# Number Factoring Game

This is an old math exercise often called the "Taxman". Here's how it works.
• Pick a number from the board

• You get the value of the number

• Your opponent, the taxman, gets all the factors of that number that remain on the board

• When no more numbers remain that have factors, the taxman gets all those too!

Think you can beat the taxman? Start by clicking a number!
Congratulations!
You beat the taxman by .
You lost by
ðŸ˜ž
You tied the taxman
Score:
0-0
Taxman's Take:
After many tries, I beat the taxman by 303. I could write a computer program to find the perfect answer, but that would ruin the fun! Is it possible to beat the taxman by more than 303?
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Jun 14, 2016
(59)
Fun! Yes, there's a way to beat him by 325 :)
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Jun 14, 2016
Impressive! I can't figure out how to do it.
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Jun 15, 2016
Another challenge is to see how much you can lose by. The most I am able to lose by is 541, but I think it's possible to do worse.
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Jun 15, 2016
(58)
2, 8, 6, 28, 25, 22, 20, 26, 18, 32, 24, 34, 38, 42, 46, 45 = -443 (worst)
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Jun 15, 2016
(72)
It's possible to lose by 633 points.
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Jun 15, 2016
(61)
I managed to lose by 549, my biggest loss so far.
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Jun 16, 2016
(7)
I lost by 435 (420-855) by picking 2, 48, 10, 14, 18, 22, 26, 30, 34, 38, 40, 42, 46, 50
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Mar 4, 2017
(72)
I found a 808-467 solution, so a difference of 341. Still not quite sure it's the best possible.
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Jun 15, 2016
(72)
After some research, I am now convinced that 808-467 (+341) is the best win and that 296-979 (-683) is the worst loss.
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Jun 16, 2016
(61)
Wow! I managed 800-475 for a 325 point win by always picking the largest number with only one factor left. How did you push it up to 808-467?
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Jun 16, 2016
(72)
I don't know how you did it, but I would say... try taking 28 instead of 20.
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Jun 16, 2016
(61)
Actually I see where my algorithm fails to be optimal. Near the end it basically trades 32 for 24 unnecessarily. I wonder if anyone has proven an optimal algorithm for this game.
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Jun 16, 2016
(72)
Maybe you take out 48 too early ?
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Jun 16, 2016
(61)
No, the last three strictly following my algorithm are 24, 48, 40, which is not as good as instead picking 32, 48, 40 which gets my score to match yours.
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Jun 16, 2016
(72)
So did you do it? I think you can just take 32 instead of 24...
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Jun 16, 2016
(61)
Yeah I did. I'm just curious as to whether or not there is an algorithm that will always give you the correct result (ie if the quiz went to 75 instead of 50).
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Jun 16, 2016
(45)
Deddrk, I used your algorithm and only won by 149 points.
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Jun 19, 2016
I used @Deddryk's algorithm and lost by 253.
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Jun 20, 2016
(37)
Yep, I agree. got 808-467, quite challenging, nice job in finding highest and lowest.
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Oct 21, 2016
(40)
There's a way to beat him by 17! 646-629
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Feb 3, 2017
(58)
To relate back to the JetPunk quiz area, could this perhaps earn you points, dspending on how many points you won by? Very cool game.
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Jun 14, 2016
(60)
By randomly choosing numbers, I only lost by 99, my closest
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Jun 15, 2016
(35)
lol, thats my best score actually trying
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Jun 24, 2017
(65)
I tried it about a dozen times before winning by 61.
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Jun 15, 2016
(1)
YES I beat him by 103 points, my trick was to start out by avoiding any numbers that are divisible by 2,3, and 4, which mostly left numbers only divisible by 5, and 7 then gradually started with numbers that were divisble by 3 then finally the ones with 2. But you always have to make sure the number you choose is the largest possible, always compare with the numbers in the diagonal row with that number. it took about 10 mins for my game so be patient
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Jun 15, 2016
(60)
I did your way, and managed to beat taxman by 1
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Jun 20, 2016
(61)
Took a few tries, but it was my 4th or 5th I finally won, by 49.
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Jun 15, 2016
(61)
Wow. That was close. Beat him by 1, 638-637. Makes it even sweeter.
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Jun 15, 2016
(58)
2nd try. I beat the taxman by 87 points. It was surprisingly mind-challenging.
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Jun 15, 2016
(35)
HOW DO YOU DO IT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
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Jun 15, 2016
(46)
Love this game!!
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Jun 15, 2016
(48)
After few times, I won by 21 points ;) 648-627
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Jun 15, 2016
(59)
first try 638-637 dub. Phew!
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Jun 15, 2016
(72)
