B.5.QD - Develop a New Question from a Result or Construction

Type: Method-description pattern Status: Draft Normativity: Normative unless marked informative

B.5.QD:1 - Problem frame

Use this when a result, counterexample or newly available construction changes what you could usefully investigate, but the next question is still unclear. A failed conjecture may reveal a missing condition. A successful calculation may supply an operation that makes a previously impractical question approachable. The work is to turn that change into a question you can begin to answer.

First useful move. Recover the step at which the earlier reasoning failed, or the operation that has become available. Ask what it now lets you distinguish, construct, explain or change. For example, a triangle refutes the claim that every connected graph can be divided into two groups with every edge crossing between groups. It also suggests asking for a construction that returns either the division or an obstruction.

The result is a question with an understood answer form, a useful first attempt and a reason to pursue or retain it. The answer may support further theory, a different investigation or an action in the world. Developing a concept or a reusable operation can be that contribution even before its eventual applications are known.

Use an already adequate question directly. If the question is clear and the missing contribution is a way to answer it, C.39 supplies that search. This pattern is useful when the result sought or the question’s conditions themselves need to be developed.

B.5.QD:2 - Problem

“Find the next interesting problem” leaves the crucial work undone. Many variations merely change a number or a name. Others repeat a failed claim with a convenient exception, ask for information the available description cannot determine, or generate questions whose answers would add nothing to the inquiry.

A useful next question grows from a consequential change: an exposed dependency, a new object or operation, a newly distinguished case, or a result that can be used elsewhere. The difficulty is to construct that question while preserving what the earlier work established and keeping a feasible point of entry.

B.5.QD:3 - Forces

B.5.QD:4 - Solution

Recover what changed → locate the dependency or new operation → construct a consequential question → work a revealing case → choose the next inquiry and keep what remains useful.

Enter with the result that actually changed the inquiry. Reuse a known formulation or derivation when it already supplies the needed step. Return to an earlier step if an attempted answer reveals that the question is ambiguous, underdetermined or more expensive than its use warrants.

B.5.QD:4.1 - Recover the contribution of the earlier work

State the earlier question and what the work actually obtained. Identify the part that matters now: a counterexample, an unresolved inference, a constructed object, a computational operation or a distinction the earlier description omitted.

For a failure, locate its scope. A counterexample to a lemma can expose a defect in one argument while leaving the main conjecture undecided. A counterexample satisfying the main conjecture’s premises refutes that conjecture. A program’s failure on a case may instead concern its implementation. Recover the relevant reasoning under B.5.RA or B.5.RR when this difference is unclear.

For a success, recover what can now be done with the result. Explain its inputs, output and application conditions. A computation that produces cumulative totals, for example, may provide a way to answer many interval questions. Its reusable contribution is the relation between those totals and the intervals.

Keep independently supported results available. They can supply the construction, limiting case or partial answer for the next question.

B.5.QD:4.2 - Find the relation that can change the question

Follow the earlier result back to a consequential dependency, or forward to an operation it enables.

If a description gives the same information for cases that need different answers, construct two such cases. Their difference identifies information a stronger question may need. If an inference depends on an unproved step, state the missing step as a possible question. If an operation succeeds, ask which other inputs, outputs or combinations preserve its useful relation.

Use variations that have a reason in the work. The following are common ways to construct them.

A variation can change the original task. Preserve that change explicitly: restricting a claim to trees may give a valid result while leaving the original request about all graphs unanswered. If the broader request still matters, keep it as an unresolved question.

These variations can be combined or repeated. Their purpose is to expose a useful answer, not to fill a catalogue of question types.

When requirements appear incompatible, B.5.QD.CF recovers the premises producing the conflict and asks what other construction could satisfy the retained need. It distinguishes changing an assumed means or representation from changing a requirement, and can return a useful impossibility when the conditions must remain.

B.5.QD:4.3 - Give the new question an answer form

Say what would answer the question: for example, a construction, an explanatory relation, a bound, a counterexample or a condition under which an operation works. State the objects, allowed changes and premises that can alter the answer.

Connect that answer to its possible use. “Find a partition” and “return a partition or an odd-cycle witness” make different contributions: the second also explains failure. “Predict the position” and “bound the reachable positions” may serve the same present decision at different effort.

