Mathematical Structures

Order matters — swapping two elements creates a different arrangement.

Arrangements around a circle have no fixed starting point; rotations are considered the same.

Required neighbors can be bound into one block, then arrange blocks.

Forbidden neighbors are handled by placing main items first, then using gaps.

When some items are identical, divide by the factorial of each group of identical items.

Count arrangements where no item appears in its original position.

If there are m ways to do one thing and n ways to do another, there are m × n ways to do both.

Split the problem into non-overlapping cases and count each separately.

Count the total first, then subtract what is NOT wanted.

Some objects are counted more than once; duplicates must be identified and removed.

When order matters, count all possible arrangements of selected items.

When order doesn't matter, count sets of items without regard to arrangement.

Count valid paths on a grid under specific movement rules.

Distribute identical items into distinct groups by placing dividers among them.

Parity (odd/even) often determines what is possible before exact computation.

Operations preserve or change parity in predictable ways.

Use divisibility conditions as filters to narrow candidates instead of testing blindly.

Some quantities maintain their parity regardless of allowed operations.

Use parity to prove impossibility or to narrow down possibilities.

A complicated region becomes simple after splitting into or subtracting familiar shapes.

Volume and surface area of 3D shapes follow specific formulas.

Angles in geometric figures follow specific relationships (sum, equality, complementary, etc.).

Angle values are linked by structural relationships (supplementary, vertical, triangle sum).

Similar figures have proportional sides and equal angles.

In a right triangle, a² + b² = c² where c is the hypotenuse.

Break complex shapes into simple parts, find each area, then combine.

Perimeter depends on the boundary only, not the interior.

The degree of a vertex is the number of edges connected to it.

A graph's structure becomes clear when you count each node's degree (number of connections).

The number of odd-degree nodes determines whether a path using every edge is possible.

Assign colors to vertices so that no two adjacent vertices share the same color.

A graph whose vertices can be divided into two sets with no edges within the same set.

Use AND, OR, NOT operations to evaluate compound logical statements.

Use given conditions one by one to rule out impossible options until only the answer remains.

From 'If P then Q', we can deduce certain conclusions but not others.

Use constraints to systematically eliminate impossible options until only the answer remains.

If 'P implies Q' is true, then 'not Q implies not P' is also true (and vice versa).

Make the locally optimal choice at each step, hoping to find the global optimum.

Look at the maximum or minimum case to gain insight or prove a result.

For non-negative numbers, arithmetic mean ≥ geometric mean, with equality when all numbers are equal.

The difference between consecutive terms is constant.

Two separate rules interleave — odd and even positions follow different patterns.

Each term is obtained by multiplying the previous term by a constant ratio.

A visual configuration follows a structural rule that can be counted.

Each term depends on one or more previous terms according to a fixed rule.

The pattern repeats after a fixed number of terms.

The shortest route should be compared structurally, not guessed visually.

Find the path with minimum total distance or cost between two points.

Count or find paths that satisfy specific rules (only right/down, avoid obstacles, etc.).

Find a path that visits every vertex exactly once.

Remainders follow predictable patterns when adding, subtracting, or multiplying.

Remainders cycle in predictable patterns for powers and sequences.

The last digit of powers follows a short repeating cycle.

Shortcuts to check if a number is divisible by common divisors without full division.

Find a number that satisfies multiple remainder conditions simultaneously.

A figure has reflection symmetry if it looks the same when flipped across a line.

A figure has rotational symmetry if it looks the same after being rotated by less than 360°.

Numbers or sequences that read the same forwards and backwards.

Distance = Rate × Time. Use this relationship to model motion problems.

If someone completes a job in T time, their rate is 1/T per unit time. Rates add when working together.

Track total amount and concentration separately. Final amount = sum of parts; final concentration = weighted average.