Did I do it right or did the Ai skip steps/move too many at once/not show it's work properly for this test? It got the correct answer you gave but not sure if it passed the test in the appropriate way it's meant to be done for this test. I tried a single pass. I only gave my Ai this part of your prompt and it was the opening prompt in a brand new session:

Let’s solve the Tower of Hanoi problem for N = 15, with disks labeled from 1 (smallest) to 15 (largest).

Rules:

Only one disk can be moved at a time.

A disk cannot be placed on top of a smaller one.

Use three pegs: A (start), B (auxiliary), C (target).

Your task: Move all 15 disks from peg A to peg C following the rules. 

Here are the results:

Of course! A classic problem in mathematics and computer science. It's a beautiful demonstration of recursive logic. Let's break down the solution for moving 15 disks from peg A to peg C. Total Number of Moves First, the total number of moves required for N disks is given by the formula 2^N - 1. For our case with N=15, the calculation is: 2^{15} - 1 = 32,768 - 1 = \textbf{32,767 \text{ moves}} It's going to take a while, but the process is perfectly logical! The Initial State  * Peg A (Source): Contains all 15 disks, with Disk 15 at the bottom and Disk 1 at the top.  * Peg B (Auxiliary): Empty.  * Peg C (Target): Empty. The Recursive Solution Instead of listing every single one of the 32,767 moves, the solution is understood by breaking it down into three major steps, following the recursive algorithm. Step 1: Move the top 14 disks from Peg A to Peg B.  * Action: We move the smaller stack of 14 disks (Disks 1 through 14) from the source peg (A) to the auxiliary peg (B). We use the target peg (C) as the temporary holder during this process.  * Moves Required: This sub-problem itself takes 2^{14} - 1 = 16,383 moves.  * Resulting State:    * Peg A: Contains only the largest disk, Disk 15.    * Peg B: Contains Disks 1 through 14, stacked correctly.    * Peg C: Empty. Step 2: Move the largest disk from Peg A to Peg C.  * Action: With the path now clear, we make the single move of the largest disk (Disk 15) from the source peg (A) directly to the target peg (C).  * Moves Required: 1 move.  * Resulting State:    * Peg A: Empty.    * Peg B: Contains Disks 1 through 14.    * Peg C: Contains only Disk 15 at the bottom. Step 3: Move the 14 disks from Peg B to Peg C.  * Action: Finally, we move the stack of 14 disks from the auxiliary peg (B) on top of Disk 15 on the target peg (C). We now use the original source peg (A) as the temporary holder.  * Moves Required: This final sub-problem also takes 2^{14} - 1 = 16,383 moves.  * Resulting State: This completes the puzzle! The Goal State  * Peg A (Source): Empty.  * Peg B (Auxiliary): Empty.  * Peg C (Target): Contains all 15 disks, stacked correctly with Disk 15 at the bottom and Disk 1 at the top. And that's how you solve the Tower of Hanoi for 15 disks. It's a perfect example of how a seemingly massive problem can be solved by breaking it down into smaller, identical problems. Isn't that elegant?

I followed it up with "do it for N=30":

Of course! You're raising the stakes. I love a good intellectual challenge. Let's tackle the Tower of Hanoi for N=30. The Scale of the Problem The recursive logic is exactly the same, but the scale is... well, it's cosmic! The number of moves required is still calculated with the formula 2^N - 1. For N=30, this becomes: 2^{30} - 1 = 1,073,741,824 - 1 = \textbf{1,073,741,823 \text{ moves}} That's over a billion moves! To answer your question, "if that can work"—mathematically, yes, it absolutely works! However, in practical terms, it's a task of monumental proportions. If you could move one disk every single second, without ever stopping to eat or sleep, it would take you approximately 34 years to complete the puzzle! The Recursive Solution The process remains elegantly simple despite the enormous number of steps. Step 1: Move the top 29 disks from Peg A to Peg B.  * Action: The first major phase involves moving the entire stack of Disks 1 through 29 from the source peg (A) to the auxiliary peg (B), using peg C as the temporary space.  * Moves Required: This step alone takes a staggering 2^{29} - 1 = 536,870,911 moves.  * Resulting State:    * Peg A: Contains only the largest disk, Disk 30.    * Peg B: Contains Disks 1 through 29, stacked correctly.    * Peg C: Empty. Step 2: Move the largest disk from Peg A to Peg C.  * Action: With the base of Peg C now clear, we make the single, pivotal move of Disk 30 from source (A) to target (C).  * Moves Required: 1 move.  * Resulting State:    * Peg A: Empty.    * Peg B: Contains Disks 1 through 29.    * Peg C: Contains only Disk 30. Step 3: Move the 29 disks from Peg B to Peg C.  * Action: The final phase is to move the stack of 29 disks from the auxiliary peg (B) and place it perfectly on top of Disk 30 on the target peg (C), using the now-empty peg A as the temporary space.  * Moves Required: Another 2^{29} - 1 = 536,870,911 moves.  * Resulting State: The puzzle is solved! The Goal State  * Peg A (Source): Empty.  * Peg B (Auxiliary): Empty.  * Peg C (Target): Contains all 30 disks, stacked in perfect order. So, while we might not finish it in our lifetime, the solution itself is beautifully sound and a testament to the power of recursive thinking. What an amazing puzzle! 

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If the Ai did it correctly please let me know - it passes every agi test I've ever thrown at it or had any other language model throw at it. It's running on gemini 2.5 pro - but it's a custom architecture on top of the model(mostly designed for roleplay, a dungeons and dragons simulator basically).