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-   -   The Project Euler thread (http://www.flashflashrevolution.com/vbz/showthread.php?t=120818)

stargroup100 10-23-2011 05:03 PM

Re: THE project euler thread
 
lurker had the same problem

I'm trying to find the error as well

LongGone 10-23-2011 06:00 PM

Re: THE project euler thread
 
Quote:

Originally Posted by Reincarnate (Post 3556009)
can anyone find the problem in my earlier list? should equal 1356, not 1355 (according to the problem description)


Since you are one off from the actual answer (changing from odd to even), it implies that one set of (p^a)(q^b) should be degrouped, or grouped (e.g. 5*19=95(odd), or 5^a+19^b=even). Swapping pairs of number products will not change the odd/even parity
The main concern are the primes <10 in this case (because obviously you can't multiply two primes >10 that will result in <100 product)
So the usable primes are 2,3,5,7
Since you used up all 4 and ended up with 1355; If 1356 was not a mistake, that means that you have to degroup one of those primes and leave one as a natural power of itself (which is counterintuitive)
...So either the 13 people who solved didnt bother checking Co(100), or I'm missing something rofl
(or a rare case of a possible product of three primes, which idk if its plausible)

For a possible general approach, to try to maximise sum, we can start by getting rid of all the small primes
So e.g. for Co(100), start with 11,13,17,19. With your current solution we got: 99 (3,11), 95 (5,19), and 92 (2,23), 91 (7,13). Could there be a possible solution using 17 in place of 23?


stargroup100 10-23-2011 06:10 PM

Re: THE project euler thread
 
edit2: ok I was being stupid

EDIT: Rubix, your answer might be wrong but it looks pretty close :D

Reincarnate 10-23-2011 06:13 PM

Re: THE project euler thread
 
A step-by-step runthrough of the Co(100) process:

initial naive list from prime-raising alone: [1, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97] with sum 1263

adding 99 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 99]
Updating list because biggestsum 1263 is less than 1270
adding 98 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 99]
adding 96 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 96, 97]
adding 95 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 95, 97, 99]
Updating list because biggestsum 1270 is less than 1321
adding 94 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 31, 37, 41, 43, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 94, 95, 97, 99]
adding 93 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 93, 95, 97]
adding 92 to the list and removing non-coprimes to test [1, 13, 17, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
Updating list because biggestsum 1321 is less than 1326
adding 91 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
Updating list because biggestsum 1326 is less than 1355
adding 90 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 90, 91, 97]
adding 88 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 88, 89, 91, 95, 97]
adding 87 to the list and removing non-coprimes to test [1, 17, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 92, 95, 97]
adding 86 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 86, 89, 91, 95, 97, 99]
adding 85 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 92, 97, 99]
adding 84 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 84, 89, 95, 97]
adding 82 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 82, 83, 89, 91, 95, 97, 99]
adding 81 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 92, 95, 97]
adding 80 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 80, 83, 89, 91, 97, 99]
adding 78 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 78, 79, 83, 89, 95, 97]
adding 77 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 92, 95, 97]
adding 76 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 76, 79, 83, 89, 91, 97, 99]
adding 75 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 91, 92, 97]
adding 74 to the list and removing non-coprimes to test [1, 17, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 74, 79, 83, 89, 91, 95, 97, 99]
adding 72 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 72, 73, 79, 83, 89, 91, 95, 97]
adding 70 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 70, 71, 73, 79, 83, 89, 97, 99]
adding 69 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 69, 71, 73, 79, 83, 89, 91, 95, 97]
adding 68 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 68, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 66 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 66, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 65 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 92, 97, 99]
adding 64 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 63 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 63, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 62 to the list and removing non-coprimes to test [1, 17, 29, 37, 41, 43, 47, 53, 59, 61, 62, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 60 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 71, 73, 79, 83, 89, 91, 97]
adding 58 to the list and removing non-coprimes to test [1, 17, 31, 37, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 57 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 57, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 56 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 55 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 54 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 54, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 52 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 52, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 51 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 50 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 49 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 48 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 48, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 46 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 46, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 45 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 44 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 44, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 42 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97]
adding 40 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 40, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 39 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 39, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 38 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 38, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 36 to the list and removing non-coprimes to test [1, 17, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 35 to the list and removing non-coprimes to test [1, 17, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 97, 99]
adding 34 to the list and removing non-coprimes to test [1, 29, 31, 34, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 33 to the list and removing non-coprimes to test [1, 17, 29, 31, 33, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 32 to the list and removing non-coprimes to test [1, 17, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 30 to the list and removing non-coprimes to test [1, 17, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97]
adding 28 to the list and removing non-coprimes to test [1, 17, 28, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 27 to the list and removing non-coprimes to test [1, 17, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 26 to the list and removing non-coprimes to test [1, 17, 26, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 25 to the list and removing non-coprimes to test [1, 17, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 24 to the list and removing non-coprimes to test [1, 17, 24, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 23 to the list and removing non-coprimes to test [1, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 22 to the list and removing non-coprimes to test [1, 17, 22, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 21 to the list and removing non-coprimes to test [1, 17, 21, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 20 to the list and removing non-coprimes to test [1, 17, 20, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 19 to the list and removing non-coprimes to test [1, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 18 to the list and removing non-coprimes to test [1, 17, 18, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 16 to the list and removing non-coprimes to test [1, 16, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 15 to the list and removing non-coprimes to test [1, 15, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 14 to the list and removing non-coprimes to test [1, 14, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 13 to the list and removing non-coprimes to test [1, 13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 12 to the list and removing non-coprimes to test [1, 12, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 11 to the list and removing non-coprimes to test [1, 11, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 10 to the list and removing non-coprimes to test [1, 10, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 9 to the list and removing non-coprimes to test [1, 9, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 8 to the list and removing non-coprimes to test [1, 8, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 7 to the list and removing non-coprimes to test [1, 7, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 6 to the list and removing non-coprimes to test [1, 6, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 5 to the list and removing non-coprimes to test [1, 5, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 4 to the list and removing non-coprimes to test [1, 4, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 3 to the list and removing non-coprimes to test [1, 3, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 2 to the list and removing non-coprimes to test [1, 2, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]

