i figured out how to do this the game is to draw 8 cards and combines them arithmetically to get any 4 digit number consider one puzzle with 1, 4, 5, 6, 8, 9, 11, 12 and target 7217 a solution is 7217=9*(6*(11*12)+5*8/4)-1 first step is to realize that this problem is not only mostly solvable (98% of puzzle has a solution), there are usually many ways to solve it for example, 7217=(12×9-5)×(11×6+4)+8-1 =11×8×(12×6+9+5-4)+1 =11×(12+4)×(9×6-8-5)+1 in fact, most puzzles each has 100+ unique solutions (here a+b*c and c*b+a count as one solution) i know this by writing code to analyze large amount of puzzles, but you can just play the game long enough to notice as well once you build the confidence, here's one practical methodology i found by playing a bit a typical solution consists of the main multiplicative stem and some corrections along the way you can typically get 3-4 out of the largest 5 numbers form the stem, say the numbers are a1, a2, ..., a8 ranked from large to small so final solution typically looks something like a1 * a2 * a3 with some corrections along the way like ((a1 + a5) * a3 + a7) * a8 + a4 + a2 now you do need to be familiar with how to factorize numbers, there are simply tricks like the sum of digits of a number can be divided by 9 then that number can be divided by 9 as well, like 783 can be divided by 9 because 7+8+3=18 can be divided by 9 then you work your work backwards to see what correction is needed on the original target to divide it down you typically already have a rough sense of what the main multiplicative trunk could be take this hand as an example 1, 4, 5, 6, 8, 9, 11, 12 and target 7217 6*8*9*11 = 6*8*99 ~ 4800 which is far from 7217, which means you'll most likely need 11 & 12 in the trunk in fact 6*9*11*12 = 72*99 ~ 7200 looks close! now we work backwards and try to correct it a bit so it can be divided by 9, 11, or 12 here's where there's a bit trial & error but because there're many possible solutions, it tends to work out 7217 digits sum up to 7+2+1+7 = 17, which means with a correction of 1 it's dividable by 9 so (7217+1)/9 = 802 (you need to be able to do this kind of math in your head, which is not very hard but needs practice) now we are left with 4, 5, 6, 8, 11, 12 and we need to make 802 we try to find correction again, for example, what's the correction that makes it dividable by 12? it needs to be dividable by both 3 and 4, it's not hard to tell you can add 2 802 + 2 = 804/12 = 67 once you have that, you know you can make a 66 as well 802 - 10 = 792/12 = 66 once you have 66 we have 6 * 11, then we need to make a 10 from 4, 5, 8, so you can do 5*8/4 the last part might look like luck, but really you can try the divider number exhaustively, and because there are typically many solutions, you'll find a valid solution relatively fast this method almost always works now the deeper reason that there's nearly always a solution is related to percolation theory in math consider a sponge, it's full of holes and water can flow through it. but if the holes are sparser, then at some point it becomes very unlikely for water to flow through there's a "phase transition" at which point the sponge goes from water-permeating to not when it is water-permeating, there's many path for water to flow through in this card game, the number of cards is critical, if you can too few cards, then the probability of finding a right solution decreases dramatically but with enough cards, there's almost certainly a solution, and when there's a solution, there are typically many