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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
😞
You tied the taxman
Score:
0-0
Your Picks:
Taxman's Take:
level 55
Jun 15, 2016
2 tries, beat him by 43
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level 47
Jun 15, 2016
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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level 61
Jun 15, 2016
47, 49 is a solid start
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level 58
Jun 15, 2016
After 3 tries I got him by 125. Hard but fun!
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level 45
Jun 15, 2016
After a few tries, i won by 3 :D
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level 45
Jun 15, 2016
11
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level 42
Jun 15, 2016
After many unsuccessful tries, I finally beat the taxman by 25!!!!!!!!
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level 42
Jun 16, 2016
Now 195!!!
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level 25
Jun 15, 2016
You can't win :(
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level 40
Feb 17, 2017
47, 49, 4, 6, 25, 46, 38, 26, 50, 40, 30, 45, 44, 24, 36, 28, 34, 48, 42
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level 20
Sep 21, 2017
47, 49, 35, 21, 39, 33, 27, 45, 26, 46, 38, 18, 42, 34, 30, 50, 28, 44, 36, 24, 48, 40 - taking always the biggest number with only two factors
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level 47
Jun 15, 2016
Finally beat him, but only by 45. I'm gonna keep trying.
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level 63
Jun 15, 2016
I beat him by 7, I feel satisfied
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level 54
Jun 15, 2016
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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level 79
Jun 16, 2016
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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level 55
Jun 16, 2016
For the math challenged (like me): I would start with high numbers with very few factors. Start with 47, the highest prime. Next, take 49 because it's only factor is 7, so after those two moves, you're winning 96-8. Then grab the numbers whose only factors are prime numbers that are already off the board. After the first selection, all the primes come off the board because their only factor is 1. So take 38, because you've already lost 19. Take 46 because you've already lost 23. Take 34 because you've already lost 17. Next, be careful about the multiples of 4 and 8. It's tempting to take 48, but then you lose 4, 6, 8, 12, 16, 24, and then, because all its factors are gone, 32. That's a bad trade. Wait until only one or two of those factors are left to grab 48. Instead, think from the middle. Take 22 before 44, because the factor 4 will keep 44 alive, but if you take 44 first, you'll lose 22. I'm sure there is a more exact science, but this has been working for me.
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level 73
Jun 16, 2016
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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level 58
Jun 16, 2016
There would be no point picking 48 and 36 because the taxman can still take those numbers anyway
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level 55
Jun 19, 2016
Just lost by 1 by basically guessing the whole thing ;~;
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level 37
Jun 20, 2016
Very creative and fun quiz!
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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 73
Jun 20, 2016
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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level 58
Nov 22, 2017
Nice write-up @Arp2600. I've confirmed that your solution is optimal by writing and running a brute force algorithm. It did in fact come up with the exact answer you stated here.
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level 72
Jul 18, 2016
Third try, won by 145! Fun game!
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level 33
Jul 22, 2016
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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level 33
Jul 22, 2016
Yup, lost by 199!
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level 34
Aug 15, 2016
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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level 65
Aug 16, 2016
But can you tie?
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level 47
Oct 31, 2016
Nope. Add all the numbers together and you get an odd number.
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level 21
Aug 24, 2016
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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level 20
Aug 25, 2016
aww
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level 17
Sep 27, 2016
I DID WITH 341 POINTS
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level 65
Oct 5, 2016
Eh?
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level 6
Oct 6, 2016
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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level 35
Oct 12, 2016
Finally won 693-582
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level 48
Oct 15, 2016
Is your score the sum of your selections?
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level 66
Oct 17, 2016
641-634. I don't know how... For now.
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level 43
Oct 20, 2016
Bet him on the 3rd try by 23
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level 37
Oct 21, 2016
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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level 37
Oct 21, 2016
So far for me biggest win is 341 and biggest loss is 633, not quite optimal :(
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level 73
Jun 27, 2017
As strange as it can seem, the worst loss (683) is obtained by using the winning strategy for a 25 grid.
