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:
+1
level ∞
Jun 14, 2016
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?
+1
level 61
Jun 14, 2016
Fun! Yes, there's a way to beat him by 325 :)
+1
level ∞
Jun 15, 2016
Impressive! I can't figure out how to do it.
+1
level ∞
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.
+1
level 57
Jun 15, 2016
2, 8, 6, 28, 25, 22, 20, 26, 18, 32, 24, 34, 38, 42, 46, 45 = -443 (worst)
+1
level 75
Jun 15, 2016
It's possible to lose by 633 points.
+1
level 59
Jun 16, 2016
I managed to lose by 549, my biggest loss so far.
+1
level 25
Sep 14, 2017
I managed to lose by 683 with 23, 25, 15, 21, 14, 22, 20, 16, 18, 24, 26, 38, 34
+1
level 9
Apr 30, 2018
So close! Lost by 31 by picking 47, 49, 25, 50, 46, 44, 39, 40, 34, 38, 27, 12, 48, 42, 45, 36
+1
level 9
Apr 30, 2018
Just picked random numbers and won by 117. SPOILER ALERT 47, 49, 35, 6, 28, 33, 46, 42, 27, 24, 30, 39, 40, 44, 50, 48, 38, 34, 36
+1
level 39
Jun 28, 2018
TLThe Legend I tried your numbers and won by 17!
+1
level 22
Jul 2, 2018
After using @TLtheLegend’s numbers, I won by 117. Whoopee!
+1
level 15
Jul 12, 2018
I lost by 3. This is frustrating. I think I'm going to win but the taxman's numbers just go shooting up
+1
level 75
Jun 15, 2016
I found a 808-467 solution, so a difference of 341. Still not quite sure it's the best possible.
+1
level 75
Jun 16, 2016
After some research, I am now convinced that 808-467 (+341) is the best win and that 296-979 (-683) is the worst loss.
+1
level 62
Jun 16, 2016
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?
+1
level 75
Jun 16, 2016
I don't know how you did it, but I would say... try taking 28 instead of 20.
+1
level 62
Jun 16, 2016
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.
+1
level 75
Jun 16, 2016
Maybe you take out 48 too early ?
+1
level 62
Jun 16, 2016
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.
+1
level 75
Jun 16, 2016
So did you do it? I think you can just take 32 instead of 24...
+1
level 62
Jun 16, 2016
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).
+1
level 52
Jun 19, 2016
Deddrk, I used your algorithm and only won by 149 points.
+1
level ∞
Jun 20, 2016
I used @Deddryk's algorithm and lost by 253.
+1
level 43
Oct 21, 2016
Yep, I agree. got 808-467, quite challenging, nice job in finding highest and lowest.
+1
level 52
Apr 4, 2018
I found the worst loss fairly easy to prove. It is indeed 683. But I'm not sure about the best win yet.
+1
level 39
Feb 3, 2017
There's a way to beat him by 17! 646-629
+1
level 58
Mar 16, 2018
I won by 3... I started with doing 47, 49,21,27,33,39,45, then all the even numbers in decreasing order starting at 48. Then lastly, 50.
+1
level 33
Apr 8, 2018
Randomly picking numbers, I lost by 283.
+1
level 37
Sep 4, 2018
As far as I know, the best is 808-467
+1
level 33
Sep 13, 2018
i agree
+2
level 57
Jun 14, 2016
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.
+2
level 44
Apr 6, 2018
A badge might be cooler, though, and more practical. Being able to show off that you beat the taxman is better than getting points for doing that. Plus, someone could just look at the answers to find the best solution possible, and they would get a lot of points (although the quizzes aren’t safe from that, either)
+2
level 22
Jul 3, 2018
Badges are a big plus. But make the margin, like, 100 or something to get it. 1-point wins earning you badges is no fun.
+1
level 44
Sep 27, 2018
Good point
+1
level 64
Jun 15, 2016
By randomly choosing numbers, I only lost by 99, my closest
+1
level 39
Jun 24, 2017
lol, thats my best score actually trying
+1
level 68
Jun 15, 2016
I tried it about a dozen times before winning by 61.
+1
level 59
Jun 15, 2016
Took a few tries, but it was my 4th or 5th I finally won, by 49.
