Question 7650: Same-Shape Matrix Sum and Both Transpose Routes

Original question and givens

Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1], [-4, -3, -2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

• G001 — Use complete nonempty integer matrices with 1–5 rows and columns and 1-based external indices. Transposition is not inversion or reversal of element order. Same-shape M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]], N=[[0, 1, 2, 3, 4], [-1, 0, 1, 2, 3], [-2, -1, 0, 1, 2], [-3, -2, -1, 0, 1], [-4, -3, -2, -1, 0]]. Give (shape of M, M+N, (M+N)ᵀ, Mᵀ+Nᵀ).

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-matrix

Step-by-step derivation

M=[[12, 11, 10, 9, 8], [13, 12, 11, 10, 9], [14, 13, 12, 11, 10], [15, 14, 13, 12, 11], [16, 15, 14, 13, 12]];shape=[5, 5]

external indices start at1;transpose is not inverse

M transpose:source(1,1)=12

target(1,1)=12

M transpose:source(1,2)=11

target(2,1)=11

M transpose:source(1,3)=10

target(3,1)=10

M transpose:source(1,4)=9

target(4,1)=9

M transpose:source(1,5)=8

target(5,1)=8

M transpose:source(2,1)=13

target(1,2)=13

M transpose:source(2,2)=12

target(2,2)=12

M transpose:source(2,3)=11

target(3,2)=11

M transpose:source(2,4)=10

target(4,2)=10

M transpose:source(2,5)=9

target(5,2)=9

M transpose:source(3,1)=14

target(1,3)=14

M transpose:source(3,2)=13

target(2,3)=13

M transpose:source(3,3)=12

target(3,3)=12

M transpose:source(3,4)=11

target(4,3)=11

M transpose:source(3,5)=10

target(5,3)=10

M transpose:source(4,1)=15

target(1,4)=15

M transpose:source(4,2)=14

target(2,4)=14

M transpose:source(4,3)=13

target(3,4)=13

M transpose:source(4,4)=12

target(4,4)=12

M transpose:source(4,5)=11

target(5,4)=11

M transpose:source(5,1)=16

target(1,5)=16

M transpose:source(5,2)=15

target(2,5)=15

M transpose:source(5,3)=14

target(3,5)=14

M transpose:source(5,4)=13

target(4,5)=13

M transpose:source(5,5)=12

target(5,5)=12

M shape=[5, 5];N shape=[5, 5]

same shape;entrywise sum permitted

M+N(1,1) := sum(12, 0)

M+N(1,1) = 12

M+N(1,2) := sum(11, 1)

M+N(1,2) = 12

M+N(1,3) := sum(10, 2)

M+N(1,3) = 12

M+N(1,4) := sum(9, 3)

M+N(1,4) = 12

M+N(1,5) := sum(8, 4)

M+N(1,5) = 12

M+N(2,1) := sum(13, -1)

M+N(2,1) = 12

M+N(2,2) := sum(12, 0)

M+N(2,2) = 12

M+N(2,3) := sum(11, 1)

M+N(2,3) = 12

M+N(2,4) := sum(10, 2)

M+N(2,4) = 12

M+N(2,5) := sum(9, 3)

M+N(2,5) = 12

M+N(3,1) := sum(14, -2)

M+N(3,1) = 12

M+N(3,2) := sum(13, -1)

M+N(3,2) = 12

M+N(3,3) := sum(12, 0)

M+N(3,3) = 12

M+N(3,4) := sum(11, 1)

M+N(3,4) = 12

M+N(3,5) := sum(10, 2)

M+N(3,5) = 12

M+N(4,1) := sum(15, -3)

M+N(4,1) = 12

M+N(4,2) := sum(14, -2)

M+N(4,2) = 12

M+N(4,3) := sum(13, -1)

M+N(4,3) = 12

M+N(4,4) := sum(12, 0)

M+N(4,4) = 12

M+N(4,5) := sum(11, 1)

M+N(4,5) = 12

M+N(5,1) := sum(16, -4)

M+N(5,1) = 12

M+N(5,2) := sum(15, -3)

M+N(5,2) = 12

M+N(5,3) := sum(14, -2)

M+N(5,3) = 12

M+N(5,4) := sum(13, -1)

M+N(5,4) = 12

M+N(5,5) := sum(12, 0)

