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A group of five meerkats (M1, M2, M3, M4, and M5) need to share information about predators they’ve spotted during the week. To avoid attracting attention, no more than two meerkats can meet at a time. The meerkats always communicate face-to-face to ensure accurate information transfer.
Each day, the meerkats conduct several rounds of meetings at various locations in their territory. Each round consists of two simultaneous meetings, involving four meerkats in total. There is always one meerkat not participating in a round.
During each meeting, meerkats share all the information they know, including both their own observations and information they’ve learned from others in previous meetings that day.
Determine the minimum number of rounds of meetings required for each meerkat to learn all the information about predators observed by each of the other meerkats during the week. The information includes:
1. Aerial predators (like eagles and hawks)
2. Ground predators (like jackals and snakes)
We will prove that the minimum number of rounds required is four.
1. Proof that at least four rounds are needed:
– After the first round, at least one meerkat (let’s say M5) hasn’t shared its information.
– In the second round, even if M5 meets another meerkat, only two meerkats know M5’s information.
– In the third round, these two meerkats can meet with at most two others, so after three rounds, at most four meerkats know M5’s information.
– Therefore, at least one more round is needed, making the minimum four rounds.
2. Proof that four rounds are sufficient:
Round 1: (M1, M2), (M3, M4), M5 sits out
| Meerkat | Knows information from |
|---|---|
| M1 | M1, M2 |
| M2 | M1, M2 |
| M3 | M3, M4 |
| M4 | M3, M4 |
| M5 | M5 |
Round 2: (M1, M5), (M2, M3), M4 sits out
| Meerkat | Knows information from |
|---|---|
| M1 | M1, M2, M5 |
| M2 | M1, M2, M3, M4 |
| M3 | M1, M2, M3, M4 |
| M4 | M3, M4 |
| M5 | M1, M2, M5 |
Round 3: (M1, M3), (M4, M5), M2 sits out
| Meerkat | Knows information from |
|---|---|
| M1 | M1, M2, M3, M4, M5 |
| M2 | M1, M2, M3, M4 |
| M3 | M1, M2, M3, M4, M5 |
| M4 | M1, M2, M3, M4, M5 |
| M5 | M1, M2, M3, M4, M5 |
Round 4: (M2, any other meerkat)
After this round, all meerkats know all information.
Therefore, the minimum number of rounds required for each meerkat to learn all the information about predators observed by each of the other meerkats is four.