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level 42
Oct 29, 2016
Is this even possible? Best score: lose by 65
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level 4
Nov 1, 2016
OMG i won (by 43)
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level 26
Nov 26, 2016
win 498 loss 684
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level 42
Dec 12, 2016
I lost by just 17... So close
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level 41
Mar 10, 2017
Same
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level 33
Dec 15, 2016
Beat it by 647
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level 16
Dec 15, 2016
Beat it by 647
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level 23
Dec 15, 2016
beat it by 647 also
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level 44
Dec 16, 2016
I beat him by 15 by randomly guessing numbers. WOW
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level 44
Dec 16, 2016
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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level 39
Dec 17, 2016
lost by 83
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level 3
Dec 27, 2016
I won by 1 (638-637)
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level 43
Jan 5, 2017
Beat him by 75
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level 20
Jan 22, 2017
Weird game😕
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level 41
Jan 31, 2017
I beat the taxman by 17! 646-629
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level 33
Feb 24, 2017
I won by 115 on my third try. No real strategy, just picking high numbers with minimal factors.
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level 48
Dec 4, 2017
There is your strategy, bud.
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level 7
Mar 4, 2017
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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level 7
Mar 4, 2017
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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level 32
Mar 8, 2017
Got a cheeky 738-537 win. AFTER ABOUT 20 TRIES!!!!
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level 28
Mar 17, 2017
I finally got it but barely. Only won 665-610.
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level 12
Mar 18, 2017
I was winning at first but then lost by 297
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level 54
Apr 1, 2017
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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level 5
Apr 6, 2017
I lost by 391
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level 13
Apr 24, 2017
I beat it by 639
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level 55
Apr 25, 2017
Dangit! Closest I can get is losing by 37
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level 65
Apr 30, 2017
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
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level 33
May 11, 2017
i finally won on like my 15th try i beat the taxman by 5
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level 18
May 18, 2017
I lost by 1 after the following sequence 47, 49, 4, 6, 25, 26, 46, 50, 45, 40, 24, 42, 33, 38, 34, 48, 36, 44
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level 20
Jul 8, 2017
I beat him by 21
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level 77
Jul 15, 2017
I beat the taxman 715-560. between that and my recent improvement on levels, I now need too be a 70 or + level player to fulfill my expectations in this website.
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level 9
Jul 15, 2017
i lost by 403 and then 355
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level 14
Jul 31, 2017
I was 75 points away from my first win ever.....
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level 14
Jul 31, 2017
Now 17
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level 53
Aug 13, 2017
Closest is -17 :((
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level 26
Sep 6, 2017
lost by 7
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level 17
Sep 14, 2017
The best win is 808-467 winning by 341 points with 47, 49, 35, 21, 39, 33, 26, 46, 38, 34, 27, 18, 42, 28, 45, 44, 36, 30, 50, 40, 32, 48 The worst loss is 296-979 losing by 683 points with 23, 25, 15, 21, 14, 22, 20, 16, 18, 24, 26, 38, 34. This is done by using the "winning" strategy as if it was a 25 number board.
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level 28
Sep 14, 2017
You know how when a number is eliminated? It would make it a lot less confusing if you crossed out the numbers that the players get in a color (say, green) that is different from the ones that the taxman gets (say red). Just a suggestion.
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level 23
Sep 21, 2017
FINALLY WON, only by 49 but still
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level 43
Sep 22, 2017
Beat the Taxman by 31
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level 41
Sep 29, 2017
I'm honestly a really bad at math, so a win by 5 points is good enough for me.
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level 41
Sep 29, 2017
"a really bad at math". I'm also bad at spelling it seems.
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level 41
Sep 30, 2017
47, 49, 25, 39, 26, 8, 50, 45, 40, 36, 22, 28, 32, 34, 38, 42, 46, 48 - I WIN BY 35. 655-620.
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level 48
Oct 6, 2017
W by 9, 642-633. I'm so proud of myself!
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level 3
Nov 1, 2017
I seem to have double or even triple the score of the taxman before I click the last number, and when I click the last number, I lose for some reason.
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level 49
Nov 11, 2017
Because he also gets all Numbers that you haven't used.
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level 6
Nov 3, 2017
My score reads '435-217'. What does that mean?
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level 49
Nov 11, 2017
You got 435 Points and he got 217.
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level 49
Nov 11, 2017
I did it by 81.
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level 20
Nov 19, 2017
Took about 21 tries lol
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level 7
Nov 30, 2017
(not a lie ) i tried 2 times and beat the taxman the 2nd time by 209!
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level 40
Dec 7, 2017
won by 5, took way too long
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level 33
Dec 12, 2017
Lost 628-647, my closest try!
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level 33
Dec 12, 2017
Now beat him 740-535
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