There cannot be a 7-cent coin because you could make 14 cents with 2 coins (7, 7). Also, there cannot be a 13-cent coin or a 14-cent coin.

Suppose there is a 1-cent coin. Then, there cannot be a 4 because you could make 13 cents with 4 coins (1, 4, 4, 4). Also, there cannot be a 6 because you could make 13 cents with 3 coins (1, 6, 6). There cannot be a 10 because of (1, 1, 1, 10). There cannot be an 11 because of (1, 1, 11). There cannot be a 12 because of (1, 12).

Suppose there is a 1 and a 5. Then, there cannot be a 2 because you could make 13 cents with 4 coins (1, 2, 5, 5). There cannot be a 3 because you could make 13 cents with 3 coins (3, 5, 5). There cannot be an 8 because you could make 13 cents with 2 coins (5, 8). There cannot be a 9 because you could make 14 cents with 2 coins (5, 9). Therefore, if there is a 1-cent coin and a 5-cent coin, then those are the only coins less than 15 cents. However, it is impossible to make 14 cents with 4 of these coins. Therefore, there cannot be a 1 and a 5.

Suppose there is a 1 and an 8. Then, there cannot be a 2 because of (1, 2, 2, 8). There cannot be a 3 because of (3, 3, 8). The only other possibility for another coin is 9, but you cannot make 14 cents with 4 coins that are either 1, 8, or 9 cents. Therefore, there cannot be a 1 and an 8.

Suppose there is a 1 and a 9. Then, there cannot be a 2 because of (2, 2, 9). There cannot be a 3 because of (1, 3, 9). Therefore, 1 and 9 are the only coins, but you cannot make 14 cents with 4 of these coins. Therefore, there cannot be a 1 and a 9. If there is a 1, then there can only be a 2 or a 3. However, the greatest sum of 4 coins is 12<14. Therefore, there cannot be a 1-cent coin.

Since, you can make 13 cents with 5 coins, there must be a coin less than 3 cents. It must be 2 cents. Now, there cannot be a 9 because of (2, 2, 9). There cannot be a 10 because of (2, 2, 10). There cannot be an 11 because of (2, 11). There cannot be a 12 because of (2, 12).

Suppose there is a 3-cent coin. Then, there cannot be a 4 because of (3, 3, 3, 4). There cannot be a 5 because of (3, 5, 5). There cannot be a 6 because of (2, 2, 3, 6). There cannot be an 8 because of (3, 3, 8). Therefore, 2 and 3 are the only coins. However, you cannot make 14 cents with 4 of these coins. Therefore, there cannot be a 3-cent coin.

Since you can make 13 cents, there must be an odd coin. It must be 5 cents. Now, there cannot be a 4 because of (2, 2, 4, 5). There cannot be a 6 because of (2, 5, 6). There cannot be an 8 because of (2, 2, 8). Therefore, 2 and 5 are the only coins. You make 13 cents with (2, 2, 2, 2, 5). You make 14 cents with (2, 2, 5, 5).