Robert K. Moniot

• Accelerator-based mass spectrometry measurement of long-lived radioisotopes, applied to archeology, geophysics and meteoritics.
• Scale-space and equation-error minimization approaches to problems in image reconstruction and solution of partial differential equations.
• Least-squares fitting data with experimental uncertainties in all coordinates.
  1. Fit to a hyperplane having only one dependent coordinate and any number of independent coordinates. This work was done when I was a graduate student many years ago. It was never published except as a tech report. Here is that tech report in PDF format, HTML format, or the original LaTeX document.

     Download source code of an implementation of this method.

  2. Fit to a hyperplane having any number of dependent coordinates and any number of independent coordinates. Applied Numerical Mathematics, 59, (2009), pp. 135-150. doi:10.1016/j.apnum.2008.01.001. An expanded version of that paper is available in PDF format here.
• I am the developer of the ftnchek program for debugging FORTRAN programs.

• The Red-and-Blue-Balls Puzzle (Odds Inversion). The National Museum of Mathematics Varsity Math puzzle number 117 "Fifty-Fifty" asks how to fill a bag with red and blue balls such that if one draws out two balls at random, the odds that they are different colors will be exactly 50-50. One might think the answer is to have equal numbers of each color. That turns out not to be the case. The answer is quite surprising and involves triangular numbers. But then, what about other odds? Can one fill the bag to achieve any desired odds from nil to certainty? This problem turns out to have many interesting features, and can be solved completely. I wrote a paper giving the most interesting features of this problem, titled "Solution of an Odds Inversion Problem," which was published in 2021 as an open-access article in the American Mathematical Monthly, available here.
  • Here is a PDF digest of the analysis (generated from a Mathematica notebook).
  • For those who have Mathematica and would like to explore the problem themselves, here is the notebook itself, including also an appendix defining a function that solves the problem for any odds.
  • And for those who really want to dive into it, here is a Mathematica notebook with a complete analysis of the problem, including the proofs, and a great deal more. Here is a PDF printout of the notebook.
  The story did not end there. A team of three students and a math professor at San Diego State University read my article in the Monthly and followed up with an analysis of one of the open questions listed there. Their paper, Hilmer et al., “The Elliptical Case of an Odds Inversion Problem,” was published in Involve, a journal of mathematics for students, in 2023. The abstract and full citation of that article can be found here, In this paper they stated a conjecture about the number of possible solutions for a certain class of cases of this problem. The conjectured result was both surprising and elegant and seemed, in tests, to be true. I took up the challenge of proving it, and to my surprise I was successful. That proof and a complete analysis of the problem for all probabilities from 0 to 1 was published in 2025 as “Solution of an odds inversion problem,” in Notes on Number Theory and Discrete Mathematics, and can be found here.
• The Taxman Game is a mathematical amusement, often used as an assignment for classes in computer programming. I wrote an article about it aimed at math students, which appeared in Math Horizons, February 2007. This paper was awarded the 2008 Trevor Evans Prize of the Mathematical Association of America.
  • Here is a somewhat expanded version of my paper, in PDF format.
  • You can play the Taxman Game here.
  • Another name for the game is "Number Shark."
  • After the Math Horizons paper appeared, I learned that the game was invented by Diane Resek of San Francisco State University. She writes: I came up with the game when I was working at the Lawrence Hall of Science in Berkeley from about 1969 to 1972. I was coordinating a grant Leon Henkin (UC Berkeley) and Robert Davis (I think he was at U of Illinois at that time) had from NSF to work with K-6 teachers in the Berkeley Unified School District. One of the things I tried to do was to come up with interesting ways for kids to practice their skills or their facts which would involve them in some thinking and not be so boring. The Taxman was one game I came up with for multiplication facts. It was named for the Beatle's song -"Taxman". At the same time other people were working with kids on teletype machines. They taught them Basic and had games on it for them to play. When I came up with a game or an activity, they would turn it into a program. (slightly edited).
  • John A. Trono published a short analysis of the Taxman game, comparing various strategies, in "Taxman Revisited," SIGCSE Bulletin 26:4, pp. 56-58 (Dec. 1994).
  • The "Taxman sequence" is the sequence of optimal scores as a function of pot size. It is posted at The On-Line Encyclopedia of Integer Sequences. That site includes references to other studies. Using an approach by Dan Hoey (cited there) in which the pot is represented by a directed acyclic graph, one can dramatically reduce the search space for optimal sequences. The Taxman sequence has been determined out to a pot size of 1000 by Brian Chess.
  • Norman Perlmutter, then a student at Grinnell College and later at City University of New York, contacted me to share his proof of the existence of a winning strategy for the game. He proved that a winning strategy exists, at least asymptotically (i.e., for pot sizes larger than some undetermined value). He published a paper on this topic in the Pi Mu Epsilon Journal 14:3, pp. 199-204 (2015).
  • Later I learned that Douglas Hensley had previously published an asymptotically winning strategy: “A Winning Strategy at Taxman,” Fibonacci Quarterly 26:3, 262 (1988).
  • A comparison of various heuristic strategies for playing Taxman is given here.
  • In 2021 I was contacted by Atli Fannar Franklín, then a master's student in mathematics at ETH Zürich and later a PhD student at the University of Iceland. He had proven a result concerning the difficulty of finding the optimal score for Taxman, and asked me to review it. The paper triggered some ideas of my own, and we subsequently had a very fruitful collaboration, each bringing different strengths and ideas to the problem. Eventually this collaboration resulted in a publication, "The Difficulty of Beating the Taxman." A summary of the main results in that paper is given here.
• I am also an amateur Galileo scholar. Here is a talk about Galileo that I presented as part of Fordham's College At 60 lecture series.

My Erdős Number is at most 4, via the following chain:

1. Franklín, Atli Fannar; Moniot, Robert K. The difficulty of beating the taxman. Discrete Appl. Math. 339 (2023), 166–171.
2. Claesson, Anders; Dukes, Mark Franklín, Atli Fannar; Stefánsson, Sigurđur Örn. Counting tournament score sequences. Proc. Amer. Math. Soc. 151 (2023), no. 9, 3691–3704.
3. Chung, Fan; Claesson, Anders; Dukes, Mark; Graham, Ronald. Descent polynomials for permutations with bounded drop size. European J. Combin. 31 (2010), no. 7, 1853–1867.
4. Chung, F. R. K.; Erdős, P.; Graham, R. L. On the product of the point and line covering numbers of a graph. Second International Conference on Combinatorial Mathematics (New York, 1978), pp. 597–602.

My publications