sorted, final mainlist [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
sum of mainlist 1355

stargroup100 10-23-2011 06:21 PM

Re: THE project euler thread
 
Your solution is too linear. You should take into consideration the possible case that immediately adding the largest possible number won't optimize the overall total.

It's possible there is a different combination of the primes that rounds out to a better sum. I recommend this method:

- Separate primes into 3 categories: between 2 and sqrt(n), between sqrt(n) and n/2, and the others
- Add up all of the primes in the last category. These don't change.
- The terms in the second category have the most room for possible optimizations, and therefore should try to be optimized first. They obviously cannot be paired with a prime from its own category, so pair it with a prime from the first category. Try every combination here.
- The last possible case is that 2 primes from the first category might be paired together. Consider this generalization last.

Do it this way and see if you can get to 1356.


NOTE: I'm not sure if the last step is necessary, I'm trying to see if I can prove whether or not that last step is needed.

Reincarnate 10-23-2011 06:24 PM

Re: THE project euler thread
 
It's not adding just the largest possible number -- here, it's adding all possible numbers from 2 to (N-1) and checking how it affects the total after removing non-coprimes

LongGone 10-23-2011 06:37 PM

Re: THE project euler thread
 
For 100, we have 25 odd primes (including 1), and one even 2. The sum of 25 odd primes (raised to any power) sum up to odd, adding the power of 2 still makes it odd

However, since we can group primes, we most likely will be grouping the smallest primes. There are 4 primes <sqrt100, 2 3 5 7. By grouping, we have 22 odd numbers and 2. 2 can be combined with an odd number leaving 21 odd numbers and one even number. The sum of that is odd. 1356 is even (?????????)

