December 07, 2008

Taxman Game

Another gem from my neighbor Tom Sander. The "taxman" game is (apparently) an old programming exercise. But it is also a good game for practicing factors. I have been playing it a bit with Anthony (by hand). Here is a gadget that applies the rules of the game for you.

The rules are simple and fifth-grade friendly:

  • When you take a number, the taxman gets all the remaining factors.
  • You are only allowed moves that give the taxman at least one new number.
  • When you can't move any more, the taxman gets the rest.

It is worth playing without reading anything else - it is not too hard to find a heuristic that beats the taxman. The game was written up in an article by Moniot in the Feb 2007 MAA Horizons - an optimal strategy is not known.

I've gotten up to 121 points on the 20-size board; I am pretty sure this is not optimal. Can you beat the board with 100 squares? What is the best score you can get?

Posted by David at December 7, 2008 05:12 PM

David --

Highest I've been able to get to with board of 20 is 124 from drawing numbers in this order (19,14,10,20,16,15,12,18).

I suspect this is not optimal.



Posted by: Tom Sander at December 9, 2008 08:26 AM

Hm, I'm pretty sure 124 is optimal!

Posted by: David at December 11, 2008 09:22 PM

I got 2957 v 2093 using the "greedy" algorithm. I a fairly certain it isn't optimal.


Posted by: John Kemeny at January 5, 2009 04:09 PM

what's the highest possible score for Taxman 24?

Posted by: dude wheres my car? at November 4, 2009 06:57 PM

Im pretty sure the Highest possible score in Taxman is something like 1227. Thats the highest I've gotten with my friends soo... Yah...

Posted by: Dude What's mine say? at November 4, 2009 07:01 PM

isn't that..... impossible?

Posted by: dude wheres my car? at November 4, 2009 07:03 PM


Posted by: at April 22, 2010 01:41 PM

I just got Score: 3006 Taxman: 2044

Posted by: dustin at July 13, 2010 05:46 PM

124 must be optimal. 19 is the correct first move as it is the largest prime on the board. Since 11, 13, and 17 are also prime and larger than half of 20, we cannot choose those, so the taxman gets them always. After the first move, there are 15 numbers that we can choose from left on the board, as 1, 11, 13, 17, and 19 are now gone. Since the taxman must receive at least one number each time we choose a number, the greatest number of picks that we have left is the floor of 15/2, which is 7. Thus we want 19 and the highest remaining 7 numbers, excluding 11, 13, and 17. The highest 7 remaining numbers would be, in descending order, 20, 18, 16, 15, 14, 12, 10. 19+20+18+16+15+14+12+10=124. Since there is a sequence of choices that will actually give us this outcome, that score is optimal.

Posted by: tony at April 2, 2011 12:28 PM

Did you know that you can type in 999 and it will make a board with 999 numbers. You can get scores over 25000

Posted by: will at April 12, 2011 08:46 AM

Impossible to win on 3 board.

Posted by: Bacon at January 11, 2012 09:26 AM

Yeah, 999 is hard. And yes, I knew that.

Posted by: Bacon at January 11, 2012 09:28 AM

Woo cares tony

Posted by: Rinder at February 20, 2012 09:46 AM

I'm not tony

Posted by: Bacon at May 16, 2012 09:46 AM

easy to beat 2 taxman hard on 500

Posted by: Ramsey at May 31, 2012 10:05 AM


Posted by: ian at July 2, 2012 03:15 PM

love it!

Posted by: at July 31, 2012 11:12 AM

love this game

Posted by: arav at July 31, 2012 11:13 AM

First move should always be the largest prime number on the board.

Posted by: Pie Man at October 30, 2012 02:06 PM

Check here for scores taxman games up to n=50.

Alex (Germany)

Posted by: Alex at March 6, 2013 06:24 AM

The lowest number you can win at is 4

Posted by: STOP READING MY NAME at August 19, 2013 09:13 AM

Got 28403 on 947 board

Posted by: at April 1, 2014 04:02 PM


Posted by: at April 1, 2014 04:02 PM

Hey guys I'm doing this for my assignment at uni. I'd done the greedy method and according to my mate, it's optimal. Basically taking the largest number each time with the smallest divisor (there's always a number with at 1 divisor). This display really helped me out as I slowly developed and tested this with this board.

For 100, I got 2970 - 2080

For 24, I got 178 - 122

For 947, I got 275607 - 173271

Mind telling me how you did it Mr.28403?

3 is a tie and actually the smallest number you can win with is 2.

Posted by: Butters at April 21, 2016 12:45 AM

The link to doesn't work anymore. Currently, works.

Posted by: Mark S. at May 31, 2020 01:30 PM

I <3 Butter

Posted by: Kiddie at October 27, 2020 02:15 PM

Posted by: Biswaprakash sia at November 7, 2020 06:37 AM

I won with 100: 2663 to 2387

Posted by: Ben at November 21, 2020 09:15 PM

i got 1,2,3,4,5,6,7,8,9,10
14,15,16 18 20

Posted by: Maritza Mendoza at October 6, 2021 01:08 PM

3079:1921 100 by 100 board

Posted by: at October 12, 2021 10:16 PM

Love this game - tried it with my Grade 8 and 9 Math classes. They went the whole period trying to figure out better pathways; even the weaker students were getting involved. I got 775 for a board of 50 (first go). Towards the end there seemed to be a pattern of taking 2 for every Taxman's one as we finished off the numbers - I call this the 2:1 (2-fer) strategy.

Posted by: Jim Davies at February 16, 2022 01:01 AM

P_revious comment updated - Board of 50 - score Me 750 : Taxman 500.

Posted by: Jim Davies at February 16, 2022 01:03 AM

:( thank you very much so helpful ......... ):

Posted by: at April 12, 2022 07:51 AM

On board 30 I got 274 pretty sure that the optimal score. This game is so fun!

Posted by: Curtis at June 6, 2022 04:19 PM

274 is not optimal for a board of size 30; I got 286.

Posted by: AG at July 10, 2022 08:32 AM

this is really fun and im really good

Posted by: at October 18, 2022 10:58 AM

I made a program to do any number of the taxman problem and I think it works well. I did the 999-length one and there's a list of 429 options in a row with a final score of 113,948 better then the taxman

Posted by: Hortonbear at March 9, 2023 10:22 AM

I'm not so good at the game.


Posted by: Eska at April 10, 2023 02:15 PM


Posted by: Eska at May 8, 2023 02:42 PM

I got 21 points on an 8 size board!

Posted by: at November 6, 2023 02:11 PM

I beat the taxman 2756 to 2294 on a 100-square board (1st attempt) :D

Posted by: Kaleb at March 17, 2024 11:04 PM

New best on a 100-square board: 2808 (me) to 2242 (taxman) yay

Posted by: Kaleb at March 17, 2024 11:08 PM

New best on a 100-square board: 2812 (me) to 2238 (taxman)

Posted by: Kaleb at March 24, 2024 04:32 PM


Posted by: at April 26, 2024 10:08 AM
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