Explanation:

- We start with the information that the average of twelve numbers is 55.5. Therefore, the sum of all twelve numbers is \(12 \times 55.5 = 666\).

- The average of the first four numbers is 53.4, giving us a sum of \(4 \times 53.4 = 213.6\).

- The average of the next four numbers is 54.6, so their sum is \(4 \times 54.6 = 218.4\).

- Now, let’s calculate the sum of the last four numbers: \(666 - 213.6 - 218.4 = 234\).

- We have the 10th number greater than the 9th by 3, but lesser than the 11th and 12th by 2 and 3, respectively.

- Let's denote the 10th number as \(x\). Therefore, the 9th number is \(x - 3\), the 11th is \(x + 2\), and the 12th is \(x + 3\).

- The sum of these numbers is \( (x - 3) + x + (x + 2) + (x + 3) = 4x + 2 \).

- Given the sum is 234, \(4x + 2 = 234\). Solving this gives \(4x = 232\), so \(x = 58\).

- The 12th number is \(x + 3 = 58 + 3 = 61\).

- The average of the 10th and 12th numbers is \(\frac{58 + 61}{2} = 59.5\).

- Option 1: 59.5