Separate a proposed answer from the question. “Does this procedure always terminate on finite inputs?” remains a useful question when the proposed positive answer is false. The failing case can open the question of a termination condition or another procedure.

If a question combines several missing contributions, make the dependency visible. For a physical calculation, you may first need a model that determines a quantity, then a computation of that model, then an interpretation for action. B.5.MPC coordinates those contributions. Name the first unresolved connection so a person, AI agent or group can work on it using the available subject knowledge and tools.

B.5.QD:4.4 - Work a case that reveals the difficulty

Choose a small case that exercises the changed condition or new operation. Attempt the requested construction or inference and explain where it succeeds or stops.

Use the result to improve the question. A successful case can expose a reusable relation. A failed case can identify a missing premise, an incompatible demand or an operation still to be developed. If two admissible cases give different answers from the same supplied information, ask for the missing distinction or a result valid for both.

A case establishes its own result. A broader mathematical claim needs its derivation, and applying a physical model needs the relevant physical basis. The question can already be useful before those answers are established: the first attempt should make the next contribution more identifiable.

Begin with available reasoning and information. Obtain further evidence only when its possible answers can change the useful inquiry enough to justify the effort under C.11.DUA. An existing bound or conditional answer may settle the current use.

B.5.QD:4.5 - Choose an attainable inquiry

Compare the few questions that remain serious candidates. Ask what each answer would enable and where work could begin with the available capabilities, collaborators and resources. Include the cost of learning or obtaining a missing operation when it affects the choice.

A worthwhile first question may be narrower than the eventual aim: establish a limiting case, find one witness, recover a missing lemma or test a proposed connection. Explain how that result would contribute to the larger question. If the contribution is already available, use it and select the next unresolved step.

E.10.INT helps distinguish useful interest from novelty, surprise or a local scoring heuristic. A question can be worth pursuing because it opens further constructions, changes what can be explained or makes another question approachable. Its eventual practical destination may remain unknown. When problems and ways of solving them must develop together, use C.40; retain a promising alternative when its prospective use justifies doing so.

Continue with the chosen question, its first attempt and the dependency that attempt is meant to resolve. A short explanation in the work can carry this result. Stop when the current need is answered, or when another inquiry is the better use of the available effort.

B.5.QD:4.6 - Separate question recognition from answer assurance

Recognize the opening from a changed possibility of inquiry. One worked counterexample or a useful new operation can be sufficient to begin developing the question.

Judge the question by whether it has a recoverable meaning, a possible contribution and a workable point of entry. Judge its proposed answer by the claim it makes. A conjecture, a proven relation, a simulated result and an observed physical effect support different uses. Apply their subject Methods and B.3 when the receiving use needs that assurance.

Reuse the question while its purpose and conditions remain applicable. Reopen it when a result changes its premises, available operations, attainable scope or intended use. A solved question can open another valuable one; an adequate answer can also end the present work.

B.5.QD:5 - Archetypal Grounding

B.5.QD:5.1 - From a false graph claim to a construction or obstruction

An engineer represents pairwise incompatibilities by a finite simple undirected graph. The proposed claim is that connectedness suffices to divide the vertices into two groups so every edge crosses between groups.

A triangle refutes the claim. Assign A to the first group and its neighbour B to the second. The third vertex C is adjacent to both, so neither group is available. The failure concerns the universal claim, not the ability to divide any graph: a four-vertex cycle A-B-C-D-A admits groups {A,C} and {B,D}.

Follow the failed operation. Along a path, successive vertices can alternate between the two groups. Returning around an odd cycle forces its final edge to join vertices assigned to the same group. That identifies an obstruction worth seeking.

The next question is: For a given finite simple undirected graph, can we construct the division or return an odd cycle that explains why it is impossible? The answer form now serves both allocation and diagnosis.

B.5:5.1 supplies the broader construction. Traverse each component by breadth-first search and assign groups by even or odd depth. If every edge joins opposite parities, the assignment works. An edge joining equal parities combines with the two parent paths up to their last shared vertex to give a simple odd cycle. The construction therefore answers the new question for the stated graph class.

