Work Backwards is the habit of starting from the desired result or final state when forward construction is unclear. In contest math, that habit turns a crowded setup into a relation the student can test, bound, count, or compute. MathGrit teaches it as a recognizable signal, a deliberate move, and a final translation back to the original question.
$$\text{known end state} \;\xrightarrow{\text{invert each step}}\; \text{unknown start state.}$$
When the final result is given but the starting point is hidden, run the operations in reverse — undoing each step — instead of guessing the start and going forward.
Why It Works
1
Identify the final state the problem hands you and the sequence of operations that produced it.
2
Replace each forward operation by its inverse: addition $\leftrightarrow$ subtraction, multiplication $\leftrightarrow$ division, 'doubled' $\to$ 'halve'.
3
Apply the inverses in the reverse order, starting from the final value, until you reach the original unknown.
4
This collapses a search over possible starting values into a single deterministic backward computation.
Worked Examples
Example 1
A jar holds some marbles. On day one half the marbles are removed and then $4$ more are added. On day two half of the new total is removed and then $4$ more are added, leaving $20$ marbles. How many marbles were there at the start?
1
End of day two is $20$. Undo 'add $4$': $20 - 4 = 16$. Undo 'remove half' (i.e. the $16$ was half of the start of day two's pile, since removing half leaves half): the day-two pile was $16 \times 2 = 32$.
2
So the start of day two (= end of day one) was $32$. Undo day one's 'add $4$': $32 - 4 = 28$.
3
Undo day one's 'remove half': $28$ is half the original, so the original was $28 \times 2 = 56$.
A shopper spends half of her money, then $\$10$ more. She then spends a third of what remains, then $\$6$ more, leaving her with $\$14$. How much money did she start with?
1
Final amount is $\$14$. Undo the last 'spend $\$6$': $14 + 6 = 20$ — this was what remained after the third was spent.
2
Undo 'spend a third': $\$20$ is the remaining two-thirds, so the amount before that step was $20 \cdot \tfrac{3}{2} = 30$.
3
Undo 'spend $\$10$': $30 + 10 = 40$ — this was the half left after the first spend, so it is half the original.
4
Undo 'spend half': the original was $40 \cdot 2 = 80$. Check forward: $80 \to 40 \to 30 \to 20 \to 14$.
Answer:
$\$80$.
Olympiad / Challenge
A number machine takes a positive integer, then in order: multiplies by $3$, adds $5$, multiplies by $2$, and subtracts $4$. The machine outputs $66$. Searching forward over possible inputs is slow — find the input directly.
1
Output is $66$. Invert the last operation 'subtract $4$' with 'add $4$': $66 + 4 = 70$.
2
Invert 'multiply by $2$' with 'divide by $2$': $70 \div 2 = 35$.
Invert 'multiply by $3$' with 'divide by $3$': $30 \div 3 = 10$. Check forward: $10 \to 30 \to 35 \to 70 \to 66$.
Answer:
The input was $10$.
Going Deeper
Working backwards is exactly inverting a function composition: if the process is $f = f_k \circ \cdots \circ f_1$, then the start is $f_1^{-1} \circ \cdots \circ f_k^{-1}$ applied to the end — undo the last operation first.
It dominates 'I'm thinking of a number' puzzles, probability questions phrased from the final outcome, and AIME process problems where the end state is given; backward search replaces a blind forward guess-and-check.
Pitfall: an inverse step must stay legal. Halving an odd number, dividing when the value is not a multiple, or taking a square root with a sign ambiguity can introduce non-integer or extraneous predecessors — track every branch and discard the impossible ones.
Spot the Signal
Look for problems where the key step is starting from the desired result or final state when forward construction is unclear.
You can describe the hard part as starting from the desired result or final state when forward construction is unclear, but a direct attack starts producing clutter.
The problem rewards preserving structure instead of expanding, listing, or guessing too early.
Learn the Move
Start by identify the problem-solving bottleneck that calls for work backwards, then rewrite the givens around it.
Name the relation that makes Work Backwards legal before doing computation.
Use the new relation to replace the messiest part of the problem with a cleaner one.
Translate the result back to the quantity the problem actually asks for.
Avoid These Traps
Do not use Work Backwards just because the surface looks familiar; verify the required condition first.
Applying Work Backwards because it sounds relevant, without checking the trigger first.
Stopping after spotting the technique instead of finishing the calculation or proof.
MathGrit Coach Note
Let work backwards reveal the strategic structure; then compute only what remains.
Try it on:
Practice a contest problem where the key step is starting from the desired result or final state when forward construction is unclear.