+1
level 59
Jun 15, 2016
Wow. That was close. Beat him by 1, 638-637. Makes it even sweeter.
+1
level 60
Jun 15, 2016
2nd try. I beat the taxman by 87 points. It was surprisingly mind-challenging.
+1
level 34
Jun 15, 2016
HOW DO YOU DO IT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
+1
level 47
Jun 15, 2016
Love this game!!
+1
level 47
Jun 15, 2016
After few times, I won by 21 points ;) 648-627
+1
level 58
Jun 15, 2016
first try 638-637 dub. Phew!
+1
level 75
Jun 16, 2016
That's another challenge indeed. How to make the closest win and the closest loss.
+1
level 75
Jun 16, 2016
Ah ha, found both 637-638 and 638-637. It's a bit less logical than the extreme cases, but still fun.
+1
level 57
Jun 15, 2016
2 tries, beat him by 43
+1
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
+1
level 60
Jun 15, 2016
47, 49 is a solid start
+1
level 57
Jun 15, 2016
After 3 tries I got him by 125. Hard but fun!
+1
level 44
Jun 15, 2016
After a few tries, i won by 3 :D
+1
level 44
Jun 15, 2016
11
+1
level 33
Jan 9, 2018
67
+1
level 25
Jun 15, 2016
You can't win :(
+1
level 39
Feb 17, 2017
47, 49, 4, 6, 25, 46, 38, 26, 50, 40, 30, 45, 44, 24, 36, 28, 34, 48, 42
+1
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
+1
level 33
Jan 9, 2018
47, 10, 15, 20, 30, 35, 40, 45, 50, 24, 34, 39, 44, 28, 38, 48, 42, 36, 46
+1
level 21
Jul 17, 2018
47, 49, 21, 27, 45, 33, 39, all even numbers from 46 and below, 48, 50 = +67
+1
level 51
Jun 15, 2016
Finally beat him, but only by 45. I'm gonna keep trying.
+1
level 64
Jun 15, 2016
I beat him by 7, I feel satisfied
+1
level 52
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.
+1
level 33
Jan 9, 2018
I also wonder what is the biggest win you can achieve...
+1
level 25
May 30, 2018
The biggest win is 808-467. The solution was pointed out by other users already.
+1
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...
+1
level 57
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.
+1
level 75
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.
+1
level 57
Jun 16, 2016
There would be no point picking 48 and 36 because the taxman can still take those numbers anyway
+1
level 54
Jun 19, 2016
Just lost by 1 by basically guessing the whole thing ;~;
+1
level 37
Jun 20, 2016
Very creative and fun quiz!
+1
level 75
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...
+1
level 75
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.
+1
level 75
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.
+1
level 75
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.
+1
level 75
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.
+1
level 75
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.
+1
level 75
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.
+1
level 75
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
+1
level 52
Apr 4, 2018
Pretty good algorithm, but I don't think it will always be optimal. Basically the algorithm is: take the highest number with only one divisor left on the board. It may work for this specific example but for instance, in the case where only 3, 9 and 27 are in the board your algorithm would lead to us taking 9. The taxman would end up with 3 and 27, and your algorithm would lose. The optimal answer is to take 27 and beat the taxman 27-12, of course. I am not sure if there is any algorithm that will get the right answer for every case.
+1
level 52
Apr 5, 2018
Oh actually it isn't quite that, but still it is in this case better to take the number with two factors left.
+1
level 60
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.
+1
level 75
Jul 18, 2016
Third try, won by 145! Fun game!
+1
level 32
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!
+1
level 32
Jul 22, 2016
Yup, lost by 199!
+1
level 32
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
+1
level 67
Aug 16, 2016
But can you tie?
+1
level 47
Oct 31, 2016
Nope. Add all the numbers together and you get an odd number.
+1
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.
+1
level 25
Aug 25, 2016
aww
+1
level 17
Sep 27, 2016
I DID WITH 341 POINTS
+1
level 67
Oct 5, 2016
Eh?
+1
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.
+1
level 35
Oct 12, 2016
Finally won 693-582
+1
level 51
Oct 15, 2016
Is your score the sum of your selections?
