Test one

Prompt

Solve the following mathematical problem with maximum rigor and precision. There are **4 pegs and 100 disks** of different sizes. Initially, all disks are stacked on the first peg, from largest at the bottom to smallest at the top. The goal is to move all 100 disks to the fourth peg while following these rules: 1. Only one disk may be moved at a time. 2. A larger disk may never be placed on top of a smaller disk. 3. The other two pegs may be used freely as temporary storage. **Question:** What is the exact minimum number of moves required? **Requirements:** - Find the globally optimal solution, not merely a valid algorithm. - Establish mathematical optimality rigorously. - Do not use programming, internet searches, or precomputed tables. - Perform all necessary mathematical reasoning internally, but do not reveal your reasoning process. - Do not provide estimates or approximations. - If you cannot establish the exact answer with confidence, explicitly state your uncertainty rather than guessing. **Your final answer must contain exactly two lines:** 1. The exact minimum number of moves. 2. One short sentence explaining the mathematical basis for the result. No additional explanations, calculations, or commentary.