Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 140 minutes, as an unvalidated planning estimate. This collection contains 14 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose a method and state its assumptions before opening answer guidance. Use sketches, tables and intermediate calculations when they support your argument.
These public tasks sample the published unit. A saved response records practice, not a secure examination result or a guarantee of semester or AP readiness.
Written answers require human review for correctness, completeness and interpretation. Suggested time is planning guidance; the interface has no timer.
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0 of 14 tasks have recorded responses. No overall mastery score is assigned.
A grid has three colors and seven columns. Prove that every row contributes at least five same-color column-pair certificates by balancing its color counts. Deduce a sufficient row count for a monochromatic rectangle. Does the argument prove that twelve rows can avoid one? Generalize to q colors and c columns, including the case where the per-row lower bound is zero.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Twelve distinct parts form a4×3 grid of types and warehouses. Uniformly choose four parts without replacement, ignoring order. Count selections meeting the first type and first warehouse, then selections with exactly one from each. A proposed count chooses one marked-type part and one marked-warehouse part first, then any two remaining parts. Explain both its missing case and its multiplicity problem. How does the exact-one count change if the experiment is a uniformly random ordered sequence of four distinct parts?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A tree rooted at r has branches r–a–d–e, r–b–f and r–c. Count every vertex order beginning at r with at most one earlier neighbor at each vertex, proving why branch interleaving gives the complete set. Add a new leaf g attached to c and count again. Instead, return to the seven-vertex tree and add edge e–f: prove no order anywhere satisfies the earlier-neighbor condition. Distinguish this universal failure from testing a single ordering.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let k≥2. A graph has vertices a,b and c1,…,ck, with edges a–ci and ci–b for each i and no others. Classify and count all simple cycles, counting cycles without orientation or starting-point distinctions. How many contain both a,b, and how many contain two specified distinct branch vertices ci,cj? Prove completeness using degrees. Add a leaf t attached only to c1: give two distinct simple t-to-b paths but prove no simple cycle contains t. Explain why a generic two-path cycle argument loses the required endpoint.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that subsets of a nine-element labeled set with exactly four elements correspond bijectively to length-nine binary strings with exactly four ones. Count them. Separately count all nonempty subsets and explain why the two answers differ.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Find the coefficient of x³ in (2+x)^6. In a separate survey of 50 people, sets A,B,C have sizes 24,22,18; pairwise intersections have sizes 10,8,7 respectively and the triple intersection has size 4. Find the number in at least one and in none; justify every sign in the count.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that among any eight integers, two have a difference divisible by 7. Show that seven integers need not have this property and explain what your proof establishes about the smallest guaranteed sample size.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A sequence has a₀=3 and aₙ=2aₙ₋₁+1 for n≥1. Derive a closed formula and its ordinary generating function A(z)=Σₙ≥0 aₙzⁿ. Check the initial term and explain whether your coefficient derivation requires assigning a numeric value to z.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A connected simple undirected graph has seven vertices and degree sequence 3,2,2,2,1,1,1. Determine its edge count. Prove it is a tree using a valid general theorem or argument, and explain why dropping connectedness would invalidate that conclusion.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For the complete bipartite graph K₂,₄, determine whether it has an Euler circuit, an open Euler trail, a Hamilton cycle, and a proper two-coloring. Justify each conclusion rather than inferring one kind of traversal from another.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A task DAG has edges A→C, B→C, B→D, C→E, D→E. Give a topological order. Explain the invariant and termination argument for repeatedly removing a zero-indegree vertex. Then add E→B and explain why the same algorithm can get stuck with vertices remaining.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that T(n)=Σₖ₌₁ⁿ(2k+3) is Θ(n²) for positive integers n. Give explicit constants and explain why Θ(n²) says neither the exact elapsed time nor that all implementations run equally fast.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Use the Euclidean algorithm and back-substitution to find gcd(43,17), express it as an integer combination, and find the inverse of 17 modulo 43. Explain why the same method cannot produce an inverse of 18 modulo 42.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Solve x≡3 (mod 5), x≡4 (mod 7), giving every integer solution. Separately determine all integer solutions to 12u+15v=9. Explain why the first system has uniqueness modulo a product while the second has infinitely many pairs.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A grid has three colors and seven columns. Prove that every row contributes at least five same-color column-pair certificates by balancing its color counts. Deduce a sufficient row count for a monochromatic rectangle. Does the argument prove that twelve rows can avoid one? Generalize to q colors and c columns, including the case where the per-row lower bound is zero.