+1
level 67
Oct 17, 2016
641-634. I don't know how... For now.
+1
level 48
Oct 20, 2016
Bet him on the 3rd try by 23
+1
level 43
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
+1
level 43
Oct 21, 2016
So far for me biggest win is 341 and biggest loss is 633, not quite optimal :(
+1
level 75
Jun 27, 2017
As strange as it can seem, the worst loss (683) is obtained by using the winning strategy for a 25 grid.
+1
level 7
Nov 1, 2016
OMG i won (by 43)
+1
level 26
Nov 26, 2016
win 498 loss 684
+1
level 42
Dec 12, 2016
I lost by just 17... So close
+1
level 39
Mar 10, 2017
Same
+1
level 34
Dec 15, 2016
Beat it by 647
+1
level 15
Dec 15, 2016
Beat it by 647
+1
level 44
Dec 16, 2016
I beat him by 15 by randomly guessing numbers. WOW
+1
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
+1
level 40
Dec 17, 2016
lost by 83
+1
level 3
Dec 27, 2016
I won by 1 (638-637)
+1
level 44
Jan 5, 2017
Beat him by 75
+1
level 21
Jan 22, 2017
Weird game😕
+1
level 36
Feb 24, 2017
I won by 115 on my third try. No real strategy, just picking high numbers with minimal factors.
+1
level 53
Dec 4, 2017
There is your strategy, bud.
+1
level 32
Mar 8, 2017
Got a cheeky 738-537 win. AFTER ABOUT 20 TRIES!!!!
+1
level 29
Mar 17, 2017
I finally got it but barely. Only won 665-610.
+1
level 12
Mar 18, 2017
I was winning at first but then lost by 297
+1
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
+1
level 5
Apr 6, 2017
I lost by 391
+1
level 13
Apr 24, 2017
I beat it by 639
+1
level 57
Apr 25, 2017
Dangit! Closest I can get is losing by 37
+1
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
+1
level 34
May 11, 2017
i finally won on like my 15th try i beat the taxman by 5
+1
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
+1
level 25
Aug 11, 2018
Same but with 47, 49, 15, 26, 46, 42, 34, 33, 44, 27, 36, 30, 40, 32, 38, 48, 50
+1
level 20
Jul 8, 2017
I beat him by 21
+1
level 81
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.
+1
level 20
Jul 15, 2017
i lost by 403 and then 355
+1
level 14
Jul 31, 2017
I was 75 points away from my first win ever.....
+1
level 14
Jul 31, 2017
Now 17
+1
level 57
Aug 13, 2017
Closest is -17 :((
+1
level 26
Sep 6, 2017
lost by 7
+1
level 25
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.
+1
level 65
Jan 6, 2018
so the best winning strategy is to tell the taxman he has to pick and convince him you only have half the numbers you do... sounds like real life
+1
level 65
Jan 6, 2018
ps. i've seen this on other sites and 808 is the most likely best score stated there (haven't seen a complete proof against a higher one but seems unlikely there is one), kudos for working it out!
+1
level 25
Feb 20, 2018
I don't understand what you mean by that but the using the strategy for a 25 number board is the way to lose by the most.
+1
level 75
Mar 1, 2018
Yes, that's what I stated above (three months before you :p). It's mostly because, playing like that, you prevent yourself from taking any high number (for example, eliminating 25 with 5 gives the 50 to the taxman).
+1
level 30
Apr 19, 2018
The strategy to win is starting off with 47 and 49, and just going from there.
+1
level 32
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.
+1
level 23
Sep 21, 2017
FINALLY WON, only by 49 but still
+1
level 48
Sep 22, 2017
Beat the Taxman by 31
+1
level 41
Sep 29, 2017
I'm honestly a really bad at math, so a win by 5 points is good enough for me.
+1
level 41
Sep 29, 2017
"a really bad at math". I'm also bad at spelling it seems.
+1
level 39
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.
+1
level 52
Oct 6, 2017
W by 9, 642-633. I'm so proud of myself!
+1
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.
+1
level 57
Nov 11, 2017
Because he also gets all Numbers that you haven't used.
+1
level 6
Nov 3, 2017
My score reads '435-217'. What does that mean?
