/li>\n
          2. Discrete Mathematics<\/em><\/strong><\/span>
            The Great Courses,, Chantilly, VA (24 thirty minute video lectures), 2009.\u00a0
            (Order here!)<\/a><\/li>\n
          3. The Secrets of Mental Math<\/em>\u00a0<\/strong><\/span>
            The Great Courses,, Chantilly, VA (12 thirty minute video lectures), 2011.\u00a0
            (Order here!)<\/a><\/li>\n
          4. The Mathematics of Games and Puzzles: From Cards to Sudoku\u00a0<\/strong><\/em><\/span>
            The Great Courses, Chantilly, VA (12 forty-five minute video lectures), 2013.\u00a0
            (Order here!)<\/a><\/li>\n
          5. Math and Magic<\/strong><\/em><\/span>
            The Great Courses, Chantilly, VA (12 thirty minute video lectures), 2018.\u00a0
            (Order here!)<\/a><\/li>\n
          6. Fibonacci Numbers and the Golden Ratio<\/em><\/strong><\/span>
            The Great Courses,, Chantilly, VA (12 thirty minute video lectures), 2024.\u00a0
            (Order here!)<\/a><\/li>\n<\/ol>\n\n\n

            <\/p>\n","protected":false},"excerpt":{"rendered":"

            Color scheme for paper themes:             Papers on combinatorics are RED.             Papers on mathematical games are BLUE.             Papers on miscellaneous topics are GREEN.  * Stars denote undergraduate co-authors. Published Works The Bisection Method: Which Root?\u00a0The American Mathematical Monthly, Vol. 94, No. 9, pp. 861-863, November 1987.\u00a0[PDF file] Turnpike Structures for Optimal Maneuvers\u00a0Ph.D. Thesis, The Johns Hopkins University, Department … Continue reading “Research Papers”<\/span><\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-32","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/pages\/32","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/comments?post=32"}],"version-history":[{"count":46,"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/pages\/32\/revisions"}],"predecessor-version":[{"id":348,"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/pages\/32\/revisions\/348"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/media?parent=32"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/benjamin\/wp-json\/wp\/v2\/tags?post=32"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}