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Taxman (mathematical game)

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Taxman, also known as Tax Factor, Number Shark, The Factor Game, Factor Blast, Factor Blaster, or Dr. Factor, is a mathematical game invented by mathematician Diane Resek.

Description

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The game is played between two players on a board consisting of whole numbered tokens labeled 1 through N, where N is any positive whole number. During each turn, one player (deemed the tax payer) takes a number from the board, and the other player (deemed the taxman) removes all remaining factors of the tax payer's number from the board. The game ends when there are no legal moves left, and each player's score is calculated by adding up the values of the numbers they have collected. The player with the highest score wins.[1][2][3][4][5][6][7][8]

Single-Player Version

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In the single-player version, (Taxman, Tax Factor, Number Shark), the human player assumes the role of the tax payer each turn while the computer player is always the taxman. In addition, the human player may only collect a number that still has proper factors remaining on the board. When there are no legal moves left, the taxman collects all of the remaining tokens on the board.[1][2][3]

Two-Player Versions

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In all two-player versions of the game, (The Factor Game, Factor Blast, Factor Blaster, Dr. Factor), the two players swap roles each turn, so that whoever is playing as the taxman during one turn will be the tax payer during the next turn, and vice versa.[4][5][6][7][8]

  • In some versions, the tax payer may only collect a number that still has proper factors remaining on the board. If a player picks a number incorrectly, they may or may not lose their next turn, and they may or may not keep the number.[8]
  • In some versions, the taxman may neglect to collect all of the factors of the tax payer's number, or may attempt to collect a factor incorrectly. The taxman may or may not lose points for missing factors or choosing incorrectly, and the tax payer may or may not be able to steal a factor that the taxman misses.[4][5][6][8]
  • In some versions, there is the option to remove one or more numbers from the board before the game starts.[4][5][6]

Origin and Spread

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Taxman was invented by Resek sometime in the late 60's or early 70's while working at the Lawrence Hall of Science.[9] It was published as a BASIC program in the September 1973 issue of the People's Computer Company Newsletter,[1] and later appeared in the 1975 programming anthology book What to Do After You Hit Return.[10]

In 1980, Taxman appeared as part of the software collection MECC - Elementary Volume 1 for the Apple II.[11][12] The concept was later reused in other MECC titles, such as Wonderland Puzzles (as Hedgehog Croquet) and The Secret Island of Dr. Quandary (as Tax Factor) in 1992.[13][14][2][15]

Starting in 1984, Taxman appeared as a coding exercise in a series of programming textbooks written by Lowell Carmony, a professor at Lake Forest College (and Berkeley alumnus).[16][17][18][19] Carmony was part of the writing group for the 1993 NRC publication Measuring Up: Prototypes for Mathematics Assessment, which included Taxman as one of its prototypes.[20] Carmony also described Taxman in an article for SIGCSE.[21]

In 1996, a list of the best possible scores in Taxman, (called the Taxman sequence), was uploaded to the On-Line Encyclopedia of Integer Sequences.[22] As of 2022, the sequence has been calculated out to a board size of 1000.[23]

Around 2000, a version of Taxman was uploaded to the NRW's learn:line educational server under the name Der Zahlenhai (or Number Shark in English).[3] A version of Number Shark was later added to CrypTool in 2006.[24]

In 2015, Taxman appeared in the New York Times' Numberplay column as The Tax Collector.[25]

Two-Player Versions

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A two-player version of Taxman, known simply as The Factor Game, was described in an article for the November 1973 issue of The Arithmetic Teacher, a publication of the National Council of Teachers of Mathematics.[26] The article was later reprinted in the 1975 anthology Games and Puzzles for Elementary and Middle School Mathematics.[4]

In 1983, Factor Blast by Joe DeMuth was published by Hayden Software.[27][5] Around 2000, educator Terry Kawas developed teaching materials for a similar variant called Factor Blaster which was later uploaded to Mathwire, a math education resource website.[28][6][29]

In 1985, Dr. Factor appeared as one of four games in Playing To Learn by Antonia Stone, Joshua Abrams, and Ihor Charischak of HRM Software.[30][31][7]

