1. The problem asks us to find numbers with rotational symmetry of order 2, meaning when rotated 180 degrees, the digits look the same. 2. We analyze digits 0-9 for rotational symmetry: - Digits 0, 1, 8 are symmetric when rotated 180 degrees and look the same. - Digit 6 becomes 9, and 9 becomes 6. - Digits 2, 3, 4, 5, 7 do not form valid digits when rotated. 3. (a) Two-digit number with rotational symmetry order 2: - Let the two digits be $AB$. - After 180-degree rotation, the number appears as $BA$ but rotated individually. - Digit $A$ must match rotated $B$, and digit $B$ must match rotated $A$. From the digit transformation: - 0 ↔ 0 - 1 ↔ 1 - 8 ↔ 8 - 6 ↔ 9 - 9 ↔ 6 Possible pairs $(A,B)$ are: - (0,0), (1,1), (8,8) - (6,9), (9,6) Thus, the two-digit numbers include: 00, 11, 88, 69, 96. Since 00 is not a typical two-digit number, we usually consider 11, 88, 69, 96. 4. (b) Three-digit number with rotational symmetry order 2: - Let the digits be $ABC$. - Rotated, the number becomes $C'B'A'$ where $A', B', C'$ are rotated digits of $A, B, C$ respectively. - For rotational symmetry of order 2, the number $ABC$ must equal $C'B'A'$. From this, - $A = C'$ and $B = B'$ and $C = A'$ Possible digits for $B$ are symmetric digits: 0,1,8 Possible pairs for $A$ and $C$ from digit rotation: - $(0,0), (1,1), (8,8), (6,9), (9,6)$ Example numbers: - 101 (rotates to 101) - 609 (rotates to 906, so invalid) - 689 (6→9, 8→8, 9→6, so 689 rotated is 986, not same) - 818 (rotates to 818) - 906 (rotates to 609, so invalid) - 619 (rotates to 916, invalid) - 181 (rotates to 181) Valid three-digit numbers include 101, 181, 808, 818, 689 is invalid because rotated not same. Similarly, try 696 (rotates to 969), so not same. Therefore, valid examples: 101, 181, 808, 818. Final answers: (a) Two-digit numbers with rotational symmetry order 2 include 11, 69, 88, 96 (b) Three-digit numbers with rotational symmetry order 2 include 101, 181, 808, 818