Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 120 minutes, as an unvalidated planning estimate. This collection contains 3 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.
State whether a claim concerns one preference completion, every completion or one specified full profile.
Check both endpoints of every candidate blocking pair.
Separate termination, stability, cardinality and the identities of matched participants.
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0 of 3 tasks have recorded responses. No overall mastery score is assigned.
Fix the matching A-X,B-Y,C-Z. The proposer lists are A:Y>X>Z, B:Z>Y>X and C:X>Z>Y. Receiver comparisons are only X:B>A and B>C; Y:B>A; Z:C>B. Every receiver independently completes these comparisons to a strict ranking of A,B,C. Count all complete receiver profiles and those in which the fixed matching is stable. Identify every possible blocking pair, and give one additional receiver comparison that makes the matching stable under all remaining completions. Explain why omitted comparisons are not unacceptable 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.
There are five participants per side. Two blocks have proposers p_i,q_i and receivers a_i,b_i for i=1,2; one singleton block has proposer s and receiver t. Everyone ranks all opposite-side members of their own block above all outsiders, whose order is arbitrary but strict. Within each two-by-two block, p_i ranks a_i>b_i, q_i ranks b_i>a_i, a_i ranks q_i>p_i, and b_i ranks p_i>q_i. Determine exactly how many stable matchings exist for every allowed outsider ordering. Prove no stable matching can cross blocks. Identify the proposer-proposing and receiver-proposing deferred-acceptance outcomes and count stable outcomes produced by neither direction.
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.
Three proposers rank A:X>Y, B:X>Y, C:Y>X; receivers rank X:C>B>A and Y:A>C>B. Run deferred acceptance, always selecting the alphabetically first free proposer who still has an untried receiver. List every proposal, held pair and rejection through termination. Verify stability by checking all unpaired pairs, and state the unmatched participant. Explain why B must make no further proposals and why this run now stops: no free proposer has an untried receiver. Then remove X from A’s acceptable list and remove A from X’s list, with all other comparisons unchanged: does the same trace remain legal, and which step must change?
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
Fix the matching A-X,B-Y,C-Z. The proposer lists are A:Y>X>Z, B:Z>Y>X and C:X>Z>Y. Receiver comparisons are only X:B>A and B>C; Y:B>A; Z:C>B. Every receiver independently completes these comparisons to a strict ranking of A,B,C. Count all complete receiver profiles and those in which the fixed matching is stable. Identify every possible blocking pair, and give one additional receiver comparison that makes the matching stable under all remaining completions. Explain why omitted comparisons are not unacceptable pairs.
There are five participants per side. Two blocks have proposers p_i,q_i and receivers a_i,b_i for i=1,2; one singleton block has proposer s and receiver t. Everyone ranks all opposite-side members of their own block above all outsiders, whose order is arbitrary but strict. Within each two-by-two block, p_i ranks a_i>b_i, q_i ranks b_i>a_i, a_i ranks q_i>p_i, and b_i ranks p_i>q_i. Determine exactly how many stable matchings exist for every allowed outsider ordering. Prove no stable matching can cross blocks. Identify the proposer-proposing and receiver-proposing deferred-acceptance outcomes and count stable outcomes produced by neither direction.
Three proposers rank A:X>Y, B:X>Y, C:Y>X; receivers rank X:C>B>A and Y:A>C>B. Run deferred acceptance, always selecting the alphabetically first free proposer who still has an untried receiver. List every proposal, held pair and rejection through termination. Verify stability by checking all unpaired pairs, and state the unmatched participant. Explain why B must make no further proposals and why this run now stops: no free proposer has an untried receiver. Then remove X from A’s acceptable list and remove A from X’s list, with all other comparisons unchanged: does the same trace remain legal, and which step must change?