In 1986, another variant, also called The Factor Game, appeared as the first activity in the Factors and Multiples module of the Middle Grades Mathematics Project curriculum, and later appeared as part of the Connected Mathematics Project in 1996.[32][33][34] Interactive versions were developed for Macintosh and Windows, and eventually a web version was developed for the NCTM's Illuminations website in 2001.[35][36]

In Education

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Taxman and its variants have been studied and used as tools in mathematics and computer science education.[37][21][38][20][26][39]

Analysis

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The winnability, strategy, and optimal score for the single-player version of Taxman have been studied.[40][21][38][22][41][42]

References

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  1. ^ a b c "Taxman". People's Computer Company. Vol. 2, no. 1. September 1973. pp. 6–7.
  2. ^ a b c MECC (1992). The Secret Island of Dr. Quandary. Level/area: Tax Factor. 'I'll take some of the tokens from your bag and put them out on the trail. Next, you pick up one of the tokens. Then, I'll pick up all the FACTORS of the token you picked up... High score wins... You can't take that token. No other tokens are factors of that token... That's it! The west are mine!!'
  3. ^ a b c Carl, Lothar (October 12, 2000). "Der Zahlenhai" [Number Shark]. learn:line - Bildungsserver NRW (in German). Archived from the original on February 10, 2001.
  4. ^ a b c d e National Council of Teachers of Mathematics (1975). "The factor game". Games and Puzzles for Elementary and Middle School Mathematics - Readings from the Arithmetic Teacher (PDF). pp. 145–147.
  5. ^ a b c d e Joe DeMuth (1983). Factor Blast (Manual). Lowell, Massachusetts: Hayden Software Company.
  6. ^ a b c d e Kawas, Terry. "Games to Practice Multiplication Facts". Mathwire. Archived from the original on May 8, 2021.
  7. ^ a b c Elliott, John C. (October 1987). "Reviewing and Viewing Computer Materials - Playing To Learn: Math/Logic Games". The Arithmetic Teacher. 35 (2): 57.
  8. ^ a b c d "Factor Game". NCTM Illuminations. Archived from the original on August 7, 2023.
  9. ^ Moniot, Robert (2008). "Robert Moniot's Research". Fordham University. 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.'
  10. ^ People's Computer Company (January 1975). What to Do After You Hit Return. Menlo Park, California: Nowels Publications. pp. 112–113, 153.
  11. ^ MECC (1981). MECC - Elementary Volume 1 (Apple II) (3.4 ed.).
  12. ^ "Educational Excellence". Creative Computing. Vol. 8, no. 1. Morristown, New Jersey: Creative Computing. January 1982. p. 61.
  13. ^ MECC (1992). Wonderland Puzzles (Apple II). Level/area: Hedgehog Croquet. 1. You may take any numbered hedgehog if it has a factor (other than itself) showing. 2. The queen gets all the factors of your number. 3. The queen gets all the leftover hedgehogs.
  14. ^ Brown, Cathy; Clements, Douglas H. (October 1993). "Reviewing and Viewing Computer Materials - Wonderland Puzzles". The Arithmetic Teacher. 41 (2): 121–122. doi:10.5951/AT.41.2.0121.
  15. ^ David Sears (September 1992). "Discovery Choice - The Secret Island of Dr. Quandary". Compute!. Vol. 14, no. 8 #144. Broadway, New York: COMPUTE Publications International. pp. 82–83.
  16. ^ Carmony, Lowell A.; McGlinn, Robert J.; Millman, Ann Miller; Becker, Jerry P. (1984). Problem Solving in Apple Pascal. Rockville, Maryland: Computer Science Press. pp. 175–185. ISBN 0-88175-006-9.
  17. ^ Carmony, Lowell A.; Holliday, Robert L. (1985). Macintosh Pascal. Rockville, Maryland: Computer Science Press. pp. 433–438. ISBN 0-88175-081-6.
  18. ^ Carmony, Lowell A.; Holliday, Robert L. (1990). A First Course in Modula 2. New York, New York: Computer Science Press. pp. 364–367. ISBN 0-7167-8229-4.
