We are tasked with determining how many times we can split a group of 12 people into 3 groups of 4, ensuring that every person has a new set of teammates each time and that no one has a repeated teammate.

Step-by-step analysis:

1. Initial Grouping:

   • We have 12 people, which we will denote as $P_1, P_2, \dots, P_{12}$.
   • In each round, we will split these 12 people into 3 groups of 4.

2. No Teammate Repetition Rule:

   • The requirement is that in each round, no person can have a teammate from a previous round. This means after a round of grouping, in the next round, a person should be paired with entirely new people.

3. Total Number of Possible Pairings:

   • In each group, every person will have 3 teammates. If a person can never be paired with the same person twice, the maximum number of teammates they can have across all rounds is $11$ (since each person can be paired with any of the other 11 people exactly once).

4. Counting Groupings:

   • After each round, every person will have interacted with 3 new people. If we do this for multiple rounds, each person will eventually be paired with all other people.
   • Therefore, each person can participate in at most $\frac{11}{3} \approx 3$ rounds before they have been paired with every other participant.

Number of Unique Groupings:

Given the conditions:

• After 3 rounds, each person will have been paired with 9 different people.
• We can continue for one more round, but in the fourth round, some people will end up repeating teammates, which violates the problem's condition.

Thus, you can split the group exactly 3 times while ensuring that no one is paired with a teammate more than once.

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Let me know if you would like more details on this, or if you have any questions! Here are some further exploration questions:

1. What is the mathematical approach behind determining the number of groupings in larger sets?
2. How do combinatorial designs like block designs apply to problems like this?
3. Could you devise a different algorithm to generate unique groupings under relaxed conditions?
4. What happens if we allow at least one teammate to be repeated once?
5. How can we compute the number of ways to split a group without the "new teammates" restriction?

Tip: Problems like this often relate to graph theory, where people are represented by nodes and pairings by edges in a graph. Exploring such representations can offer deeper insights into grouping challenges.