Reach the number, using what you have
Six tiles, four operations, one target. Every puzzle is generated from what the tiles can actually reach, so there is always a way — the only question is whether you find it.
Generated on this device
How a target is chosen
Every value the tiles can reach is enumerated first, by working through all 64 subsets of six tiles and combining each pair of disjoint subsets with every permitted operator. That produces a map of reachable value to the cheapest expression making it.
The target is then drawn from that map, which is why there is always a solution. The alternative — picking a number and hoping — is how these puzzles end up quietly impossible.
Trivial targets are filtered out: anything equal to a tile, anything reachable in a single operation, and anything outside the range. Targets needing more of the tiles are preferred, which is what makes a puzzle feel designed rather than accidental.
Your expression is parsed into a tree and evaluated in exact fractions — never with eval(), and never in floating point. That matters: in floating point 0.1 + 0.2 is not 0.3, and a puzzle that rejects a correct answer over a rounding error in the fifteenth decimal place is worse than no puzzle.
The rulesets differ in more than tile count
Beginner gives four tiles and only + − ×. You may leave tiles unused, which makes the search much smaller and means a first solution is usually visible within a few seconds.
Standard is the classic form: five tiles, all four operations, and every tile used exactly once. Every intermediate step must be a positive whole number, so you cannot pass through −3 or 7½ on the way.
Hard gives six tiles and lifts the whole-number rule, so fractions are allowed mid-calculation as long as the final answer is exact. That opens routes that are invisible under the standard rules, and it is why hard puzzles often look impossible until you try dividing something awkward.
Hints narrow rather than reveal
The first press names two tiles that appear together in a solution. The second gives an intermediate value worth aiming for. Only the third shows an expression — and it is an expression, not the expression, because most targets have several routes.
Related
Mental Math Arena for speed rather than search, and the worksheet generator for printable practice.
Questions people actually ask
Is every puzzle solvable?
Yes, and by construction rather than by luck. Every value the tiles can reach is enumerated before a target is chosen, and the target is drawn from that list. Picking a number and hoping is how these puzzles end up impossible, and it is a common way to build them.
Can I use a tile more than once?
Only as many times as it appears. If you are given two 5s you may use 5 twice; if you are given one, you may not. The tiles grey out as you consume them so you can see what is left.
Why was my correct-looking answer rejected?
Usually a ruleset rule rather than the arithmetic. Standard play requires every intermediate step to be a positive whole number, so a route passing through −3 or 7½ is refused even if it lands on the target. The message says which step broke it. Hard rules allow fractions part-way.
Is there only one solution?
Usually not. Most targets can be reached several ways, which is why the third hint says 'one route of several'. A single-solution mode would need proving no other expression family exists, which is a much stronger claim than the game needs.
Does it use eval() on what I type?
No. Your expression is parsed into a tree and walked, so nothing you type is ever executed. It is also evaluated in exact fractions rather than floating point, because 0.1 + 0.2 is not 0.3 in floating point and a puzzle that rejects a correct answer over that would be maddening.