The earlier connectedness requirement can be dropped: work through each component, including isolated vertices. For a real allocation, establish that the graph represents the relevant pairwise incompatibilities. Three-way constraints would change that application question.

B.5.QD:5.2 - From a position to a sufficient state or useful bound

A model describes a point moving along one line in an inertial frame. During the next two seconds there is no net force; use the classical relation q(t) = q(0) + v(0)t. The supplied position is q(0) = 0. Someone asks where the point will be two seconds later.

Construct two cases allowed by this description. With v(0) = +1 metre per second, q(2) = +2 metres. With v(0) = -1 metre per second, q(2) = -2 metres. Position alone leaves the requested answer undetermined.

One next question is what information completes the state for this prediction? Under this model, initial velocity together with position suffices. Obtaining velocity could use an already available displacement over a known interval of constant velocity. A.3.3 and C.16 develop state and measurement when those contributions are needed.

Another question is cheaper if the current decision only asks whether the point stays within three metres of its starting position for the next two seconds. Suppose the available information bounds the initial velocity between -1 and +1 metre per second. Then the displacement magnitude is at most (1 metre per second)t throughout that interval, so the point stays within two metres. The bound answers the decision without acquiring a more specific velocity.

The two questions support different uses. If an actuator must meet the point at a specified position, the bound can be insufficient and the state question becomes useful. If the net force becomes nonzero during the interval, revise the model and its prediction. The construction of the question has located the relevant missing contribution in each case.

B.5.QD:5.3 - A successful cumulative computation opens an interval question

A program can construct cumulative totals for an integer array a. It starts with S[0] = 0 and sets S[i+1] = S[i] + a[i]. For a = [2,-1,3,4], the result is S = [0,2,1,4,8].

The original task needed the total 8. Recovering the operation reveals that each S[i] already gives the sum before position i. This opens the question: Can we answer many interval-sum queries from the same cumulative totals?

Specify an interval by indices l and r, with 0 ≤ l ≤ r ≤ n; it includes l and ends just before r. Splitting the first r elements at l gives S[r] = S[l] + sum(a[l],…,a[r-1]). Hence the interval sum is S[r] - S[l]. For the last two elements, l = 2 and r = 4, so the answer is 8 - 1 = 7. For l = r the answer is 0.

This is a general derivation under integer arithmetic with enough capacity to avoid overflow. Each query uses two stored totals and one subtraction after the array has been prepared. Whether preparation is worthwhile depends on how many queries and changes the application requires.

If a[i] changes by d, every S[j] with j > i changes by d. Frequent changes therefore open a further question: which data organization supports the required mixture of updates and interval queries? That question can be developed when the workload makes it relevant. The successful cumulative operation already answers the unchanged-array question.

B.5.QD:6 - Bias-Annotation

The original formulation can anchor subsequent questions. Recovering its failed condition or unused operation helps reveal alternatives, but the new question should still explain what it contributes. A more familiar or easier formulation can silently abandon the unresolved use.

Mathematical search often favors statements that can be proved. Question development also benefits from locating a false conjecture, an underdetermined description or a construction that cannot meet its conditions. Their usefulness comes from the inquiry they enable.

B.5.QD:7 - Conformance Checklist

• The earlier work and its consequential result are recoverable.
• A failure is located at the claim, premise, inference or implementation it affects; a success exposes its reusable operation or relation.
• The new question states an answer form and the conditions needed to understand it.
• A changed formulation preserves the still-unanswered part of the earlier task when that part remains useful.
• A worked attempt reveals a consequence, a missing distinction or the operation to obtain next.
• The question’s possible contribution and its attainable first step justify pursuing or retaining it.
• The proposed answer is used with the scope and support appropriate to its claim.
• Further work can stop when an available answer suffices; explanations and records are only as extensive as their receiving use requires.

B.5.QD:8 - Common Anti-Patterns and How to Avoid Them

B.5.QD:9 - Consequences

Question development turns both failure and success into possible further work. It can preserve a useful construction, make a missing premise investigable or reveal a result that serves another practice. Contributors can divide the work around those identified dependencies.

There is a cost to opening alternatives. A few consequential variations and a revealing attempt often suffice. Some questions remain worth keeping until a needed operation, collaborator or use becomes available; others can be set aside after their first attempt shows little prospective contribution.

