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There may be multiple solutions but here is one. Number the cows 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Put them into 3 groups : A(1,2,3,4), B(5,6,7,8) and C(9,10,11,12). Weigh group A and B. There are 3 possible outcomes:

Scenario 1: If they balance then group C has the prized cow, so weigh 6,7,8 vs 9,10,11 with 3 possible outcomes:

1A) If 6,7,8 vs 9,10,11 balances, 12 is the prized cow. Weigh it against any other cow to determine if heavy or light.

1B) If 9,10,11 is heavy then they contain a heavy cow. Weigh 9 vs 10, if balanced then 11 is the odd heavy cow, else the heavier of 9 or 10 is the odd heavy cow.

1C) If 9,10,11 is light then they contain a light cow. Weigh 9 vs 10, if balanced then 11 is the odd light cow, else the lighter of 9 or 10 is the odd light cow.

Scenario 2: If group B > group A then either 5,6,7,8 contains a heavy cow or 1,2,3,4 contains a light cow so weigh 1,2,5 vs 3,6,12 with 3 possible outcomes:

2A) If 1,2,5 vs 3,6,12 balances, then either 4 is the odd light cow or 7 or 8 is the odd heavy cow. Weigh 7 vs 8, if they balance then 4 is the odd light cow, or the heaviest of 7 vs 8 is the odd heavy cow.

2B) If 3,6,12 is heavy then either 6 is the odd heavy cow or 1 or 2 is the odd light cow. Weigh 1 vs 2, if balanced then 6 is the odd heavy cow, or the lightest of 1 vs 2 is the odd light cow.

2C) If 3,6,12 is light then either 3 is light or 5 is heavy. Weigh 3 against any other cow, if balanced then 5 is the odd heavy cow else 3 is the odd light cow.

Scenario 3: If group A > group B then either 1,2,3,4 contains a heavy cow or 5,6,7,8 contains a light cow so weigh 5,6,1 vs 7,2,12 with 3 possible outcomes:

3A) If 5,6,1 vs 7,2,12 balances, then either 8 is the odd light cow or 3 or 4 is the odd heavy cow. Weigh 3 vs 4, if they balance then 8 is the odd light cow, or the heaviest of 3 vs 4 is the odd heavy cow.

3B) If 7,2,12 is heavy then either 2 is the odd heavy cow or 5 or 6 is the odd light cow. Weigh 5 vs 6, if balanced then 2 is the odd heavy cow, or the lightest of 5 vs 6 is the odd light cow.

3C) If 7,2,12 is light then either 7 is light or 1 is heavy. Weigh 7 against any other cow, if balanced then 1 is the odd heavy cow else 7 is the odd light cow.