Ordered Coordinate Fields, Axes, and Equality
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261011-advance-20261015-r07-coordinate-plane-foundations · 2026-10-11 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[0, 0], [1, 0], [0, -1], [-1, 1]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([0, 1, 0, -1], [0, 0, -1, 1], [Yes, Yes, No, No], [Yes, No, Yes, No], [origin, x-axis, y-axis, II])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([0, 1, 0, -1], [0, 0, -1, 1], [Yes, Yes, No, No], [Yes, No, Yes, No], [origin, x-axis, y-axis, II]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
2. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[1, 1], [-1, 1], [-1, -1], [1, -1]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([1, -1, -1, 1], [1, 1, -1, -1], [No, No, No, No], [No, No, No, No], [I, II, III, IV])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([1, -1, -1, 1], [1, 1, -1, -1], [No, No, No, No], [No, No, No, No], [I, II, III, IV]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
3. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-10, 1], [1, 0], [0, 1], [-1, -2]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-10, 1, 0, -1], [1, 0, 1, -2], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-10, 1, 0, -1], [1, 0, 1, -2], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
4. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-9, 1], [1, 0], [0, 1], [-1, -3]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-9, 1, 0, -1], [1, 0, 1, -3], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-9, 1, 0, -1], [1, 0, 1, -3], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
5. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-8, 1], [1, 0], [0, 1], [-1, -4]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-8, 1, 0, -1], [1, 0, 1, -4], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-8, 1, 0, -1], [1, 0, 1, -4], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
6. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-7, 1], [1, 0], [0, 1], [-1, -5]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-7, 1, 0, -1], [1, 0, 1, -5], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-7, 1, 0, -1], [1, 0, 1, -5], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
7. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-6, 1], [1, 0], [0, 1], [-1, -6]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-6, 1, 0, -1], [1, 0, 1, -6], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-6, 1, 0, -1], [1, 0, 1, -6], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
8. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-5, 1], [1, 0], [0, 1], [-1, -7]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-5, 1, 0, -1], [1, 0, 1, -7], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-5, 1, 0, -1], [1, 0, 1, -7], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
9. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-4, 1], [1, 0], [0, 1], [-1, -8]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-4, 1, 0, -1], [1, 0, 1, -8], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-4, 1, 0, -1], [1, 0, 1, -8], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
10. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-3, 1], [1, 0], [0, 1], [-1, -9]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-3, 1, 0, -1], [1, 0, 1, -9], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-3, 1, 0, -1], [1, 0, 1, -9], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
11. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-2, 1], [1, 0], [0, 1], [-1, -10]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-2, 1, 0, -1], [1, 0, 1, -10], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-2, 1, 0, -1], [1, 0, 1, -10], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
12. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[-1, 1], [1, 0], [0, 1], [-1, -11]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([-1, 1, 0, -1], [1, 0, 1, -11], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([-1, 1, 0, -1], [1, 0, 1, -11], [No, Yes, No, No], [No, No, Yes, No], [II, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
13. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[0, 1], [1, 0], [0, 1], [-1, -12]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([0, 1, 0, -1], [1, 0, 1, -12], [No, Yes, No, No], [Yes, No, Yes, No], [y-axis, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([0, 1, 0, -1], [1, 0, 1, -12], [No, Yes, No, No], [Yes, No, Yes, No], [y-axis, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
14. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[1, 1], [1, 0], [0, 1], [-1, -13]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([1, 1, 0, -1], [1, 0, 1, -13], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([1, 1, 0, -1], [1, 0, 1, -13], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
15. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[2, 1], [1, 0], [0, 1], [-1, -14]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([2, 1, 0, -1], [1, 0, 1, -14], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([2, 1, 0, -1], [1, 0, 1, -14], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
16. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[3, 1], [1, 0], [0, 1], [-1, -15]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([3, 1, 0, -1], [1, 0, 1, -15], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([3, 1, 0, -1], [1, 0, 1, -15], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
17. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[4, 1], [1, 0], [0, 1], [-1, -16]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([4, 1, 0, -1], [1, 0, 1, -16], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([4, 1, 0, -1], [1, 0, 1, -16], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
18. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[5, 1], [1, 0], [0, 1], [-1, -17]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([5, 1, 0, -1], [1, 0, 1, -17], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([5, 1, 0, -1], [1, 0, 1, -17], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
19. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[6, 1], [1, 0], [0, 1], [-1, -18]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([6, 1, 0, -1], [1, 0, 1, -18], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([6, 1, 0, -1], [1, 0, 1, -18], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
20. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[7, 1], [1, 0], [0, 1], [-1, -19]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([7, 1, 0, -1], [1, 0, 1, -19], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([7, 1, 0, -1], [1, 0, 1, -19], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
21. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[8, 1], [1, 0], [0, 1], [-1, -20]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([8, 1, 0, -1], [1, 0, 1, -20], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([8, 1, 0, -1], [1, 0, 1, -20], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
22. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[9, 1], [1, 0], [0, 1], [-1, -21]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([9, 1, 0, -1], [1, 0, 1, -21], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([9, 1, 0, -1], [1, 0, 1, -21], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
23. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[10, 1], [1, 0], [0, 1], [-1, -22]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([10, 1, 0, -1], [1, 0, 1, -22], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([10, 1, 0, -1], [1, 0, 1, -22], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
24. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[11, 1], [1, 0], [0, 1], [-1, -23]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([11, 1, 0, -1], [1, 0, 1, -23], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([11, 1, 0, -1], [1, 0, 1, -23], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
25. Four Ordered Coordinate Columns, Axes, and Locations
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Four ordered points=[[12, 1], [1, 0], [0, 1], [-1, -24]]. Give (x column, y column, x-axis mask, y-axis mask, origin/x-axis/y-axis/I/II/III/IV classifications).
