{
  "2257": {
    "model": "google/gemini-3.8-flash",
    "response": "To find the fewest number of moves to transition from the initial configuration to the final configuration, we track the positions of the four disks ordered by size:\n- Disk 0: Blue (largest)\n- Disk 1: Green\n- Disk 2: Orange\n- Disk 3: Red (smallest)\n\n### Initial State:\n- Peg 1: [Blue (0), Orange (2)]\n- Peg 2: [Red (3)]\n- Peg 3: [Green (1)]\n\n### Goal State:\n- Peg 1: [Green (1)]\n- Peg 2: [Orange (2), Red (3)]\n- Peg 3: [Blue (0)]\n\n### Optimal Move Sequence:\n1. Move disk 3 from Peg 2 to Peg 1 (`3: 2->1`)\n2. Move disk 1 from Peg 3 to Peg 2 (`1: 3->2`)\n3. Move disk 3 from Peg 1 to Peg 3 (`3: 1->3`)\n4. Move disk 2 from Peg 1 to Peg 2 (`2: 1->2`)\n5. Move disk 3 from Peg 3 to Peg 2 (`3: 3->2`)\n6. Move disk 0 from Peg 1 to Peg 3 (`0: 1->3`)\n7. Move disk 3 from Peg 2 to Peg 1 (`3: 2->1`)\n8. Move disk 2 from Peg 2 to Peg 3 (`2: 2->3`)\n9. Move disk 3 from Peg 1 to Peg 3 (`3: 1->3`)\n10. Move disk 1 from Peg 2 to Peg 1 (`1: 2->1`)\n11. Move disk 3 from Peg 3 to Peg 1 (`3: 3->1`)\n12. Move disk 2 from Peg 3 to Peg 2 (`2: 3->2`)\n13. Move disk 3 from Peg 1 to Peg 2 (`3: 1->2`)\n\nAnswer: {'moves': '3: 2->1,1: 3->2,3: 1->3,2: 1->2,3: 3->2,0: 1->3,3: 2->1,2: 2->3,3: 1->3,1: 2->1,3: 3->1,2: 3->2,3: 1->2', 'num_moves': '13'}"
  },
  "2258": {
    "model": "google/gemini-3.8-flash",
    "response": "An optimal sequence of moves to transition from the starting configuration to the ending configuration using the minimum number of moves is:\n\n1. Disk 3 (red) from Peg 1 to Peg 2\n2. Disk 2 (orange) from Peg 3 to Peg 1\n3. Disk 3 (red) from Peg 2 to Peg 1\n4. Disk 0 (blue) from Peg 2 to Peg 3\n5. Disk 3 (red) from Peg 1 to Peg 2\n6. Disk 2 (orange) from Peg 1 to Peg 3\n7. Disk 3 (red) from Peg 2 to Peg 3\n8. Disk 1 (green) from Peg 1 to Peg 2\n9. Disk 3 (red) from Peg 3 to Peg 1\n10. Disk 2 (orange) from Peg 3 to Peg 2\n\nAnswer: {'moves': '3: 1->2,2: 3->1,3: 2->1,0: 2->3,3: 1->2,2: 1->3,3: 2->3,1: 1->2,3: 3->1,2: 3->2', 'num_moves': '10'}"
  },
  "2259": {
    "model": "google/gemini-3.8-flash",
    "response": "To transition from the starting configuration to the ending configuration in the fewest number of moves, we can track the positions of the disks (where 0 is the largest disk and 3 is the smallest disk: 0 = Blue, 1 = Green, 2 = Orange, 3 = Red):\n\n- **Initial State:**\n  - Peg 1: [Orange (2)]\n  - Peg 2: [Blue (0), Green (1), Red (3)]\n  - Peg 3: [ ]\n\n- **Moves:**\n  1. Move disk 3 (Red) from Peg 2 to Peg 1 (`3: 2->1`)\n  2. Move disk 1 (Green) from Peg 2 to Peg 3 (`1: 2->3`)\n  3. Move disk 3 (Red) from Peg 1 to Peg 2 (`3: 1->2`)\n  4. Move disk 2 (Orange) from Peg 1 to Peg 3 (`2: 1->3`)\n  5. Move disk 3 (Red) from Peg 2 to Peg 1 (`3: 2->1`)\n\n- **Final State:**\n  - Peg 1: [Red (3)]\n  - Peg 2: [Blue (0)]\n  - Peg 3: [Green (1), Orange (2)]\n\nAnswer: {'moves': '3: 2->1,1: 2->3,3: 1->2,2: 1->3,3: 2->1', 'num_moves': '5'}"
