Question 2800: Area of a Half-Open Unit-Cell Union

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2), (0, 2), (1, 2), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. The half-open unit cells are listed by lower-left coordinates as [(-2, -2), (-1, -2), (0, -2), (1, -2), (2, -2), (-2, -1), (-1, -1), (0, -1), (1, -1), (2, -1), (-2, 0), (-1, 0), (0, 0), (1, 0), (2, 0), (-2, 1), (-1, 1), (0, 1), (1, 1), (2, 1), (-2, 2), (-1, 2), (0, 2), (1, 2), (-2, -2)]; each (i,j) represents [i,i+1)×[j,j+1). Find the area of their union in squared units.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-area

Step-by-step derivation

orthonormal equal-length coordinate units;half-open regions

area in squared length units

[[-2, -2], [-1, -2], [0, -2], [1, -2], [2, -2], [-2, -1], [-1, -1], [0, -1], [1, -1], [2, -1], [-2, 0], [-1, 0], [0, 0], [1, 0], [2, 0], [-2, 1], [-1, 1], [0, 1], [1, 1], [2, 1], [-2, 2], [-1, 2], [0, 2], [1, 2], [-2, -2]]

[[-2, -2], [-2, -1], [-2, 0], [-2, 1], [-2, 2], [-1, -2], [-1, -1], [-1, 0], [-1, 1], [-1, 2], [0, -2], [0, -1], [0, 0], [0, 1], [0, 2], [1, -2], [1, -1], [1, 0], [1, 1], [1, 2], [2, -2], [2, -1], [2, 0], [2, 1]]

[[-2, -2], [-2, -1], [-2, 0], [-2, 1], [-2, 2], [-1, -2], [-1, -1], [-1, 0], [-1, 1], [-1, 2], [0, -2], [0, -1], [0, 0], [0, 1], [0, 2], [1, -2], [1, -1], [1, 0], [1, 1], [1, 2], [2, -2], [2, -1], [2, 0], [2, 1]]

24 distinct, pairwise disjoint unit cells

unit-cell union area := sum(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

unit-cell union area = 24

Reverse checks

• S004 · premise_and_result_audit — Recompute unit-cell union area with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

24

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.