  19. ^ "Obituary for Lowell A. Carmony". Davenport Family Funeral Homes and Crematory. September 2020. Archived from the original on November 25, 2023.
  20. ^ a b National Research Council (1993). Measuring Up: Prototypes for Mathematics Assessment. Washington, DC: National Academy Press. pp. vii, 101–114. doi:10.17226/2071. ISBN 0-309-04845-1.
  21. ^ a b c Carmony, Lowell A.; Holliday, Robert L. (March 1993). "An example from artificial intelligence for CS1". ACM SIGCSE Bulletin. 25 (1): 1–5. doi:10.1145/169073.169077.
  22. ^ a b Sloane, N. J. A. (ed.). "Sequence A019312 (Taxman Sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  23. ^ Chess, Brian (September 10, 2022). "Optimal Taxman scores for N=1 through N=1000". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Archived from the original on April 30, 2023.
  24. ^ Esslinger, Bernhard (July 27, 2006). "Readme". CrypTool. Archived from the original on February 20, 2007.
  25. ^ Antonick, Gary (April 13, 2015). "The Tax Collector". Numberplay. The New York Times. Archived from the original on April 30, 2023.
  26. ^ a b Harkin, J. B.; Martin, D. S. (November 1973). "The factor game". The Arithmetic Teacher. 20 (7): 580–582. doi:10.5951/AT.20.7.0580.
  27. ^ Banks, Michael A. (January 1984). "Marketalk Reviews - Factor Blast". Softalk. Vol. 4, no. 5. North Hollywood, California: Softalk Publishing Inc. pp. 156–157.
  28. ^ Kawas, Terry (2000). "Factor Blaster Directions" (PDF). Mathwire. Archived from the original (PDF) on May 13, 2006. Kawas, 2000
  29. ^ Stoloff, D. L. (March 2011). "Mathwire". Choice: Current Reviews for Academic Libraries. Vol. 48, no. 7. American Library Association CHOICE. p. 1347.
  30. ^ Martinez, Marcy A. (November 1986). "Courseware Reviews - Playing To Learn: Math/Logic Games". The Mathematics Teacher. 79 (8): 665–666.
  31. ^ Steiner, Carol (November 1986). "Reviewing and Viewing Computer Materials - Playing To Learn: Math/Logic Games". The Arithmetic Teacher. 34 (5): 40–41.
  32. ^ Huinker, DeAnn (September 1987). "Reviewing and Viewing New Books - Middle Grades Mathematics Project: Factors and Multiples, Mouse and Elephant, Similarity and Equivalent Fractions, Spatial Visualization, Probability". The Arithmetic Teacher. 35 (1): 54–55.
  33. ^ Fitzgerald, William M.; Boyd, Jane U. (March 1994). "A number line with character". The Arithmetic Teacher. 41 (7): 368–369. doi:10.5951/AT.41.7.0368.
  34. ^ Connected Mathematics Project. "Prime Time". Michigan State University. Archived from the original on November 12, 1996.
  35. ^ Connected Mathematics Project. "Connected Mathematics Project Software". Michigan State University. Archived from the original on November 12, 1996.
  36. ^ National Council of Teachers of Mathematics (March 24, 2001). "The Factor Game: Credits". NCTM Illuminations. Archived from the original on April 20, 2001.
  37. ^ O'Brien, Thomas C. (September 1983). "Software of the Second-and-a-Half Kind". Classroom Computer Learning. 4 (2): 33–34, 36.
  38. ^ a b Trono, John A. (December 1994). "Taxman revisited". ACM SIGCSE Bulletin. 26 (4): 56–58. doi:10.1145/190650.190663. S2CID 2445603.
  39. ^ Michigan State University Department of Mathematics (1988). The Middle Grades Mathematics Project. The Challenge: Good Mathematics--Taught Well. Final Report to the National Science Foundation (PDF) (Report). pp. 140–142.
  40. ^ Hensley, Douglas (August 1988). "A Winning Strategy at Taxman" (PDF). The Fibonacci Quarterly. 26 (3): 262–270.
  41. ^ Moniot, Robert K. (February 2007). "The Taxman Game" (PDF). Math Horizons. 14 (3): 18–21. doi:10.1080/10724117.2007.11974693. S2CID 115624319.
  42. ^ Franklín, Atli Fannar; Moniot, Robert K. (November 15, 2023). "The difficulty of beating the Taxman". Discrete Applied Mathematics. 339: 166–171. arXiv:2211.00461. doi:10.1016/j.dam.2023.06.019.
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