  },
  "2260": {
    "model": "google/gemini-3.8-flash",
    "response": "To determine the minimal sequence of moves to reach the goal configuration, we analyze the constraints of the puzzle:\n- **Disk sizes**: 0 (Blue, largest), 1 (Green), 2 (Orange), 3 (Red, smallest).\n- **Initial configuration**:\n  - Peg 1: empty\n  - Peg 2: [0, 2]\n  - Peg 3: [1, 3]\n- **Goal configuration**:\n  - Peg 1: [0, 2]\n  - Peg 2: [1, 3]\n  - Peg 3: empty\n\n### Step-by-step Solution:\n1. Move disk 3 (Red) to Peg 1: `3: 3->1`\n2. Move disk 2 (Orange) to Peg 3: `2: 2->3`\n3. Move disk 3 (Red) to Peg 3: `3: 1->3`\n4. Move disk 0 (Blue) to Peg 1: `0: 2->1`\n5. Move disk 3 (Red) to Peg 2: `3: 3->2`\n6. Move disk 2 (Orange) to Peg 1: `2: 3->1`\n7. Move disk 3 (Red) to Peg 1: `3: 2->1`\n8. Move disk 1 (Green) to Peg 2: `1: 3->2`\n9. Move disk 3 (Red) to Peg 2: `3: 1->2`\n\nAnswer: {'moves': '3: 3->1,2: 2->3,3: 1->3,0: 2->1,3: 3->2,2: 3->1,3: 2->1,1: 3->2,3: 1->2', 'num_moves': '9'}"
  },
  "2261": {
    "model": "google/gemini-3.8-flash",
    "response": "To transition from the starting configuration to the ending configuration in the fewest number of moves, we need to move the Green disk (1) from Peg 1 to Peg 2, while ending with the Orange disk (2) on Peg 1 and the Blue (0) and Red (3) disks on Peg 3.\n\nThe optimal 7-move sequence is:\n1. Move disk 3 (Red) from Peg 3 to Peg 2\n2. Move disk 2 (Orange) from Peg 1 to Peg 3\n3. Move disk 3 (Red) from Peg 2 to Peg 3\n4. Move disk 1 (Green) from Peg 1 to Peg 2\n5. Move disk 3 (Red) from Peg 3 to Peg 2\n6. Move disk 2 (Orange) from Peg 3 to Peg 1\n7. Move disk 3 (Red) from Peg 2 to Peg 3\n\nAnswer: {'moves': '3: 3->2,2: 1->3,3: 2->3,1: 1->2,3: 3->2,2: 3->1,3: 2->3', 'num_moves': '7'}"
  },
  "2262": {
    "model": "google/gemini-3.8-flash",
    "response": "An optimal sequence of moves to reach the ending configuration from the starting configuration is:\n\n1. Move disk 3 from peg 2 to peg 1\n2. Move disk 3 from peg 1 to peg 3\n3. Move disk 1 from peg 1 to peg 2\n4. Move disk 3 from peg 3 to peg 1\n5. Move disk 2 from peg 3 to peg 2\n6. Move disk 3 from peg 1 to peg 2\n7. Move disk 0 from peg 3 to peg 1\n8. Move disk 3 from peg 2 to peg 1\n9. Move disk 2 from peg 2 to peg 3\n10. Move disk 3 from peg 1 to peg 3\n11. Move disk 1 from peg 2 to peg 1\n12. Move disk 3 from peg 3 to peg 1\n13. Move disk 2 from peg 3 to peg 2\n\nAnswer: {'moves': '3: 2->1, 3: 1->3, 1: 1->2, 3: 3->1, 2: 3->2, 3: 1->2, 0: 3->1, 3: 2->1, 2: 2->3, 3: 1->3, 1: 2->1, 3: 3->1, 2: 3->2', 'num_moves': '13'}"
