Sum Fold answer

How Sum Fold puzzles are built

The board is generated forwards, not backwards. Six numbers are drawn — one or two large ones from 25, 50, 75 and 100, the rest from one to ten — and the generator then applies a random chain of legal operations to them. Whatever that chain lands on becomes the target. This guarantees a solution exists before the puzzle is ever shown, and the length of that chain becomes par.

Operations that add nothing are filtered out: multiplying or dividing by one, and subtractions that return the number you started with. They are legal arithmetic but they pad the step count without creating a puzzle.

A method that works most days

1. Read the target's neighbourhood. Round it to the nearest hundred, or to the nearest multiple of 25. That rounded number is what your big multiplication should aim at.
2. Find the big jump. Which pair of your numbers multiplies to something near that neighbourhood? This uses up your large number and one small one.
3. Close the gap. Subtract the target from your big number. Now you only need the remaining numbers to make that difference, which is a far smaller search.

Three steps solves a majority of boards. When it does not, the usual culprit is that the target is not near a clean multiple, and the fix is a division early on to reshape one of your numbers.

Why every value must stay a whole number

Allowing fractions would make almost every board trivially solvable and impossible to do in your head. Restricting intermediate results to positive integers is what turns arithmetic into a search problem — and it is why division is the operation that most often unlocks a stubborn board, because the divisions that work are rare enough to be informative.