Four Fours Problem Answers

Working Through the Four Fours Puzzle

Four Fours Problem Answers come together when you stack basic arithmetic, factorials, and the occasional decimal point in ways that aren't immediately obvious. The setup is simple: take four 4s and figure out how to make any given target number. That's it. The execution is where people hit walls. Here's how I actually approach this. Start by listing out the operations you have available. Addition and subtraction are free. Multiplication and division cost you nothing conceptually but get you only so far. Once you start needing factorials, square roots, or decimals like .4, that's when the real solutions appear. I remember trying to express the number 91 once. Standard operations get you to 88 pretty fast with things like 44 + 44, but 91 sits just outside that neighborhood. The trick was combining factorial notation with division. I used (4! - .4) / .4 which gives you (24 - 0.4) / 0.4 = 59. Wait, that's not 91. I had to go deeper. I ended up using a combination involving the gamma function extension for non-integer inputs, but honestly, most people don't need that depth. The standard set of allowed operations usually covers everything up to around 100 or so.

Let me be straight about the common operations and what they actually buy you: Basic operations (+, -, ×, ÷) get you most integers up through about 20. Factorials (4! = 24) unlock bigger numbers quickly. Square roots reduce 4 to 2, which is useful when you need smaller building blocks. Decimal points let you write .4, which is 2/5 and opens up division-based constructions. Concatenation, writing two 4s together as 44, is almost essential and usually counts as one of the four 4s used together. One thing beginners consistently miss is that order of operations matters more than they expect. Writing 4 + 4 × 4 + 4 doesn't give you 64. It gives you 24. You need parentheses to force the grouping you want. I've seen people spend twenty minutes trying to build a number only to realize they forgot a set of brackets halfway through. Double check your grouping before you move on.

Another counter-intuitive point: factorials grow too fast for most purposes. 4! = 24. Double factorial 4!! = 8 if you're using the double factorial variant, or 24 if you mean factorial of a factorial. This gets confusing fast. Stick to standard factorial unless your ruleset explicitly allows otherwise. Single factorial combined with square roots and decimals covers the vast majority of targets. Here are some working examples that demonstrate the patterns: Zero: 4 - 4 + 4 - 4. Trivial but necessary to establish the baseline.

One: (4 + 4) / (4 + 4). Or 44 / 44. Both work. Two: 4 / 4 + 4 / 4. Clean and straightforward. Ten: (44 - 4) / 4. This one trips people up because they try to add their way there instead of using concatenation first.

Eleven: 44 / 4 + 4. Again, concatenation does heavy lifting. Twenty-five: (4 × 4) + (4 / 4). You're building from 16 and adding 1 four times. The bigger numbers require more creativity. Target 100 is (4 + 4) × 4 × 4 - 4 - 4 which equals 64 minus 8, no that's wrong. Let me recalculate. (4 + 4) × 4 × 4 is 128. Subtract 4 and subtract 4 gives 120. I keep making arithmetic errors under pressure. The actual solution for 100 uses (44 - 4) / .4 which is 40 divided by 0.4 equals 100. That decimal trick is the key insight most tutorials skip over.

If you're building a solver or writing code to generate these, the brute force approach explores all possible expressions with four 4s and evaluates them. It's computationally feasible for small targets. A well-implemented DFS with memoization on intermediate results runs in reasonable time for targets up to 200. Beyond that you're looking at exponential growth in expression possibilities and the search space gets unwieldy without pruning strategies. One honest limitation worth noting: the Four Fours puzzle becomes significantly harder if you restrict the allowed operations. Some rule sets don't permit decimal points or concatenation. In those versions, numbers like 10 and 11 become much harder or sometimes impossible depending on how strict you are. Know your ruleset before you start solving. I once spent an afternoon trying to find a solution for a target number only to discover the rules I was following didn't allow the operation I needed. That wasted time costs you nothing except the time itself. For people who want to practice, the standard reference covers integers from 0 to about 150 or 200 with solutions. Some extended lists go higher using square roots applied to factorials in nested ways, but those solutions get obscure fast and aren't particularly useful for learning the core techniques.

The practical takeaway is that concatenation and decimals do more work than people expect. If you're stuck on a target number, check whether writing two 4s together or using .4 as a divisor gets you into a productive range. That single insight solves roughly half the problems that stump beginners.