  },
  "2263": {
    "model": "google/gemini-3.8-flash",
    "response": "An analysis of the disk sizes and colors:\n- **Disk 0 (largest):** Blue\n- **Disk 1:** Green\n- **Disk 2:** Orange\n- **Disk 3 (smallest):** Red\n\nStarting configuration:\n- Peg 1: `[2]` (Orange)\n- Peg 2: `[1, 3]` (Green, Red)\n- Peg 3: `[0]` (Blue)\n\nTarget configuration:\n- Peg 1: `[0]` (Blue)\n- Peg 2: `[2]` (Orange)\n- Peg 3: `[1, 3]` (Green, Red)\n\nThe sequence of optimal moves:\n1. Move disk 3 from peg 2 to peg 3 (`3: 2->3`)\n2. Move disk 2 from peg 1 to peg 2 (`2: 1->2`)\n3. Move disk 3 from peg 3 to peg 2 (`3: 3->2`)\n4. Move disk 0 from peg 3 to peg 1 (`0: 3->1`)\n5. Move disk 3 from peg 2 to peg 3 (`3: 2->3`)\n6. Move disk 2 from peg 2 to peg 1 (`2: 2->1`)\n7. Move disk 3 from peg 3 to peg 1 (`3: 3->1`)\n8. Move disk 1 from peg 2 to peg 3 (`1: 2->3`)\n9. Move disk 3 from peg 1 to peg 3 (`3: 1->3`)\n10. Move disk 2 from peg 1 to peg 2 (`2: 1->2`)\n\nAnswer: {'moves': '3: 2->3,2: 1->2,3: 3->2,0: 3->1,3: 2->3,2: 2->1,3: 3->1,1: 2->3,3: 1->3,2: 1->2', 'num_moves': '10'}"
  },
  "2264": {
    "model": "google/gemini-3.8-flash",
    "response": "An analysis of the puzzle:\n\n1. **Disk identification and sizes** (from largest to smallest):\n   - Disk 0: Blue (largest)\n   - Disk 1: Green\n   - Disk 2: Orange\n   - Disk 3: Red (smallest)\n\n2. **Configurations**:\n   - **Start**: \n     - Peg 1: empty\n     - Peg 2: [Orange (2)]\n     - Peg 3: [Blue (0), Green (1), Red (3)]\n   - **Goal**:\n     - Peg 1: [Orange (2)]\n     - Peg 2: [Blue (0)]\n     - Peg 3: [Green (1), Red (3)]\n\n3. **Optimal sequence of moves**:\n   - To move Disk 0 from Peg 3 to Peg 2, Peg 2 must be empty and Disks 1, 2, and 3 must all be cleared onto Peg 1:\n     1. `3: 3->2` (Move Red onto Orange on Peg 2)\n     2. `1: 3->1` (Move Green to Peg 1)\n     3. `3: 2->1` (Move Red onto Green on Peg 1)\n     4. `3: 1->3` (Move Red onto Blue on Peg 3)\n     5. `2: 2->1` (Move Orange onto Green on Peg 1)\n     6. `3: 3->1` (Move Red onto Orange on Peg 1)\n   - Now Peg 2 and the top of Peg 3 are clear, so Disk 0 can move:\n     7. `0: 3->2` (Move Blue to Peg 2)\n   - Now transfer Disks 1 and 3 to Peg 3, while keeping Disk 2 on Peg 1:\n     8. `3: 1->3` (Move Red to Peg 3)\n     9. `2: 1->2` (Move Orange onto Blue on Peg 2)\n     10. `3: 3->2` (Move Red onto Orange on Peg 2)\n     11. `1: 1->3` (Move Green to Peg 3)\n     12. `3: 2->3` (Move Red onto Green on Peg 3)\n     13. `2: 2->1` (Move Orange to Peg 1)\n\nThis requires a minimum of 13 moves.\n\nAnswer: {'moves': '3: 3->2,1: 3->1,3: 2->1,3: 1->3,2: 2->1,3: 3->1,0: 3->2,3: 1->3,2: 1->2,3: 3->2,1: 1->3,3: 2->3,2: 2->1', 'num_moves': '13'}"
