Problem-Solving StrategyDifficulty 1-450 tagged problems

Make a Table

Make a Table is the habit of organizing small cases in rows and columns until a pattern or invariant appears. 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.

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The Key Result

$$\begin{array}{c|c} n & f(n) \\ \hline 1 & \cdots \\ 2 & \cdots \\ \vdots & \vdots \end{array} \;\Rightarrow\; \text{a pattern you can read down the column.}$$

Lay out small cases in an organized grid of rows and columns so a repeating cycle, recurrence, or invariant jumps out where guessing would fail.

Why It Works

1
Pick the input you want to vary (an exponent, a step number, a case label) as the rows of your table.
2
Compute the output for the first several inputs and record each in its own row so nothing is lost or miscounted.
3
Scan down the output column for repetition: a fixed cycle length, a constant difference, or each term built from earlier ones.
4
Once the rule is clear, project it to the target input without computing every intermediate row.

Worked Examples

Example 1

What is the units digit of $7^{2025}$?

1
Tabulate units digits of powers of $7$: $7^1 \to 7$, $7^2 \to 9$, $7^3 \to 3$, $7^4 \to 1$, $7^5 \to 7$.
2
The units digit column reads $7, 9, 3, 1, 7, 9, 3, 1, \dots$ — a cycle of length $4$.
3
So the units digit of $7^n$ depends only on $n \bmod 4$; compute $2025 = 4\cdot 506 + 1$, so $2025 \equiv 1 \pmod 4$.
4
The exponent matches the first position in the cycle, whose units digit is $7$.

Answer:

$7$

Contest level

What is the units digit of the sum $3^1 + 3^2 + 3^3 + \cdots + 3^{2024}$?

1
Tabulate units digits of powers of $3$: $3^1 \to 3$, $3^2 \to 9$, $3^3 \to 7$, $3^4 \to 1$, $3^5 \to 3$ — a cycle of length $4$.
2
The units digits of one full block of four consecutive powers sum to $3 + 9 + 7 + 1 = 20$, which contributes units digit $0$.
3
There are $2024$ terms, and $2024 = 4 \cdot 506$, so the sum splits into exactly $506$ complete blocks with no leftover terms.
4
Each block contributes units digit $0$, so the whole sum ends in $0$.

Answer:

$0$

Olympiad / Challenge

Find the last two digits of $2^{2026}$.

1
Tabulate $2^n \bmod 100$ from $n = 2$: $4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52$ for $n = 2, \dots, 21$.
2
At $n = 22$ the value is $04$ again, matching $n = 2$, so $2^n \bmod 100$ repeats with period $20$ for $n \ge 2$.
3
Reduce the exponent within the cycle: $2026 = 2 + (2024)$ and $2024 \equiv 4 \pmod{20}$, so $2^{2026} \equiv 2^{2+4} = 2^6 \pmod{100}$.
4
From the table $2^6 = 64$, so the last two digits are $64$.

Answer:

$64$

Going Deeper

A table that repeats is a hidden modular cycle: powers, last-digit problems, and clock/calendar questions all reduce to 'find the period, then take the exponent mod the period.'
On AIME and AMC, organizing casework into a table (by residue, by case label, by step number) is what prevents the off-by-one and double-counting errors that sink otherwise-correct solutions.
Pitfall: confirm the cycle truly repeats before extrapolating. A column can look periodic for several rows and then break (e.g. carries in $\bmod 100$ shift things), and the cycle of $2^n \bmod 100$ only stabilizes from $n \ge 2$, not from $n = 0$.

Spot the Signal

Look for problems where the key step is organizing small cases in rows and columns until a pattern or invariant appears.
You can describe the hard part as organizing small cases in rows and columns until a pattern or invariant appears, 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 make a table, then rewrite the givens around it.
Name the relation that makes Make a Table 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 Make a Table just because the surface looks familiar; verify the required condition first.
Applying Make a Table 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 make a table reveal the strategic structure; then compute only what remains.

Try it on:

Practice a contest problem where the key step is organizing small cases in rows and columns until a pattern or invariant appears.