Target Forge: a negative intermediate can preserve a positive plan
Build the same target through positive and negative branches, keeping operand order and the identity of every consumed token visible.
Worked examples use the Standard board unless a different size is stated. Hard and Expert may change the board dimensions or amount of information to track. Teaching positions are explained separately from the playable board.
Keep the sign attached to the resource
It is tempting to treat a negative result as a failed branch when the target is positive. Target Forge does not impose that restriction. Subtraction may produce a negative integer, and two negative intermediate values can later multiply to a positive result. The question is whether the remaining resources can legally complete the expression, not whether every intermediate value looks like the final target.
Consider a separate four-token teaching configuration with starting values 2, 3, 5 and 9, and target 18. Four tokens match the resource count used on Standard, but this is an explanatory setup rather than a promise about the puzzle currently generated in your browser. Every route below consumes all four original tokens once. We will track the whole expression, including minus signs, to make that requirement visible.
Construct two negative factors
First combine 2 minus 5 to obtain −3. The order matters: selecting 5 minus 2 would produce positive 3. Next combine the untouched 3 minus the untouched 9 to obtain −6. Now the only active values are −3 and −6. Their product is positive 18, so the route finishes with (2 − 5) × (3 − 9). Both subtractions are legal, and no original token is left over.
The original token showing 3 has a different identity from the intermediate value −3. They also have different signs. Once the original 3 is consumed in 3 minus 9, it cannot participate in a later independent branch. Reading expressions rather than isolated numbers prevents a mental solution from silently borrowing a token twice. This matters even more when two starting tokens happen to display equal values.
| Accepted step | Active values afterward | Original values inside the new result |
|---|---|---|
| 2 − 5 = −3 | 3, 9, −3 | 2 and 5 |
| 3 − 9 = −6 | −3, −6 | 3 and 9 |
| −3 × −6 = 18 | 18 | 2, 3, 5 and 9 |
Compare a route that stays positive
The same setup also permits 9 minus 5 equals 4, then 4 plus 2 equals 6, and finally 6 times 3 equals 18. Written as an expression, this is ((9 − 5) + 2) × 3. It uses the same four resources and the same number of accepted combinations. Its shape is different: one growing branch is combined with a remaining token at every step, whereas the negative route builds two independent branches before joining them.
With four starting tokens, every completed binary-operation route has three accepted combinations because each combination reduces the active token count by exactly one. That observation does not tell you how many unsuccessful ideas or Undo actions you will consider. It also does not make the two expressions interchangeable midway through play: after consuming 2 and 5 in the first negative branch, those original tokens are unavailable until you undo that combination.
Distinguish a legal negative from an illegal division
A negative integer is permitted, but not every numerical expression is available. Division must have a nonzero divisor and an integer result. For example, −6 divided by −3 is the legal integer 2. Dividing 3 by 2 would require a fraction and is rejected. Dividing by zero is rejected even when another part of a written expression might appear to cancel that zero later.
The game checks each chosen operation when you make it. A mathematically meaningful final expression that passes through a fractional intermediate is therefore outside these rules. Do not solve a different arithmetic game in your head and expect its intermediate states to transfer. Likewise, very large results must remain safe integers; choosing a clever sign pattern does not bypass that numerical boundary.
Plan backward, then check forward
A productive planning question is which factor pairs could make the target from disjoint groups of the available values. For 18, both 3 × 6 and (−3) × (−6) are possibilities. Then ask whether the resources can create those factors without overlap. This backward view suggested the two subtraction branches, but the forward table is what verifies the actual legal sequence.
Use Undo to reconsider an accepted combination and Restart to recover the original resources and target. A successful route need not match the generator’s construction. Its proof is simpler: each operation is allowed, each original token appears once, and the single remaining value equals the target. A minus sign can be part of that proof rather than a reason to abandon an otherwise useful branch.
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