View answer and explanation
([12, 1, 0, -1], [1, 0, 1, -24], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III])
Read both ordered columns, check zero components before quadrants, and then use signs; component data do not determine distances or rendered dimensions. The exact result, retaining the requested order, is ([12, 1, 0, -1], [1, 0, 1, -24], [No, Yes, No, No], [No, No, Yes, No], [I, x-axis, y-axis, III]).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-four-point-axis-masks · candidate
26. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[0, 0], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(0, 0, Yes)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (0, 0, Yes).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
27. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[1, 2], Q=[2, 1]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-1, 1, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-1, 1, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
28. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-10, 3], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-10, 3, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-10, 3, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
29. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-9, 4], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-9, 4, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-9, 4, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
30. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-8, 5], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-8, 5, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-8, 5, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
31. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-7, 6], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-7, 6, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-7, 6, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
32. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-6, 7], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-6, 7, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-6, 7, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
33. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-5, 8], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-5, 8, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-5, 8, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
34. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-4, 9], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-4, 9, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-4, 9, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
35. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-3, 10], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-3, 10, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-3, 10, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
36. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-2, 11], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-2, 11, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-2, 11, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
37. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-1, 12], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(-1, 12, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (-1, 12, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
38. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[0, 13], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(0, 13, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (0, 13, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
39. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[1, 14], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(1, 14, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (1, 14, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
40. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[2, 15], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(2, 15, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (2, 15, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
41. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[3, 16], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(3, 16, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (3, 16, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
42. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[4, 17], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(4, 17, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (4, 17, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
43. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[5, 18], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(5, 18, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (5, 18, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
44. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[6, 19], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(6, 19, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (6, 19, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
45. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[7, 20], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(7, 20, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (7, 20, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
46. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[8, 21], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(8, 21, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (8, 21, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
47. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[9, 22], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(9, 22, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (9, 22, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
48. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[10, 23], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(10, 23, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (10, 23, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
49. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[11, 24], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(11, 24, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (11, 24, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
50. Componentwise Coordinate Differences and Equality
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[12, 25], Q=[0, 0]. Give (P.x−Q.x, P.y−Q.y, are the points equal?), not an L2 distance.
View answer and explanation
(12, 25, No)
Subtract corresponding components and require both defects to vanish for equality in the same frame. The exact result, retaining the requested order, is (12, 25, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-ordered-pair-compare · candidate
51. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[0, 0], candidate=[0, 0], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([0, 0], [0, 0], Yes, Yes)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([0, 0], [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
52. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[1, 1], candidate=[1, 1], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([1, 1], [0, 0], Yes, Yes)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([1, 1], [0, 0], Yes, Yes).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
53. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[1, 2], candidate=[2, 1], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([2, 1], [0, 0], Yes, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([2, 1], [0, 0], Yes, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
54. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[1, 2], candidate=[1, 2], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([2, 1], [-1, 1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([2, 1], [-1, 1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
55. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-8, 5], candidate=[6, -9], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([5, -8], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([5, -8], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
56. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-7, 6], candidate=[7, -8], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([6, -7], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([6, -7], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
57. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-6, 7], candidate=[8, -7], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([7, -6], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([7, -6], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
58. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-5, 8], candidate=[9, -6], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([8, -5], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([8, -5], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
59. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-4, 9], candidate=[10, -5], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([9, -4], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([9, -4], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
60. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-3, 10], candidate=[11, -4], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([10, -3], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([10, -3], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
61. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-2, 11], candidate=[12, -3], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([11, -2], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([11, -2], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
62. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[-1, 12], candidate=[13, -2], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([12, -1], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([12, -1], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
63. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[0, 13], candidate=[14, -1], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([13, 0], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([13, 0], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
64. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[1, 14], candidate=[15, 0], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([14, 1], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([14, 1], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
65. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[2, 15], candidate=[16, 1], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([15, 2], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([15, 2], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
66. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[3, 16], candidate=[17, 2], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([16, 3], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([16, 3], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
67. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[4, 17], candidate=[18, 3], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([17, 4], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([17, 4], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
68. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[5, 18], candidate=[19, 4], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([18, 5], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([18, 5], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
69. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[6, 19], candidate=[20, 5], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([19, 6], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([19, 6], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
70. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[7, 20], candidate=[21, 6], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([20, 7], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([20, 7], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
71. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[8, 21], candidate=[22, 7], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([21, 8], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([21, 8], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
72. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[9, 22], candidate=[23, 8], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([22, 9], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([22, 9], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
73. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[10, 23], candidate=[24, 9], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([23, 10], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([23, 10], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
74. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[11, 24], candidate=[25, 10], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([24, 11], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([24, 11], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
75. Coordinate-Field Swap Candidate Audit
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. P=[12, 25], candidate=[26, 11], expected=(P.y,P.x). Give (expected list, candidate−expected component pair, is the candidate the true swap?, does P itself equal expected?).