  },
  "2265": {
    "model": "google/gemini-3.8-flash",
    "response": "An analysis of the disk sizes and initial/target configurations:\n- **Disk 0 (Blue)**: Largest disk\n- **Disk 1 (Green)**: Second largest disk\n- **Disk 2 (Orange)**: Third largest disk\n- **Disk 3 (Red)**: Smallest disk\n\n**Initial State:**\n- Peg 1: [Disk 0, Disk 1, Disk 3]\n- Peg 2: [Disk 2]\n- Peg 3: []\n\n**Goal State:**\n- Peg 1: [Disk 0, Disk 2]\n- Peg 2: [Disk 1]\n- Peg 3: [Disk 3]\n\n**Optimal Move Sequence (6 moves):**\n1. Move disk 2 from peg 2 to peg 3 (`2: 2->3`)\n2. Move disk 3 from peg 1 to peg 3 (`3: 1->3`)\n3. Move disk 1 from peg 1 to peg 2 (`1: 1->2`)\n4. Move disk 3 from peg 3 to peg 2 (`3: 3->2`)\n5. Move disk 2 from peg 3 to peg 1 (`2: 3->1`)\n6. Move disk 3 from peg 2 to peg 3 (`3: 2->3`)\n\nAnswer: {'moves': '2: 2->3,3: 1->3,1: 1->2,3: 3->2,2: 3->1,3: 2->3', 'num_moves': '6'}"
  },
  "2266": {
    "model": "google/gemini-3.8-flash",
    "response": "An analysis of the disk sizes from largest (0) to smallest (3):\n- **Disk 0 (Blue):** Largest disk\n- **Disk 1 (Green):** Second largest disk\n- **Disk 2 (Orange):** Third largest disk\n- **Disk 3 (Red):** Smallest disk\n\n**Initial State:**\n- Peg 1: Green (1)\n- Peg 2: Orange (2)\n- Peg 3: Blue (0), Red (3)\n\n**Goal State:**\n- Peg 1: Orange (2)\n- Peg 2: Blue (0), Red (3)\n- Peg 3: Green (1)\n\n**Optimal Sequence of Moves:**\n1. Move disk 2 from Peg 2 to Peg 1 (`2: 2->1`)\n2. Move disk 3 from Peg 3 to Peg 1 (`3: 3->1`)\n3. Move disk 0 from Peg 3 to Peg 2 (`0: 3->2`)\n4. Move disk 3 from Peg 1 to Peg 3 (`3: 1->3`)\n5. Move disk 2 from Peg 1 to Peg 2 (`2: 1->2`)\n6. Move disk 3 from Peg 3 to Peg 2 (`3: 3->2`)\n7. Move disk 1 from Peg 1 to Peg 3 (`1: 1->3`)\n8. Move disk 3 from Peg 2 to Peg 3 (`3: 2->3`)\n9. Move disk 2 from Peg 2 to Peg 1 (`2: 2->1`)\n10. Move disk 3 from Peg 3 to Peg 2 (`3: 3->2`)\n\nAnswer: {'moves': '2: 2->1,3: 3->1,0: 3->2,3: 1->3,2: 1->2,3: 3->2,1: 1->3,3: 2->3,2: 2->1,3: 3->2', 'num_moves': '10'}"
  },
  "2267": {
    "model": "google/gemini-3.8-flash",
    "response": "To determine the minimum number of moves to reach the target configuration from the initial configuration, we identify the disks by size (from largest to smallest):\n- **0: Blue** (largest)\n- **1: Green**\n- **2: Orange**\n- **3: Red** (smallest)\n\n### Initial State:\n- **Peg 1:** [Orange (2)]\n- **Peg 2:** [Red (3)]\n- **Peg 3:** [Blue (0), Green (1)]\n\n### Target State:\n- **Peg 1:** [Green (1), Red (3)]\n- **Peg 2:** [Orange (2)]\n- **Peg 3:** [Blue (0)]\n\n### Sequence of Optimal Moves:\n1. Move disk **3** from Peg **2** to Peg **3** (`3: 2->3`)\n2. Move disk **2** from Peg **1** to Peg **2** (`2: 1->2`)\n3. Move disk **3** from Peg **3** to Peg **2** (`3: 3->2`)\n4. Move disk **1** from Peg **3** to Peg **1** (`1: 3->1`)\n5. Move disk **3** from Peg **2** to Peg **1** (`3: 2->1`)\n\nAnswer: {'moves': '3: 2->3,2: 1->2,3: 3->2,1: 3->1,3: 2->1', 'num_moves': '5'}"