That's another challenge indeed. How to make the closest win and the closest loss.
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Jun 16, 2016
(72)
Ah ha, found both 637-638 and 638-637. It's a bit less logical than the extreme cases, but still fun.
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Jun 16, 2016
(55)
2 tries, beat him by 43
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Jun 15, 2016
(47)
I know the first number should be the highest prime number as you pretty much have to lose all the prime numbers... Took me about 5 goes to win by 25
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Jun 15, 2016
(61)
47, 49 is a solid start
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Jun 15, 2016
(58)
After 3 tries I got him by 125. Hard but fun!
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Jun 15, 2016
(46)
After a few tries, i won by 3 :D
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Jun 15, 2016
(46)
11
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Jun 15, 2016
(42)
After many unsuccessful tries, I finally beat the taxman by 25!!!!!!!!
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Jun 15, 2016
(42)
Now 195!!!
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Jun 16, 2016
(25)
You can't win :(
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Jun 15, 2016
(40)
47, 49, 4, 6, 25, 46, 38, 26, 50, 40, 30, 45, 44, 24, 36, 28, 34, 48, 42
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Feb 17, 2017
(46)
Finally beat him, but only by 45. I'm gonna keep trying.
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Jun 15, 2016
(60)
I beat him by 7, I feel satisfied
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Jun 15, 2016
(54)
Beat him by 21 points in three tries. It was easy to win, but I'm more interested in figuring out the algorithm to beat him consistently instead of winging it on what seemed like a good choice. Obviously that was good enough to work, but it isn't the true answer to this problem.
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Jun 15, 2016
(79)
It seems that there is one important piece of information missing from the directions--that when a number has no more factors (regardless of whether you have selected it or a number it is a factor of) it becomes unavailable to pick. I could not figure out why when I selected 36, 24 was eliminated...
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Jun 16, 2016
(53)
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Jun 16, 2016
(72)
47-49 is the obvious beginning. After that, what you say is a winning strategy, but the best is a bit subtler.
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Jun 16, 2016
(58)
There would be no point picking 48 and 36 because the taxman can still take those numbers anyway
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Jun 16, 2016
(55)
Just lost by 1 by basically guessing the whole thing ;~;
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Jun 19, 2016
(36)
Very creative and fun quiz!
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Jun 20, 2016
(72)
Ok, a few days have passed, so let's explain how to make the maximum. So : SPOILER ALERT. If you don't want to know, don't read...
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Jun 20, 2016
(72)
As already said above, there are a few rules to follow, basically to take only numbers with one divisor left. The first number will always take the divisor 1, so you have to begin with a prime number, logically the largest available : 47. After that, all of the other prime numbers become grey, which means you can't take them. It's not a problem for those inferior to 25, you can use them to grab their doubles later. But 29, 31, 37, 41, 43 will remain on the board and will be taken by the taxman in the end.
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Jun 20, 2016
(72)
Those 5 numbers are lost, thus only 45 can be used, which makes a maximum of 22 pairs, plus a wasted number. Either the taxman takes a sixth number in the end or you pair a number with two divisors instead of one at some point, but 22 is definitely the maximum number of numbers you can take during the game. This sets an upper bound to the score : the sum of the 22 biggest numbers is 818.
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Jun 20, 2016
(72)
So you have to try to take most of the biggest numbers, which requires a bit of thinking and a good ordering. Deddryk's method is almost perfect, because it's what you want to do : pick the highest numbers with only one divisor left... So, after 47, you must take the biggest square of a prime number, namely 49. After that you have to take the highest multiple of 7 and another prime number : 35.
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Jun 20, 2016
(72)
The next step is a little bit trickier... the fact is that, at this stage, all numbers greater than 21 have at least two divisors left. You would want to remove something larger than that but you can't really do that. 21 is the best choice, better than 15 anyway (both 21 and 15 remove only the divisor 3, but... 21 is bigger). After 21, things become more open... 27, 33 and 39 remove only 9, 11 and 13 respectively, and when 9 is gone, you can also take 45.
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Jun 20, 2016
(72)
After that, it's time to remove 2. The best number to do that is 26, since 13 is already taken, (and you will need 22 to remove 44). Then you can take all of the remaining doubles of prime numbers left : 46, 38 and 34. So, if we recapitulate, we have removed 12 big numbers : 47-49-35-21-39-33-27-45-26-46-38-34.
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Jun 20, 2016
(72)
Then, the most frustrating step... at this stage, most numbers have two or more divisors left, so you have to take a rather low number... the best thing to do is to remove 18 with the divisor 6. Back to the big ones next : 42 with 14, 28 with 4, 30 with 10, 44 with 22, 50 with 25, 36 with 12.