View answer and explanation
([25, 12], [1, -1], No, No)
Swap data fields and check the candidate componentwise, then independently check whether P’s components were equal; this is not a formal geometric-reflection bridge. The exact result, retaining the requested order, is ([25, 12], [1, -1], No, No).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-swap-candidate-audit · candidate
76. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[0, 0], [0, 0], [0, 0]], [Yes, Yes, Yes], none)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[0, 0], [0, 0], [0, 0]], [Yes, Yes, Yes], none).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
77. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[1, 0], [0, 0]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[1, 0], [0, 0], [0, 0]], [No, Yes, Yes], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[1, 0], [0, 0], [0, 0]], [No, Yes, Yes], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
78. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[0, 0], [0, 0]], [[0, -1], [0, 0]], [[0, 0], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[0, 0], [0, -1], [0, 0]], [Yes, No, Yes], 1)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[0, 0], [0, -1], [0, 0]], [Yes, No, Yes], 1).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
79. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[1, 1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[0, 0], [0, 0], [1, 1]], [Yes, Yes, No], 2)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[0, 0], [0, 0], [1, 1]], [Yes, Yes, No], 2).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
80. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-8, 1], [0, 0]], [[0, 4], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-8, 1], [0, 4], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-8, 1], [0, 4], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
81. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-7, 1], [0, 0]], [[0, 5], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-7, 1], [0, 5], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-7, 1], [0, 5], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
82. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-6, 1], [0, 0]], [[0, 6], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-6, 1], [0, 6], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-6, 1], [0, 6], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
83. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-5, 1], [0, 0]], [[0, 7], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-5, 1], [0, 7], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-5, 1], [0, 7], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
84. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-4, 1], [0, 0]], [[0, 8], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-4, 1], [0, 8], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-4, 1], [0, 8], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
85. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-3, 1], [0, 0]], [[0, 9], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-3, 1], [0, 9], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-3, 1], [0, 9], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
86. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-2, 1], [0, 0]], [[0, 10], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-2, 1], [0, 10], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-2, 1], [0, 10], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
87. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[-1, 1], [0, 0]], [[0, 11], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[-1, 1], [0, 11], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[-1, 1], [0, 11], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
88. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[0, 1], [0, 0]], [[0, 12], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[0, 1], [0, 12], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[0, 1], [0, 12], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
89. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[1, 1], [0, 0]], [[0, 13], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[1, 1], [0, 13], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[1, 1], [0, 13], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
90. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[2, 1], [0, 0]], [[0, 14], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[2, 1], [0, 14], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[2, 1], [0, 14], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
91. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[3, 1], [0, 0]], [[0, 15], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[3, 1], [0, 15], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[3, 1], [0, 15], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
92. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[4, 1], [0, 0]], [[0, 16], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[4, 1], [0, 16], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[4, 1], [0, 16], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
93. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[5, 1], [0, 0]], [[0, 17], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[5, 1], [0, 17], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[5, 1], [0, 17], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
94. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[6, 1], [0, 0]], [[0, 18], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[6, 1], [0, 18], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[6, 1], [0, 18], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
95. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[7, 1], [0, 0]], [[0, 19], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[7, 1], [0, 19], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[7, 1], [0, 19], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
96. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[8, 1], [0, 0]], [[0, 20], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[8, 1], [0, 20], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[8, 1], [0, 20], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
97. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[9, 1], [0, 0]], [[0, 21], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[9, 1], [0, 21], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[9, 1], [0, 21], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
98. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[10, 1], [0, 0]], [[0, 22], [0, 0]], [[0, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[10, 1], [0, 22], [0, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[10, 1], [0, 22], [0, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
99. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[11, 1], [0, 0]], [[0, 23], [0, 0]], [[1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[11, 1], [0, 23], [1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[11, 1], [0, 23], [1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
100. Three Ordered Coordinate Comparisons and First Mismatch
Use ordered integer pairs in the same orthogonal directed xy frame. The origin lies on both axes; axis points are not in quadrants. Three ordered [P,Q] pairs=[[[12, 1], [0, 0]], [[0, 24], [0, 0]], [[-1, -1], [0, 0]]]. Give (each P−Q component residual pair, pair-equality mask, first unequal zero-based index or none).
View answer and explanation
([[12, 1], [0, 24], [-1, -1]], [No, No, No], 0)
Check both components in each pair and take the first actual mismatch in original order; all equal gives none, and components are not distances. The exact result, retaining the requested order, is ([[12, 1], [0, 24], [-1, -1]], [No, No, No], 0).
Reviewed step-by-step derivation
Coordinate plane · qf-density15-coordinate-first-mismatch-index · candidate
Canonical question JSONL · English JSONL · Validation report