Solving a small timing constraint problem

Step 1 of 2

The request

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Solve this fictional timing-constraint problem directly as text, with no tools, files, code or external research. Event times A,B,C,D,E are integers; time differences are inclusive and events have zero duration. A=0; 2<=B-A<=4; 3<=C-A<=5; 2<=D-B<=3; 1<=D-C<=2; E-D=2; E<=7. List EVERY feasible tuple (A,B,C,D,E), then the tight earliest/latest interval for each event. Provide a witness attaining the minimum E and one attaining the maximum E. Explain why individual feasible intervals do not imply that every combination of values within those intervals is jointly feasible, giving one concrete counterexample. Audit the coordinator's claim "Because B may be 4 and D may be 4, choosing B=4 and D=4 together is feasible." Use only the stated constraints, not scheduling conventions or unstated resource limits.

The journey

Read the requestTurned it into a list of things it would have to prove before calling the work done.
Did the workPlanned the pieces, built them and checked the result as it went.
DeliveredThe answer is written below.

The result

{ "enumeration": { "method": "A=0 gives B∈{2,3,4} and C∈{3,4,5}. Since E=D+2≤7, D≤5. Also D∈[B+2,B+3]∩[C+1,C+2].", "pair_check": { "(B,C)=(2,3)": [ 4, 5 ], "(2,4)": [ 5 ], "(2,5)": [], "(3,3)": [ 5 ], "(3,4)": [ 5 ], "(3,5)": [], "B=4": "none, because D≥6>5" }, "feasible_tuples": [ [ 0, 2, 3, 4, 6 ], [ 0, 2, 3, 5, 7 ], [ 0, 2, 4, 5, 7 ], [ 0, 3, 3, 5, 7 ], [ 0, 3, 4, 5, 7 ] ], "constraint_check": [ { "tuple": [ 0, 2, 3, 4, 6 ], "B-A": 2, "C-A": 3, "D-B": 2, "D-C": 1, "E-D": 2, "E≤7": true }, { "tuple": [ 0, 2, 3, 5, 7 ], "B-A": 2, "C-A": 3, "D-B": 3, "D-C": 2, "E-D": 2, "E≤7": true }, { "tuple": [ 0, 2, 4, 5, 7 ], "B-A": 2, "C-A": 4, "D-B": 3, "D-C": 1, "E-D": 2, "E≤7": true }, { "tuple": [ 0, 3, 3, 5, 7 ], "B-A": 3, "C-A": 3, "D-B": 2, "D-C": 2, "E-D": 2, "E≤7": true }, { "tuple": [ 0, 3, 4, 5, 7 ], "B-A": 3, "C-A": 4, "D-B": 2, "D-C": 1, "E-D": 2, "E≤7": true } ] }, "tight_intervals": { "A": "[0,0]", "B": "[2,3]", "C": "[3,4]", "D": "[4,5]", "E": "[6,7]" }, "extremal_witnesses": { "minimum_E": { "value": 6, "witness": [ 0, 2, 3, 4, 6 ] }, "maximum_E": { "value": 7, "witness": [ 0, 2, 3, 5, 7 ] } }, "dependence_counterexample": "(0,3,3,4,6) lies within every marginal interval, but D-B=1 violates 2≤D-B≤3; therefore marginal intervals do not imply arbitrary joint feasibility.", "claim_audit": "The claim that B=4 and D=4 can be chosen together is false: D-B=0, violating 2≤D-B≤3. More generally, B=4 forces D≥6, contradicting D≤5 (and hence E≤7)." }

Time2 min 52 s

Cost$0.12

Finished2026-10-03

Step 2 of 2

The request

Deadline change feasibility check for event scheduling

Read the full request

Update this integer event timing problem after changing only the final deadline to E<=5: A=0; 2<=B-A<=4; 3<=C-A<=5; 2<=D-B<=3; 1<=D-C<=2; E-D=2. Events have zero duration and differences are inclusive. Determine feasibility, give a short proof using the stated constraints, and find the smallest replacement upper deadline for E that restores feasibility, with a complete witness tuple (A,B,C,D,E). Return only the changed feasibility result, proof, minimum feasible deadline and witness, directly as text. Do not repeat the former solution table or invent resource constraints. No tools, files, code or external research.

The journey

Read the requestTurned it into a list of things it would have to prove before calling the work done.
Did the workPlanned the pieces, built them and checked the result as it went.
DeliveredThe answer is written below.

The result

Changed result: infeasible with E<=5. Since A=0 and B-A>=2, B>=2; then D-B>=2 gives D>=4, so E=D+2>=6. Minimum feasible replacement deadline: E<=6. Witness: (A,B,C,D,E)=(0,2,3,4,6), satisfying all constraints.

Time1 min 07 s

Cost$0.07

Finished2026-10-03

Solving a small timing constraint problem

Step 1 of 2

The request

The journey

The result

Step 2 of 2

The request

The journey

The result

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