"The Mirzakhani Wand: A Magical Gamble"

"The legendary Mirzakhani wand stands as the most potent magical artifact ever forged, holding immense power within its core. It is this very power that the malevolent sorcerer Moldevort seeks to harness in his bid to conquer the world. In your quest, alongside the enigmatic Drumbledrore, you have finally unveiled the concealed cave where this formidable wand lies hidden. However, accessing it is not as straightforward as it seems.

The wand's secrecy is guarded by a labyrinthine arrangement of a hundred magical stones, among which there is a radiant keystone, and a hundred platforms. To reveal the wand, the keystone must be precisely placed on the correct platform. An incorrect placement, however, would trigger a catastrophic cave collapse. The keystone itself is impervious to all magical manipulation, while the other stones are susceptible. When a stone is lifted and the placement spell is cast, the corresponding platform will glow. The challenge is to accurately place all 99 stones, with the final one being the keystone.

Just as you're on the cusp of commencing this high-stakes endeavor, an agent of Moldevort arrives, maliciously sealing one stone to a randomly chosen platform, a move that cannot be undone. If you need to place a stone on a platform already occupied, your spell will cause another unoccupied platform to illuminate. The odds of positioning the keystone correctly are at stake.

Take a moment to ponder the probabilities.

In a world of perfect knowledge, one could systematically label the stones from 1 to 100 based on the order of placement and assign matching numbers to the platforms. The stone the henchman placed is designated as '1,' symbolizing its intended platform, and the keystone as '100,' reserved for platform 100. Yet, the platforms themselves remain obscured in mystery, their numbering eluding you.

Three scenarios emerge:

First, if the initial stone was haphazardly placed on its own platform, you're assured of success.

Second, if it landed on the keystone's platform, your fate is sealed, and you are destined to fail.

But the most likely scenario unfolds in the third case, where the henchman's stone ends up on a platform different from its intended placement. For instance, if stone 1 finds itself on platform 45, your journey continues. You systematically place stones 2 through 44 on their corresponding platforms, only to encounter a random platform lighting up when you reach stone 45. Here, you again confront three possibilities:

If it's platform 1, victory is yours, as the remaining stones will all align with their correct platforms. If it's platform 100, your fate is sealed, as the keystone's spot is occupied. Any other platform, and you return to a state akin to your initial scenario, now dealing with 54 remaining stones and one incorrectly placed. The cycle repeats.

This cyclic process continues, with each iteration offering an equal chance of success or failure, and the possibility of further delay. Ultimately, you will either place a stone on platform 1 or platform 100 before reaching the keystone. This singular moment determines your fate, and crucially, the odds of both outcomes are evenly matched.

This concept may initially seem perplexing, but let's envision a parallel scenario. Imagine Drumbledrore conjures numbers ranging from 1 to 100. If a 1 is selected, you win; if it's 100, you lose. Any other number prompts a reselection. Since the chances of winning with a 1 are identical to losing with a 100, this is a game where your odds of winning or losing are perfectly balanced. Although it might involve some delays, these do not skew the balance toward either outcome.

This is precisely the scenario you face now: a precarious 50/50 chance of triggering a cave-in. But Drumbledrore reveals his trump card, a rare felush felucious potion, renowned for bestowing extraordinary luck for a fleeting moment. There exists a 1 in 100 possibility that the keystone's platform was compromised from the start, rendering your efforts in vain. However, beyond this, your chances remain evenly poised, offering an enticing opportunity for success, and at this moment, fortune smiles upon you."