Make 24

Working with Make 24 Without Losing Your Mind

Make 24 is a number puzzle. You get four digits and need to combine them using addition, subtraction, multiplication, and division so the final result is exactly 24. That's the whole premise. Most people pick it up in five minutes. Solving the harder cards consistently is where things get tricky. The standard version uses a deck of cards where Ace equals 1 and face cards are 11, 12, or you remove them entirely depending on the variant. You flip four cards, then try to find an expression. The solution space is small enough that you can do it in your head for simple sets, but some combinations genuinely have no solution. About 2-3% of random four-card draws are unsolvable with basic arithmetic, and people waste a long time on them before accepting that fact. I learned this the hard way during a road trip back in 2019. My kids pulled out a Make 24 app and we spent about twenty minutes on a hand that turned out to be impossible. Eventually I just wrote down all the intermediate values systematically instead of guessing randomly, and sure enough no combination worked. The workaround was straightforward: pick one pair of numbers, compute every possible result from those two (a+b, a-b, b-a, a*b, a/b, b/a), then treat that result as a new number and repeat until you've used all four. When you hit dead ends at every path, you move on.

How to actually solve these puzzles efficiently

Start by grouping. Pick any two of the four numbers and reduce them to a single value using all valid operations. Now you have three numbers instead of four. Repeat until you're down to one. If that one equals 24, you're done. The key insight most beginners miss is that order matters with subtraction and division, so always calculate both a-b and b-a, both a/b and b/a, even if one seems redundant. That doubles your branches at each step but catches solutions people routinely overlook. There's a pattern worth knowing by heart. If two of your numbers multiply to 24 directly, check whether the remaining two can cancel each other out to become 1 through division, or 0 through subtraction. Similarly, if you can form 6 and 4 from your four cards, multiplying them gives you the answer. Building toward factor pairs like 6*4, 8*3, or 12*2 covers the majority of solvable hands. Fractions are where things get interesting though, and that's also where most people quit. Consider a hand like 3, 3, 8, 8. The solution is 8 divided by (3 minus 8 over 3), which equals 24. You can't see that unless you allow intermediate fractions and work backward from 24 = 8 / (1/3). I've seen plenty of experienced players stare at this exact combination and declare it unsolvable because they only think in whole numbers during intermediate steps. Keep fractions in play and the puzzle opens up considerably.

Common pitfalls and what to do instead

The biggest mistake is working forward from the given numbers without a target in mind. You'll generate random intermediates and hope something sticks. Instead, work backward from 24 and ask what pairs of values could produce it. What multiplies to 24? What adds to 24? What divides to 24? Then check whether your four cards can form those component values. This reverses the search space dramatically and cuts down blind guessing. Another trap is stopping too early on apparent dead ends. A hand might look unsolvable through integers alone but has a fraction-based path. I recommend keeping a scratchpad with every intermediate fraction you encounter rather than trying to hold them all in your head. Mental math gets unreliable once you're juggling thirds and eighths simultaneously. The brute-force approach I described earlier — pairing numbers and exhaustively computing all outcomes — is actually the most reliable method if you're patient. There are only 3! = 6 possible orders to process the four numbers, and at each step you have at most 6 operations on a pair. That's roughly 6 times 6 times 6 = 216 paths maximum, which a human can trace in under two minutes if they stay organized. A computer does it in milliseconds, which is why every Make 24 solver app on the market just implements this exact algorithm.

When Make 24 breaks down completely

The standard operations don't cover everything. If a puzzle requires exponentiation, square roots, or concatenation, you're outside the normal ruleset and most published answer keys won't include those solutions. Some apps add these as optional modes, but they change the difficulty curve significantly and introduce ambiguity about what counts as a valid expression. I avoid those variants unless I'm explicitly playing for fun rather than practice. There's also the issue of time pressure. In a casual setting the puzzle is fine, but if you're training for speed or competition, the unsolvable hands become a real problem. They waste time and erode confidence. The only real fix is memorizing the known unsolvable four-card combinations. There aren't that many of them — roughly a few dozen unique sets out of the total draw space — and once you recognize the patterns you can skip them immediately instead of grinding through false attempts. If you want a solver, any basic implementation of the exhaustive pairing algorithm will handle standard Make 24 correctly. The logic is simple enough that you could write one in an afternoon, and there are plenty of free open-source versions online if you don't feel like building your own. The real value isn't in the tool though. It's in recognizing when to switch between forward and backward reasoning, when to entertain fractions, and when to accept that a hand has no solution and move on.