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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!

  • Answer in: 3

  • Answer in: 2

  • Answer in: 1

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

  • it might help to start with a simpler scenario.

  • 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

  • by adding more terms until we get to our answer,

  • 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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A2 US TED-Ed egg floor thief replica interval

【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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