How To Solve the Mathler Puzzle

Sometimes people who are new to Mathler think the point is to find an expression that equals the target number, but that’s actually not enough! To “win” you need to find the expression the puzzle constructor was thinking of; we’ll call that the target solution. That’s where logic comes in – applying the information from the green/yellow/gray colors Mathler returns on your guess and making
that information fit into each new guess until you get the target solution*.
*Or arithmetically VERY close to it! You’ll remember that 3 + 5 = 5 + 3 and this is true not just for 3 and 5, but for any two numbers. This is called the commutative law, and multiplication works that way too; division and subtraction do not. Mathler will accept variations on the target solution that have this kind of reversal; it will rewrite them in the target solution order, mark them all green and say something like “commutative solution accepted.” Suppose the Easy Mathler target number is 33 and the target expression is 21 + 12. If you enter 12 + 21, Mathler will rearrange it to 21 + 12 and mark all the spaces green and say “commutative solution accepted”; you win! However, Mathler scores incorrect guesses only against the target solution:

21 + 12 The target solution.
11 + 22 Your guess.
YG G YG The score for your guess.
The score would be different if it were scored against 12 + 21, the commutative alternative! Can you figure out what it would look like?
Note: A different application of commutativity is also accepted. If, for example, the target number for a medium Mathler is 8 and the target solution is 12 – 7 + 3, Mathler also accepts 12 + 3 – 7, where the operation “subtract 7” is switched with the operation “add 3”. This can be done because subtraction can be written as adding a negative number, and the commutative law applies to that addition!
12 – 7 + 3
= 12 + (-7) + 3
= 12 + 3 + (-7)
= 12 + 3 – 7
Using this kind of commutativity, can you think of two other solutions Mathler would accept? Why can’t Mathler accept -7 + 12 + 3 or -7 + 3 + 12?

Well played! Playing Mathler well does not mean just guessing the target solution. It isn’t even just applying all the information gained from previous guesses to get a new guess. It also means making guesses that will give you as much useful information as possible. Mathler Helper “scores” possible guesses by how many digits and operators they will get new information about. When Mathler Helper suggests a next guess to you, it is one that will give you a good amount of new information and, if it isn’t the right solution, will help you eliminate a lot of impossible guesses!