  },
  "2268": {
    "model": "google/gemini-3.8-flash",
    "response": "An optimal sequence of moves to transition from the starting configuration to the ending configuration using the standard Tower of Hanoi rules is as follows:\n\n1. Move disk 2 from peg 2 to peg 3 (`2: 2->3`)\n2. Move disk 3 from peg 1 to peg 3 (`3: 1->3`)\n3. Move disk 0 from peg 1 to peg 2 (`0: 1->2`)\n4. Move disk 3 from peg 3 to peg 1 (`3: 3->1`)\n5. Move disk 2 from peg 3 to peg 2 (`2: 3->2`)\n6. Move disk 3 from peg 1 to peg 2 (`3: 1->2`)\n7. Move disk 1 from peg 3 to peg 1 (`1: 3->1`)\n8. Move disk 3 from peg 2 to peg 1 (`3: 2->1`)\n9. Move disk 2 from peg 2 to peg 3 (`2: 2->3`)\n10. Move disk 3 from peg 1 to peg 3 (`3: 1->3`)\n\nAnswer: {'moves': '2: 2->3,3: 1->3,0: 1->2,3: 3->1,2: 3->2,3: 1->2,1: 3->1,3: 2->1,2: 2->3,3: 1->3', 'num_moves': '10'}"
  },
  "2269": {
    "model": "google/gemini-3.8-flash",
    "response": "An analysis of the starting and ending configurations reveals the following disk mappings based on size:\n- Disk 0: Blue (largest)\n- Disk 1: Green\n- Disk 2: Orange\n- Disk 3: Red (smallest)\n\nStarting state:\n- Peg 1: Disk 1, Disk 2\n- Peg 2: Disk 0\n- Peg 3: Disk 3\n\nGoal state:\n- Peg 1: (empty)\n- Peg 2: Disk 0, Disk 1, Disk 2\n- Peg 3: Disk 3\n\nThe optimal sequence of 7 moves is:\n1. Move Disk 3 from Peg 3 to Peg 2\n2. Move Disk 2 from Peg 1 to Peg 3\n3. Move Disk 3 from Peg 2 to Peg 3\n4. Move Disk 1 from Peg 1 to Peg 2\n5. Move Disk 3 from Peg 3 to Peg 1\n6. Move Disk 2 from Peg 3 to Peg 2\n7. Move Disk 3 from Peg 1 to Peg 3\n\nAnswer: {'moves': '3: 3->2,2: 1->3,3: 2->3,1: 1->2,3: 3->1,2: 3->2,3: 1->3', 'num_moves': '7'}"
  },
  "2270": {
    "model": "google/gemini-3.8-flash",
    "response": "An analysis of the starting and ending configurations gives:\n- **Disks by size (from largest to smallest):**\n  - Disk 0: Blue (largest)\n  - Disk 1: Green\n  - Disk 2: Orange\n  - Disk 3: Red (smallest)\n\n- **Initial configuration:**\n  - Peg 1: Disk 0, Disk 1, Disk 2\n  - Peg 2: Disk 3\n  - Peg 3: Empty\n\n- **Target configuration:**\n  - Peg 1: Disk 0\n  - Peg 2: Disk 1, Disk 3\n  - Peg 3: Disk 2\n\nThe optimal sequence of moves is:\n1. Move Disk 2 from Peg 1 to Peg 3 (`2: 1->3`)\n2. Move Disk 3 from Peg 2 to Peg 3 (`3: 2->3`)\n3. Move Disk 1 from Peg 1 to Peg 2 (`1: 1->2`)\n4. Move Disk 3 from Peg 3 to Peg 2 (`3: 3->2`)\n\nAnswer: {'moves': '2: 1->3,3: 2->3,1: 1->2,3: 3->2', 'num_moves': '4'}"
  }
}