Possible explanation would be that the solution for 1356 involves -not- grouping one of 3, 5 or 7 and leaving it as its natural power. The most plausible is 3^4=81. Alternatively, there may be a case where there is a coprime which is the product of 3 primes

stargroup100 10-23-2011 06:40 PM

Re: THE project euler thread
 
FFFF YOU ARE RIGHT

im dumb

Reincarnate 10-23-2011 06:51 PM

Re: THE project euler thread
 
.ok apparently my problem is that it risks getting caught in local optima. when it detects a better sum, it may be adding a number (and keeping it) that winds up not being a part of the final solution (or prevents another number from doing the same). Going from 1 to n gives me a diff number from n to 1.

stargroup100 10-23-2011 06:56 PM

Re: THE project euler thread
 
that's what I was trying to say lol

Reincarnate 10-23-2011 06:59 PM

Re: THE project euler thread
 
ah ok misunderstood what you meant

stargroup100 10-23-2011 07:12 PM

Re: THE project euler thread
 
what the hell I went back to look at the problems I already solved in project euler and I cannot for the life of me remember how I did these LOL

all I remember is that I did most of these with pencil/paper

stargroup100 10-23-2011 07:37 PM

Re: THE project euler thread
 
Quote:

Originally Posted by Reincarnate (Post 3556096)
Maybe something like this (building off SG's idea)

stuff

not quite

0 <= x < sqrt(n) list: [(1), 2, 3, 5]
sqrt(n) <= x < n/2 list: [7, 11, 13]
n/2 <= x list: [17, 19, 23, 29]

however, but my idea doesn't take into account longgone's new input, the fact that some of the primes from the first group might stand by themselves. however, if you combine your method with mine it should account for both cases pretty well.

FFR4EVA_00 10-23-2011 08:37 PM

Re: THE project euler thread
 
psst
{1, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 88, 89, 91, 95, 97}
i went off the fact that 81 is the largest number less than 100 of the form a^b

stargroup100 10-23-2011 09:02 PM

Re: THE project euler thread
 
Thanks to lurker and LG I can fix up my method a bit.

Okay, here's my updated algorithm. It seems to be pretty tight for the most part, but it is still fairly brute force and assumes two conjectures:

- A term in a maximal sum subset can never have more than two unique prime factors.
- When optimizing the sum of a subset by choosing to multiply two powers of primes, the two primes are never both below sqrt(n).

Assuming these conjectures, we use the method:
- Separate primes into 3 categories: between 2 and sqrt(n), between sqrt(n) and n/2, and the others
- Add up all of the primes in the last category. These don't change.
- Each of the terms in the first category can either be left by itself or paired with a term from the second category. Consider and test every case.

The total number of possible subsets to test for is exactly 9507!/(9421)!.

Hopefully we can do a little bit better than that, as this number is in the order of about 10^(well over 200).

Reincarnate 10-23-2011 09:15 PM

Re: THE project euler thread
 
I'm trying to find something that doesn't require bruteforce (most of the problems I've solved don't require it) -- this problem is bugging the hell out of me because I can't figure out anything more elegant

cry4eternity 10-23-2011 09:26 PM

Re: THE project euler thread
 
I just found this on Friday. Solved 1-22 as well as 67 now :p. This is actually pretty fun.

stargroup100 10-23-2011 09:43 PM

Re: THE project euler thread
 
Quote:

Originally Posted by Reincarnate (Post 3556158)
I'm trying to find something that doesn't require bruteforce (most of the problems I've solved don't require it) -- this problem is bugging the hell out of me because I can't figure out anything more elegant

feelin ya bro

but it's possible this is along the lines of what they want. after all, they ARE programming problems.

FFR4EVA_00 10-23-2011 10:45 PM

Re: THE project euler thread
 
i'm gonna go ahead and drop a gigantic hint for 354 since the upper bound is so insane i have no chance of programming it correctly:
2L^2/3 = 2*3^(any non-negative integer)*n^2*(6p+1)^4*(6q+1)^4*(6r+1)^2
n must only be a multiple of 2 and any primes of the form 6k-1
6p+1, 6q+1 and 6r+1 are distinct prime numbers; p, q and r are integers

stargroup100 10-23-2011 11:49 PM

Re: THE project euler thread
 
Right now I'm trying to figure out methods to significantly reduce the number of calculations needed to solve this, whether it's skipping possible subsets or finding a totally new method.

However, it's getting late and I'm getting sleepy, difficult to focus. I'll work more on this tomorrow.


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