Solve a Simpler Problem is the habit of reducing numbers, cases, or dimensions without losing the core structure. 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.
No tagged problems yet We are still curating launch-ready practice for this technique.
The Key Result
$$\text{hard case} \;\longrightarrow\; \text{strip away size/dimension} \;\longrightarrow\; \text{solvable case with the same structure.}$$
Shrink the numbers, dimensions, or number of objects to a tiny version you can solve completely, learn the mechanism, then scale the insight back up.
Why It Works
1
Replace the large or general parameter with the smallest value that still shows the problem's essential structure.
2
Solve that miniature version fully — by hand if needed — and watch how the answer is built.
3
Identify the part of the small solution that does not depend on the size, only on the structure.
4
Generalize that structural insight to the original parameter, giving a formula or argument for the full problem.
Worked Examples
Example 1
How many diagonals does a convex polygon with $20$ sides have?
1
Start small: a quadrilateral ($4$ vertices) has $2$ diagonals; a pentagon ($5$) has $5$; a hexagon ($6$) has $9$.
2
Notice the mechanism: each vertex connects to all others except itself and its $2$ neighbors, so to $n - 3$ vertices.
3
Summing over $n$ vertices counts each diagonal twice (once from each end), giving $\frac{n(n-3)}{2}$.
4
Check on the small cases ($n=5$: $\frac{5\cdot 2}{2}=5$ ✓), then set $n = 20$: $\frac{20 \cdot 17}{2} = 170$.
Answer:
$170$ diagonals.
Contest level
At a party of $12$ people, everyone shakes hands exactly once with every other person. How many handshakes occur?
1
Shrink it: $2$ people give $1$ handshake, $3$ people give $3$, $4$ people give $6$.
2
See the mechanism: each of the $n$ people shakes hands with the other $n - 1$, but each handshake is then counted twice (once per participant).
3
So the count is $\frac{n(n-1)}{2}$; check $n = 4$: $\frac{4\cdot 3}{2} = 6$ ✓.
4
Scale to $n = 12$: $\frac{12 \cdot 11}{2} = 66$.
Answer:
$66$ handshakes.
Olympiad / Challenge
Into how many regions do $6$ lines in 'general position' (no two parallel, no three through one point) divide the plane?
1
Solve tiny cases: $0$ lines give $1$ region, $1$ line gives $2$, $2$ lines give $4$, $3$ lines give $7$.
2
Find the structural rule: the $k$th line crosses the previous $k - 1$ lines in $k - 1$ points, which split it into $k$ pieces, and each piece cuts one old region into two — so adding the $k$th line adds $k$ regions.
Specializing a parameter to its smallest meaningful value is how you discover a recurrence: the jump from the $(n-1)$-case to the $n$-case ('what does one more line/vertex/object add?') often is the whole solution.
It is the universal first move on a hard AMC/AIME combinatorics or geometry problem: replace $2026$ or $n$ by $1, 2, 3$, solve completely, and let the mechanism — not the big number — guide you.
Pitfall: the small case must keep the structure that makes the problem hard. Degenerate tiny cases (e.g. $n = 1$ or $n = 2$ where lines cannot yet cross) can mislead — the genuine pattern often only appears once the interaction you care about can occur.
Spot the Signal
Look for problems where the key step is reducing numbers, cases, or dimensions without losing the core structure.
You can describe the hard part as reducing numbers, cases, or dimensions without losing the core structure, 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 solve a simpler problem, then rewrite the givens around it.
Name the relation that makes Solve a Simpler Problem 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 Solve a Simpler Problem just because the surface looks familiar; verify the required condition first.
Applying Solve a Simpler Problem 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 solve a simpler problem reveal the strategic structure; then compute only what remains.
Try it on:
Practice a contest problem where the key step is reducing numbers, cases, or dimensions without losing the core structure.