M+N(5,5) = 12

sum transpose:source(1,1)=12

target(1,1)=12

sum transpose:source(1,2)=12

target(2,1)=12

sum transpose:source(1,3)=12

target(3,1)=12

sum transpose:source(1,4)=12

target(4,1)=12

sum transpose:source(1,5)=12

target(5,1)=12

sum transpose:source(2,1)=12

target(1,2)=12

sum transpose:source(2,2)=12

target(2,2)=12

sum transpose:source(2,3)=12

target(3,2)=12

sum transpose:source(2,4)=12

target(4,2)=12

sum transpose:source(2,5)=12

target(5,2)=12

sum transpose:source(3,1)=12

target(1,3)=12

sum transpose:source(3,2)=12

target(2,3)=12

sum transpose:source(3,3)=12

target(3,3)=12

sum transpose:source(3,4)=12

target(4,3)=12

sum transpose:source(3,5)=12

target(5,3)=12

sum transpose:source(4,1)=12

target(1,4)=12

sum transpose:source(4,2)=12

target(2,4)=12

sum transpose:source(4,3)=12

target(3,4)=12

sum transpose:source(4,4)=12

target(4,4)=12

sum transpose:source(4,5)=12

target(5,4)=12

sum transpose:source(5,1)=12

target(1,5)=12

sum transpose:source(5,2)=12

target(2,5)=12

sum transpose:source(5,3)=12

target(3,5)=12

sum transpose:source(5,4)=12

target(4,5)=12

sum transpose:source(5,5)=12

target(5,5)=12

N transpose:source(1,1)=0

target(1,1)=0

N transpose:source(1,2)=1

target(2,1)=1

N transpose:source(1,3)=2

target(3,1)=2

N transpose:source(1,4)=3

target(4,1)=3

N transpose:source(1,5)=4

target(5,1)=4

N transpose:source(2,1)=-1

target(1,2)=-1

N transpose:source(2,2)=0

target(2,2)=0

N transpose:source(2,3)=1

target(3,2)=1

N transpose:source(2,4)=2

target(4,2)=2

N transpose:source(2,5)=3

target(5,2)=3

N transpose:source(3,1)=-2

target(1,3)=-2

N transpose:source(3,2)=-1

target(2,3)=-1

N transpose:source(3,3)=0

target(3,3)=0

N transpose:source(3,4)=1

target(4,3)=1

N transpose:source(3,5)=2

target(5,3)=2

N transpose:source(4,1)=-3

target(1,4)=-3

N transpose:source(4,2)=-2

target(2,4)=-2

N transpose:source(4,3)=-1

target(3,4)=-1

N transpose:source(4,4)=0

target(4,4)=0

N transpose:source(4,5)=1

target(5,4)=1

N transpose:source(5,1)=-4

target(1,5)=-4

N transpose:source(5,2)=-3

target(2,5)=-3

N transpose:source(5,3)=-2

target(3,5)=-2

N transpose:source(5,4)=-1

target(4,5)=-1

N transpose:source(5,5)=0

target(5,5)=0

MT+NT(1,1) := sum(12, 0)

MT+NT(1,1) = 12

MT+NT(1,2) := sum(13, -1)

MT+NT(1,2) = 12

MT+NT(1,3) := sum(14, -2)

MT+NT(1,3) = 12

MT+NT(1,4) := sum(15, -3)

MT+NT(1,4) = 12

MT+NT(1,5) := sum(16, -4)

MT+NT(1,5) = 12

MT+NT(2,1) := sum(11, 1)

MT+NT(2,1) = 12

MT+NT(2,2) := sum(12, 0)

MT+NT(2,2) = 12

MT+NT(2,3) := sum(13, -1)

MT+NT(2,3) = 12

MT+NT(2,4) := sum(14, -2)

MT+NT(2,4) = 12

MT+NT(2,5) := sum(15, -3)

MT+NT(2,5) = 12

MT+NT(3,1) := sum(10, 2)

MT+NT(3,1) = 12

MT+NT(3,2) := sum(11, 1)

MT+NT(3,2) = 12

MT+NT(3,3) := sum(12, 0)

MT+NT(3,3) = 12

MT+NT(3,4) := sum(13, -1)

MT+NT(3,4) = 12

MT+NT(3,5) := sum(14, -2)

MT+NT(3,5) = 12

MT+NT(4,1) := sum(9, 3)

MT+NT(4,1) = 12

MT+NT(4,2) := sum(10, 2)

MT+NT(4,2) = 12

MT+NT(4,3) := sum(11, 1)

MT+NT(4,3) = 12

MT+NT(4,4) := sum(12, 0)

MT+NT(4,4) = 12

MT+NT(4,5) := sum(13, -1)

MT+NT(4,5) = 12

MT+NT(5,1) := sum(8, 4)

MT+NT(5,1) = 12

MT+NT(5,2) := sum(9, 3)

MT+NT(5,2) = 12

MT+NT(5,3) := sum(10, 2)

MT+NT(5,3) = 12

MT+NT(5,4) := sum(11, 1)

MT+NT(5,4) = 12

MT+NT(5,5) := sum(12, 0)

MT+NT(5,5) = 12

left=[[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]];right=[[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]]

same value at every transposed coordinate

Reverse checks

• S128 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S127 · premise_and_result_audit — Recompute MT+NT(5,5) with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute MT+NT(5,4) with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute MT+NT(5,3) with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute MT+NT(5,2) with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute MT+NT(5,1) with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute MT+NT(4,5) with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute MT+NT(4,4) with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute MT+NT(4,3) with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute MT+NT(4,2) with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute MT+NT(4,1) with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute MT+NT(3,5) with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute MT+NT(3,4) with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute MT+NT(3,3) with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute MT+NT(3,2) with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute MT+NT(3,1) with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute MT+NT(2,5) with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute MT+NT(2,4) with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute MT+NT(2,3) with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute MT+NT(2,2) with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute MT+NT(2,1) with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute MT+NT(1,5) with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute MT+NT(1,4) with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute MT+NT(1,3) with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute MT+NT(1,2) with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute MT+NT(1,1) with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S098 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S097 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S096 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S095 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S094 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S092 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S086 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S084 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S083 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S082 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S080 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S079 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S078 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S077 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S073 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S072 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S071 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S070 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S068 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S067 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S066 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S065 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S062 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S061 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S060 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S058 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S055 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Recompute M+N(5,5) with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute M+N(5,4) with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute M+N(5,3) with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute M+N(5,2) with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute M+N(5,1) with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute M+N(4,5) with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute M+N(4,4) with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute M+N(4,3) with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Recompute M+N(4,2) with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute M+N(4,1) with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Recompute M+N(3,5) with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute M+N(3,4) with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute M+N(3,3) with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute M+N(3,2) with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute M+N(3,1) with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute M+N(2,5) with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute M+N(2,4) with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute M+N(2,3) with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute M+N(2,2) with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute M+N(2,1) with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute M+N(1,5) with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute M+N(1,4) with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute M+N(1,3) with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute M+N(1,2) with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute M+N(1,1) with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([5, 5], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]], [[12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12], [12, 12, 12, 12, 12]])

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.