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Jun 20, 2016
(72)
Apart from the high primes, there are 7 numbers left on the board, that can be paired to make the best score : 8, 16, 20, 24, 32, 40, 48. You can either take 32 or 40, then the other, they will remove the divisors 8, 16 and 20. And finally you reach the score 808 by taking 48 with 24. Up to a few possible permutations, the solution is : 47-49-35-21-39-33-27-45-26-46-38-34-18-42-30-28-50-44-36-40-32-48
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Jun 20, 2016
(72)
Third try, won by 145! Fun game!
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Jul 18, 2016
(33)
I won by 33 on my first attempt, without really understanding what I was doing! Beginner's luck. Having read the comments I now have a clearer idea, so will try again...and probably lose drastically!
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Jul 22, 2016
(33)
Yup, lost by 199!
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Jul 22, 2016
(34)
47, 49, 25, 39, 26, 8, 50, 45, 40, 46, 38, 34, 12, 22, 28, 32, 36, 42, 48...667-608 win
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Aug 15, 2016
(62)
But can you tie?
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Aug 16, 2016
(47)
Nope. Add all the numbers together and you get an odd number.
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Oct 31, 2016
(21)
47, 49, 35, 21, 39, 33, 27, 45, 26, 46, 38, 34, 18, 42, 30, 50, 28, 44, 40, 36, 32, 48 - 808-467. I thought I messed it up when I clicked the 40 when it had 2 factors out (8 and 20). Win by 341.
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Aug 24, 2016
(19)
aww
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Aug 25, 2016
(18)
I DID WITH 341 POINTS
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Sep 27, 2016
(59)
Eh?
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Oct 5, 2016
(6)
After a few fails, I finally beat the taxman by 17: 646-629. I used billzs protip and started with 47, 49.
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Oct 6, 2016
(34)
Finally won 693-582
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Oct 12, 2016
(47)
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Oct 15, 2016
(64)
641-634. I don't know how... For now.
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Oct 17, 2016
(33)
Bet him on the 3rd try by 23
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Oct 20, 2016
(37)
For those who are suffering, here is the solution. If i get complaints that I am giving it to them, I will delete my post. Quizmaster can remove it anytime if he feels like it, I don't mind. This is how to get 808-467 (+341) 47 49 35 21 39 33 27 45 26 46 38 34 18 42 30 28 44 40 36 32 48
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Oct 21, 2016
(37)
So far for me biggest win is 341 and biggest loss is 633, not quite optimal :(
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Oct 21, 2016
(41)
Is this even possible? Best score: lose by 65
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Oct 29, 2016
(4)
OMG i won (by 43)
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Nov 1, 2016
(26)
win 498 loss 684
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Nov 26, 2016
(42)
I lost by just 17... So close
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Dec 12, 2016
(40)
Same
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Mar 10, 2017
(33)
Beat it by 647
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Dec 15, 2016
(15)
Beat it by 647
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Dec 15, 2016
(23)
beat it by 647 also
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Dec 15, 2016
(44)
I beat him by 15 by randomly guessing numbers. WOW
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Dec 16, 2016
(44)
My picks (Backwards): 23 28 92 72 83 81 54 44 94 82 43 93 63 24 64 04 05 84
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Dec 16, 2016
(35)
lost by 83
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Dec 17, 2016
(3)
I won by 1 (638-637)
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Dec 27, 2016
(36)
Beat him by 75
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Jan 5, 2017
(16)
Weird gameðŸ˜•
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Jan 22, 2017
(40)
I beat the taxman by 17! 646-629
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Jan 31, 2017
(34)
I won by 115 on my third try. No real strategy, just picking high numbers with minimal factors.
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Feb 24, 2017
(7)
I lost by only 75, by picking 47, 46, 49, 9, 36, 38, 39, 42, 50, 48, 34, 45, 33, 44, 40 (in that order). I know that the best first number is 47 because the score will become 47-1, the biggest advantage.
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Mar 4, 2017
(7)
SPOILER ALERT! THIS COMMENT REVEALS A WAY TO BEAT THE TAXMAN BY 159! I beat the taxman by 159! I used 47, 49, 35, 21, 33, 22, 44, 42, 18, 36, 46, 39, 45, 38, 34, 30, 48, 40, 50
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Mar 4, 2017
(32)
Got a cheeky 738-537 win. AFTER ABOUT 20 TRIES!!!!
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Mar 8, 2017
(23)
I finally got it but barely. Only won 665-610.
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Mar 17, 2017
(12)
I was winning at first but then lost by 297
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Mar 18, 2017
(46)
47, 49, 4, 6, 25, 46, 48, 38, 26, 50, 40, 30, 45, 44, 36, 28, 34, 42 Beat him by one this way
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Apr 1, 2017
(5)
I lost by 391
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Apr 6, 2017
(13)
I beat it by 639
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Apr 24, 2017
(54)
Dangit! Closest I can get is losing by 37
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Apr 25, 2017
(62)
At last, I beat him by 77. I think I could have done better. As a tip, start with 47 of course and then probably 49 35 50 15 45