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Supplements to Articles in The American Mathematical Monthly



“Early Round Bluffing in Poker”

This directory includes an Excel spreadsheet, (Poker Solver Examples), which I used for most of the calculations in the paper. You can use it to calculate equilibrium strategies for games with volatility.

This directory also contains several quick and dirty routines for other calculations. They're written in ruby, which comes pre-installed on many computers. To see if it's on your machine, type "ruby -v" in a terminal window. You can download ruby from

One program (Many Game Values) calculates the VNMplus game value to Player 1 for pre-calculated equilibrium strategies at a variety of r values. Another (r to Home = r_to_hom.rb) translates r value to H O M.

To play around with various strategies, you can use the program (Try a Strategy) To run it, type "ruby try_a_strategy.rb" from the command line.

The program asks you to input values for b_bet, c_call, d_bluff, and r_value. It calculates the game value. For example if you have a situation with relative volatility of 0.6, and Player 2 (she) is only calling 30% of the time, Player 1 (he) might get very aggressive (b=0.5, c=0.3, d=0.8, r=0.6), giving a game value (average win for him) of 0.557 units.

Player 2 might realize her mistake and adjust her strategy to call 60%, changing the game value to 0.478. Maybe at this point Player 1 drops back to b=0.3, d=0.9, giving a game value of 0.529. This is close to the equilibrium strategy (b=0.25, c=0.44, d=0.92, r=0.6), which has a game value of 0.530.

~ California Jack Cassidy

All Ruby Programs:

Game Value = game_value_by_r.rb

Game = game.rb

Hand Iterators = hand_iterators.rb

Many Game Values = many_game_values.rb

R to Home = r_to_hom.rb

Strategy = strategy.rb

Try a Strategy = try_a Strategy.rb

Values by Strategy = values_by_strategy.rb


J. M. Borwein, S. T. Chapman, I Prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly, 122(2015). 

PI Star Papers Referred to in "A Brief History and Anthology of Articles in the American Mathematical Monthly”

1. 133 citations: J. M. Borwein, P. B. Borwein, D. H. Bailey, Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi, 96(1989) 201–219.


2. 119 citations: G. Almkvist, B. Berndt, Gauss, Landen, Ramanujan, the arithmeticgeometric mean, ellipses, π, and the ladies diary, 95(1988) 585–608.

3. 73 citations: A. Kufner, L. Maligrand, The prehistory of the Hardy inequality, 113(2006) 715–732.

4. 63 citations: J. M. Borwein, P. B. Borwein, K. Dilcher, Pi, Euler numbers, and asymptotic expansions, 96(1989) 681–687.

5. 56 citations: N. D. Baruah, B. C. Berndt, H. H. Chan, Ramanujan’s series for 1/π: a survey, 116(2009) 567–587.

6. 40 citations: J. Sondow, Double integrals for Euler’s constant and ln π/4 and an analog of Hadjicostas’s formula, 112(2005) 61–65.

7. 39 citations: D. H. Lehmer, On arccotangent relations for π, 45(1938) 657–664.

8. 39 citations: I. Papadimitriou, A simple proof of the formula , 80(1973) 424–425.

9. 36 citations: V. Adamchik, S. Wagon, A simple formula for π, 104(1997) 852–855.

10. 35 citations: D. Huylebrouck, Similarities in irrationality proofs for π, ln2, ζ(2), and ξ(3), 108(2001) 222–231.

11. 35 citations: L. J. Lange, An elegant continued fraction for π, 106(1999) 456–458.

12. 33 citations: S. Rabinowitz, S. Wagon, A spigot algorithm for the digits of π, 102(1995) 195–203.

13. 32 citations:W. S. Brown, Rational exponential expressions and a conjecture concerning π and e, 76(1969) 28–34.


June/July issue

Supplement to "Record Crossword Puzzles" by Kevin K. Ferland: "Proof of Crossword Puzzle Record"