B.5.QD:10 - Architectural Rationale

A new question needs an operation of construction. The instruction to find an interesting question becomes usable when it points to a dependency that failed or an operation that became available. This also connects developing a theory with applying it: an application can expose a new question, and a theoretical construction can create an application that could not previously be attempted.

The answer form matters because it determines what work counts as progress. A witness of impossibility can be as useful as a successful construction for deciding an allocation. A bound can answer an operational question while a unique prediction remains unavailable. These alternatives are obtained by examining the earlier result and its use.

The small case and the broader question develop together. Working the case tests whether the question identifies a real difference; stating the broader question prevents the case from becoming the only thing learned. Retaining unaffected results makes this development cumulative.

Practical interest includes opening future inquiry. The method asks for the next intelligible contribution while leaving later uses open.

B.5.QD:11 - SoTA-Echoing

For developing a question after a failed proof or conjecture, adapt Lakatos’s proof-analysis method in Proofs and Refutations (I) (1963), §3 and the beginning of §4. Distinguishing a counterexample to a lemma from one to the main conjecture preserves recoverable reasoning; asking for the missing construction improves on protecting the claim by convenient restrictions alone. Sections :4.1–:4.3 and :5.1 use that contribution. The historical dialogue supplies a method of criticism and question change; the resulting mathematical claims still need their own arguments. Reopen the choice when a different failure prevents recovery of the consequential dependency. Original article.

For obtaining questions from either an unfinished proof or available constructions, adapt the distinction between goal-directed lemma discovery and bottom-up theory exploration in Zhang and Tan’s Automated Conjecturing and Theorem Finding: A Survey (2026), §§3 and 5. The two approaches support different entries in :4.2. The surveyed filters for false, redundant or uninteresting conjectures help theorem search; using those filters as a general question filter would discard a useful counterexample. Section :4.3 instead evaluates the question’s contribution separately from a proposed answer’s truth. Syntactic complexity and proof-related scores remain local heuristics. Reopen when a stronger generation method offers more useful questions at comparable effort for the receiving practice. Survey.

For finding further uses of a successful operation, adopt Blelloch’s constructive treatment of all-prefix-sums in Prefix Sums and Their Applications (1993), §1.1. It exposes an operation and its application conditions, improving on retaining only a program’s final answer. Section :5.3 uses cumulative addition to develop an interval question; its range-sum derivation is worked here. Prefix operations over other associative operators can serve other questions, but subtraction requires the additional algebraic structure used in this example. The source’s parallel implementations are separate Methods. Reopen when the operations, numeric semantics or workload change. Author’s chapter.

For questions beyond a fixed objective, adapt Wang et al.‘s Enhanced POET (2020), §§2–3, through C.40’s coupled development of problems and ways. Relative difficulty and transfer can make a question worth retaining even when it is poorly served by the present best method. Against requiring an already known route to a final objective, :4.5 keeps a useful nearer contribution and examines its cost. The reported computational environments demonstrate that search approach under their conditions; they do not establish a universal measure of interest or development. E.10.INT supplies the broader distinction. Reopen when retained questions cease to enable useful transfer or the contributors’ capabilities change. Paper.

B.5.QD:12 - Relations

• B.5 coordinates the reasoning cycle. B.5.RA recovers an argument; B.5.RR revises reasoning after a premise or question changes. This pattern constructs a question that can start that revision.
• B.5.TC can reveal a newly expressible question when comparing accounts. B.5.2 generates candidate explanations once that is the needed contribution.
• B.5.MPC connects the physical, mathematical and computational contributions a question needs. A.3.3, C.16 and C.29.2 supply state, measurement and computational-formulation work where applicable.
• C.39 obtains a missing way to produce a distinguished result. C.40 develops problems and ways together; this pattern contributes the question-construction step.
• C.22.2 describes a problem claim when that description is needed. E.10.INT, C.11 and C.11.DUA help choose a worthwhile inquiry and the effort to spend.
• C.36.RP supports continuing a shared practice by recovering and changing its ways of inquiry. A.15.9 helps obtain and use a bounded contribution from another practice when needed.

B.5.QD:End