Twelve distinct parts form a4×3 grid of types and warehouses. Uniformly choose four parts without replacement, ignoring order. Count selections meeting the first type and first warehouse, then selections with exactly one from each. A proposed count chooses one marked-type part and one marked-warehouse part first, then any two remaining parts. Explain both its missing case and its multiplicity problem. How does the exact-one count change if the experiment is a uniformly random ordered sequence of four distinct parts?
A tree rooted at r has branches r–a–d–e, r–b–f and r–c. Count every vertex order beginning at r with at most one earlier neighbor at each vertex, proving why branch interleaving gives the complete set. Add a new leaf g attached to c and count again. Instead, return to the seven-vertex tree and add edge e–f: prove no order anywhere satisfies the earlier-neighbor condition. Distinguish this universal failure from testing a single ordering.
Let k≥2. A graph has vertices a,b and c1,…,ck, with edges a–ci and ci–b for each i and no others. Classify and count all simple cycles, counting cycles without orientation or starting-point distinctions. How many contain both a,b, and how many contain two specified distinct branch vertices ci,cj? Prove completeness using degrees. Add a leaf t attached only to c1: give two distinct simple t-to-b paths but prove no simple cycle contains t. Explain why a generic two-path cycle argument loses the required endpoint.
Prove that subsets of a nine-element labeled set with exactly four elements correspond bijectively to length-nine binary strings with exactly four ones. Count them. Separately count all nonempty subsets and explain why the two answers differ.
Find the coefficient of x³ in (2+x)^6. In a separate survey of 50 people, sets A,B,C have sizes 24,22,18; pairwise intersections have sizes 10,8,7 respectively and the triple intersection has size 4. Find the number in at least one and in none; justify every sign in the count.
Prove that among any eight integers, two have a difference divisible by 7. Show that seven integers need not have this property and explain what your proof establishes about the smallest guaranteed sample size.
A sequence has a₀=3 and aₙ=2aₙ₋₁+1 for n≥1. Derive a closed formula and its ordinary generating function A(z)=Σₙ≥0 aₙzⁿ. Check the initial term and explain whether your coefficient derivation requires assigning a numeric value to z.
A connected simple undirected graph has seven vertices and degree sequence 3,2,2,2,1,1,1. Determine its edge count. Prove it is a tree using a valid general theorem or argument, and explain why dropping connectedness would invalidate that conclusion.
For the complete bipartite graph K₂,₄, determine whether it has an Euler circuit, an open Euler trail, a Hamilton cycle, and a proper two-coloring. Justify each conclusion rather than inferring one kind of traversal from another.
A task DAG has edges A→C, B→C, B→D, C→E, D→E. Give a topological order. Explain the invariant and termination argument for repeatedly removing a zero-indegree vertex. Then add E→B and explain why the same algorithm can get stuck with vertices remaining.
Prove that T(n)=Σₖ₌₁ⁿ(2k+3) is Θ(n²) for positive integers n. Give explicit constants and explain why Θ(n²) says neither the exact elapsed time nor that all implementations run equally fast.
Use the Euclidean algorithm and back-substitution to find gcd(43,17), express it as an integer combination, and find the inverse of 17 modulo 43. Explain why the same method cannot produce an inverse of 18 modulo 42.
Solve x≡3 (mod 5), x≡4 (mod 7), giving every integer solution. Separately determine all integer solutions to 12u+15v=9. Explain why the first system has uniqueness modulo a product while the second has infinitely many pairs.
