🕹️ Practice arcade

⚡ Counting Toolkit Sprint

Ninety seconds of product-rule, sum-rule, permutation, and combination counts — as many as you can.

🔀 Conditional, Bayes & Expected Value Sprint

Conditional probability from a joint/marginal, Bayes' posterior, and E(X) from a table -- two minutes, mixed.

🔢 Set Cardinality Sprint

Count a power set's subsets, apply inclusion-exclusion for |A∪B|, or read off |A∩B| from two explicit sets -- all closed-form, all fast.

🧮 Functions Sprint

Evaluate a floor or ceiling, then decide whether a listed function is one-to-one -- read the outputs, don't guess.

🧩 Equivalence Classes & Fibonacci Sprint

Count a congruence-mod-n class's elements, or iterate F(n)=F(n-1)+F(n-2) from F0=0, F1=1 -- two chapters' own defined-structure counts.

🎲 Counting Toolkit Sprint

Factorials, the permutation-vs-combination call, the pigeonhole guarantee, and the Division Algorithm's own remainder -- four fast staples in one run.

🌐 Graph & Tree Structure Sprint

The Handshaking Theorem, a vertex's own degree, and a tree's edge count -- the arithmetic every graph-theory chapter leans on.

⚙️ Algorithm Arithmetic Sprint

Binary search's worst-case comparison count, a nested loop's total operations, and linearity of expectation -- three exact closing staples.

🧠 Tautology, Equivalence & Inference Sprint

Classify a formula as tautology/contradiction/contingency, check two formulas for logical equivalence, or judge an argument's validity.

🧮 Onto, Bijection & Composition Sprint

Decide whether a function is onto, whether it's a full bijection, or compose two explicit functions at a point.

🔢 Cartesian Product & Inclusion-Exclusion Sprint

Count an ordered-pair product |A×B|, or apply three-set inclusion-exclusion to find |A∪B∪C|.

🎲 Pascal's Rule & Combinations Sprint

Apply Pascal's identity to a binomial coefficient, or count an unordered selection directly.

🔁 Recurrence Iteration Sprint

Iterate a linear recurrence or a differently-shaped recursive definition from its base case up to term n.

🔄 Euler Circuit Sprint

A connected graph has an Euler circuit only when every vertex has even degree -- check as many as you can.

🌳 Tree Formulas Sprint

Cayley's spanning-tree count, a full binary tree's max node count, or a full m-ary tree's leaf count -- three closed forms, back to back.

🪞 Verdict Survival

Every question here is a single T/F or yes/no call -- a truth-table row, a conditional, a quantifier negation, a De Morgan swap, a reflexivity check, a Big-O comparison -- and one wrong verdict ends the round.

🔮 Quantifier Truth Survival

A universal or existential claim over a small finite domain -- one wrong true/false call ends the run.

🪞 Symmetric & Transitive Survival

Check a small relation for symmetry, antisymmetry, or transitivity from its explicit pairs -- one wrong verdict ends the round.

🌐 Paths, Cycles & Bipartite Survival

Is this sequence a valid path? Does this graph have a cycle? Is it bipartite? One wrong verdict ends the run.

📉 Big-Omega & Big-Theta Survival

Big-O's own lower-bound and tight-bound cousins -- one wrong growth-rate verdict ends the run.

🎯 Proof Technique Mastery

Direct, contrapositive, or contradiction -- name what each technique assumes at the start, five correct in a row.

🎯 Partial Order & Equivalence Relation Mastery

A relation is a partial order or an equivalence relation only if all three of its defining properties hold at once -- five correct in a row.