+1
level 57
Nov 11, 2017
You got 435 Points and he got 217.
+1
level 57
Nov 11, 2017
I did it by 81.
+1
level 17
Nov 19, 2017
Took about 21 tries lol
+1
level 7
Nov 30, 2017
(not a lie ) i tried 2 times and beat the taxman the 2nd time by 209!
+1
level 42
Dec 7, 2017
won by 5, took way too long
+1
level 35
Dec 12, 2017
Lost 628-647, my closest try!
+1
level 35
Dec 12, 2017
Now beat him 740-535
+1
level 35
Dec 27, 2017
Now beat him 779-496
+1
level 16
Dec 27, 2017
Won by 15!
+1
level 35
Dec 27, 2017
My biggest win is by 283 points 779-496 :)
+1
level 33
Jan 27, 2018
WON BY 96!!!!!!
+1
level 21
Feb 9, 2018
47, 49, 4, 6, 25, 46, 38, 26, 50, 40, 30, 45, 44, 24, 36, 28, 34, 48, 42, 44. Win by 49 points
+1
level 21
Feb 9, 2018
47, 49, 35, 21, 39, 33, 27, 45, 26, 46, 38, 18, 42, 34, 30, 50, 28, 44, 36, 24, 48, 40 Win by 325 points
+1
level 43
Mar 2, 2018
My first win: 47, 49, 42, 35, 28, 45, 39, 33, 48, 44, 40, 36, 50, 46, 38, 34 By 33
+1
level 12
Mar 2, 2018
Thank you for introducing the Game.its fun.but I lost by95
+1
level 12
Mar 2, 2018
Thank you to screamrearer.325!
+1
level 30
Mar 4, 2018
You have to start off with 47 49 in order to win. I won by 159
+1
level 52
Jun 13, 2018
I'm fairly sure it's possible to win no matter what number you start with. But 47 49 is best.
+1
level 38
Mar 9, 2018
lost by 203. dont quite get it, but maybe bc the explaination is in English >.
+1
level 10
Mar 14, 2018
YES won by 13 after 30mins!
+1
level 30
Apr 19, 2018
I won by 5 with the following set: 47, 49, 24, 28, 34, 10, 33, 50, 45, 38, 26, 46, 48, 44, 40, 42, 36. Is there a closer margin of victory than this?
+1
level 26
May 4, 2018
I WON the taxman by 99 - 47, 46, 49, 39, 38, 33, 16, 27, 28, 36, 42, 44, 48, 34, 25, 45, 50, 40
+1
level 29
May 8, 2018
47, 49, 26, 32, 39, 35, 38, 33, 46, 42, 45, 20, 24, 36, 34, 44, 50 Beats him by 5.
+1
level 25
Jun 4, 2018
You should add an option to restart/reset the game while playing. This is in case you accidentally clicked on a number and want to restart and try again.
+1
level 52
Jun 11, 2018
You could just click your browser's "refresh" button.
+1
level 49
Jun 6, 2018
Is it possible to get a tie? Also beat Taxman by one.
+1
level 52
Jun 11, 2018
You can't tie as all of the numbers add up to 1225. To the you would need to get 612.5 points each - obviously impossible.
+1
level 25
Aug 11, 2018
I lost by 1, see comment below
+1
level 25
Jun 17, 2018
i won by a small margin of 11. Hard to win sometimes
+1
level 53
Jun 19, 2018
YES! First try, some initial doubts, and then a great victory by 23!
+1
level 57
Jun 28, 2018
This is the most realistic video game there is.
+1
level 25
Aug 11, 2018
Lost by 1. The scenario is possible using these numbers in order. 47, 49, 15, 26, 46, 42, 34, 33, 44, 27, 36, 30, 40, 32, 38, 48, 50.
+1
level 25
Aug 11, 2018
Also if you start on the right row and work down it, the go to the row to the left and work down it, and continue repeating until there is nowhere to click, you lose by 17.
+1
level 44
Aug 12, 2018
With no strategy whatsoever, I managed to win by 69.
+1
level 27
Aug 29, 2018
Beat the taxman by 325 :-)
+1
level 15
Oct 18, 2018
326