The legendary Mirzakhani Scepter is the most powerful magical item ever created. And that's why the evil wizard Moldevort is planning to use it to conquer the world. You and Dumbledore have finally discovered his hiding place in this cave.
The wand is hidden by a system of 100 magic stones—including a glowing keystone—and 100 platforms. If the keystone is placed on the correct platform, the wand will appear. If placed incorrectly, the entire cave will collapse.
The keystone is immune to all magic,
but the other stones are not, meaning you can pick them up and cast a place spell, and the platform the stone is on will glow. Place all 99 stones correctly and the final platform should be the correct resting place of the keystone.
You're about to start when one of Moldevort's henchmen comes and stamps a random stone onto a random platform. If you need to place a stone that's on a platform that's already occupied, your spell will flash some random empty platform instead.
What are your difficulties in placing the keystone on the correct platform?
Stop by now to find out for yourself.
Answer in 3 Answer in 2 Answer in 1 Let's imagine that we know everything about this situation. With perfect knowledge, we can label stones from 1 to 100, based on the order we intend to place them, and label the platforms they are on in the same order.
We will label the stone placed by the chickens as 1, meaning it should have gone to platform 1, and the keystone as 100, which belongs to platform 100. Of course, we don't know which platform, so the number platform is actually invisible to us.
There are three possibilities: One, that first stone was randomly placed on its own platform, in which case, you are guaranteed to succeed. Two, it was placed on a keystone platform and you will fail.
But most likely - scenario three - it was placed elsewhere. Suppose the chicken places stone 1 on platform 45. Then you place stone 2 on platform 2, 3 on platform 3, and so on, until you reach stone 45. . And here, there are three possibilities:
If it's platform 1,
you win, because all the other stones will go to the correct platform. If platform 100 lights up, you lose, as the keystone will be replaced. Any other platform, and you're basically back where you started, with only 54 stones left and on the wrong platform.
In this scenario, suppose the mantra tells us to place stone 45 on platform 82. Then we place 46 to 81 correctly and 82 randomly. And here we get to the same three possibilities:
Pedestal 1, you win, Pedestal 100, you lose,
someone else, you continue the process. In other words, you are playing a game where you have equal chances of winning and losing, and there is some chance of delaying the decisive moment.
No matter how many times this process is repeated, you will place a stone on either pedestal 1 or pedestal 100 before reaching the keystone. It is what determines whether you succeed or fail, and critically, the odds of these events being equal.
This may sound implausible,
so let's imagine another, similar game. Say that Dumbledore magically generates numbers from 1 to 100. If it's 1, you win. If it's 100, you lose.
If it's something else, he picks again. Since the odds of winning by getting 1 are the same as losing by getting 100, this is a game where you are just as likely to win as you are to lose.
It may take some time, but the delay doesn't benefit from getting 1 before 100 or vice versa. The same essential reasoning applies to our situation.
You're debating whether it's worth risking a 50/50 chance of entering the cave when Dumbledore reveals his secret weapon: a rare Felosh Felicious Potion, which offers short-lived exceptional luck.
Gives There's a 1 in 100 chance that the keystone platform took the first stone and you've already lost, but otherwise, you've got even odds of winning or losing. And right now, you're feeling lucky.
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