Solving the Mystery: Distributing Tiles Among Students

Let's dive into a fun and intriguing math puzzle involving the distribution of tiles among three students. Each student has a set of tiles, and we need to figure out which student has tile number 3. The distribution conditions are as follows:

Conditions of the Puzzle

There are 1-6 tiles distributed among 3 students. Student 1’s tiles are not consecutive numbers. Student 2’s tiles produce a product of 12. Student 3’s tiles are consecutive numbers.

Analysis of Tiling

First, let’s list the possible pairs of tiles for each student

Student 2 Tiles

Since the product of Student 2’s tiles is 12, the possible pairs are (2, 6) and (3, 4).

Exclusion of Pairs

If Student 2 has tiles (2, 6), then the remaining tiles for Student 1 and Student 3 are 1, 3, and 4, 5. If Student 2 has tiles (3, 4), then the remaining tiles for Student 1 and Student 3 are 1, 2, and 5, 6.

Checking Constraints

For (3, 4) pair of Student 2:

Student 3 has consecutive numbers, so it could be (1, 2) or (5, 6). If Student 3 has (1, 2), then Student 1 would have (5, 6), which are not consecutive, satisfying all conditions. If Student 3 has (5, 6), then Student 1 would have (1, 2), which are not consecutive, satisfying all conditions.

Therefore, if Student 2 has (3, 4), all conditions are met.

Alternative Scenarios

For (2, 6) pair of Student 2:

Student 3 must have consecutive numbers, so the only possible pairs are (1, 2) and (5, 6). If Student 3 has (1, 2), then Student 1 would have (5, 6), which are not consecutive, violating the condition for Student 1’s tiles. If Student 3 has (5, 6), then Student 1 would have (1, 2), which are not consecutive, satisfying all conditions.

Final Distribution

After analyzing all possible pairs, we conclude:

Student 2 has tiles (2, 6). Student 3 has tiles (3, 4). Student 1 has tiles (1, 5).

Conclusion

The problem asks who we think has the number 3 tile. Based on our analysis:

Student 2 cannot have tiles (3, 4) as this would violate Student 1’s condition of not having consecutive numbers. Student 3 must have the tile 3 since his tiles (3, 4) are consecutive, and he might like having his number on his tile.

Therefore, we think that Student 3 has the number 3 tile because of the constraints and his preference for having his number on his tile.