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• The city has just opened its one-of-a-kind Fabergé Egg Museum

• with a single egg displayed on each floor of a 100-story building.

• And the world's most notorious jewel thief already has her eyes on the prize.

• Because security is tight and the eggs are so large,

• she'll only get the chance to steal one

• by dropping it out the window into her waiting truck

• and repelling down before the police can arrive.

• All eggs are identical in weight and construction,

• but each floor's egg is more rare and valuable than the one below it.

• While the thief would naturally like to take the priceless egg at the top,

• she suspects it won't survive a 100-story drop.

• Being pragmatic, she decides to settle for the most expensive egg she can get.

• In the museum's gift shop, she finds two souvenir eggs,

• perfect replicas that are perfectly worthless.

• The plan is to test drop them

• to find the highest floor at which an egg will survive the fall without breaking.

• Of course, the experiment can only be repeated

• until both replica eggs are smashed.

• And throwing souvenirs out the window too many times

• is probably going to draw the guards' attention.

• What's the least number of tries it would take

• to guarantee that she find the right floor?

• Pause here if you want to figure it out for yourself!

• If you're having trouble getting started on the solution,

• Imagine our thief only had one replica egg.

• She'd have a single option:

• To start by dropping it from the first floor

• and go up one by one until it breaks.

• Then she'd know that the floor below that

• is the one she needs to target for the real heist.

• But this could require as many as 100 tries.

• Having an additional replica egg gives the thief a better option.

• She can drop the first egg from different floors at larger intervals

• in order to narrow down the range where the critical floor can be found.

• And once the first breaks,

• she can use the second egg to explore that interval floor by floor.

• Large floor intervals don't work great.

• In the worst case scenario, they require many tests with the second egg.

• Smaller intervals work much better.

• For example, if she starts by dropping the first egg from every 10th floor,

• once it breaks, she'll only have to test the nine floors below.

• That means it'll take at most 19 tries to find the right floor.

• But can she do even better?

• After all, there's no reason every interval has to be the same size.

• Let's say there were only ten floors.

• The thief could test this whole building with just four total throws

• by dropping the first egg at floors four,

• seven,

• and nine.

• If it broke at floor four, it would take up to three throws of the second egg

• to find the exact floor.

• If it broke at seven,

• it would take up to two throws with the second egg.

• And if it broke at floor nine,

• it would take just one more throw of the second egg.

• Intuitively, what we're trying to do here is divide the building into sections

• where no matter which floor is correct,

• it takes up to the same number of throws to find it.

• We want each interval to be one floor smaller than the last.

• This equation can help us solve for the first floor we need to start with

• in the 100 floor building.

• There are several ways to solve this equation,

• including trial and error.

• If we plug in two for n, that equation would look like this.

• If we plug in three, we get this.

• So we can find the first n to pass 100

• which is 14.

• And so our thief starts on the 14th floor,

• moving up to the 27th,

• the 39th,

• and so on,

• for a maximum of 14 drops.

• Like the old saying goes, you can't pull a heist without breaking a few eggs.

The city has just opened its one-of-a-kind Fabergé Egg Museum

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# 【TED-Ed】Can you solve the egg drop riddle? - Yossi Elran

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Netter Lee posted on 2018/04/05
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