Target Forge: preserve the resources inside every intermediate value
Make 24 from four number tokens without leaving two behind, and distinguish a reusable result from an original number that has already been consumed.
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.
A matching value is not yet a complete solution
Target Forge asks for one final value equal to its target after every starting token has been used exactly once. A promising intermediate number can therefore be misleading. If two tokens already multiply to the target but other tokens remain, you still need a legal way to incorporate those remaining resources without changing the required final result.
Think of each active value as a small expression rather than a number printed in isolation. When you combine two values, both disappear and their result takes their place. That result contains everything used to make it. It may participate once in a later combination, while the original tokens inside it are no longer separate resources.
Make 24 without leaving two numbers behind
Consider a separate four-token teaching problem: starting numbers 2, 3, 4 and 6, with target 24. Multiplying 6 by 4 reaches the desired value but leaves 2 and 3 unused. The game would correctly refuse to declare completion while those other tokens remain.
Instead, combine 3 minus 2 to make 1. Next divide the original 6 by that result to make 6. Finally multiply the resulting 6 by the remaining 4 to make 24. Written as one expression, the route is (6 ÷ (3 − 2)) × 4. Every starting token appears exactly once, and both divisions and multiplications obey the available rules.
Equal values do not mean identical resources
The intermediate result 6 in that example has the same numeric value as the original 6, but it is a different active token. It now contains the original 6, 3 and 2 together. You cannot also use the old 3 elsewhere, because it was consumed while constructing the intermediate value. The expression display helps you keep that history attached to the number.
The same distinction applies when two starting tokens happen to show equal values. They are separate resources and can be combined with each other. What is forbidden is using the same active token on both sides of one operation, or reusing a starting token already included in another active result.
Review arithmetic legality before committing
Operand order matters for subtraction and division. Choosing 2 as the left value and 8 as the right value cannot create 4 by division; 2 divided by 8 is not an integer. Reversing those operands gives the permitted result 4. Zero may be a result or a dividend, but never a divisor. Negative intermediate values are allowed.
Undo restores the two values before the previous accepted combination, which lets you reconsider their order or choose another operation. Restart restores the same starting resources and target. A target may have several valid constructions, so a successful expression need not reproduce the generator’s route. The final check is complete resource use and arithmetic correctness, rather than matching a hidden preferred solution.
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