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A building has 100 floors. One of the floors is the highest floor an egg can be dropped from without breaking.
If an egg is dropped from above that floor, it will break. If it is dropped from that floor or below, it will be completely undamaged and you can drop the egg again.
Given two eggs, find the highest floor an egg can be dropped from without breaking, with as few drops as possible.
We can do better than a binary ↴ approach, which would have a worst case of 50 drops.
We can even do better than 19 drops!
We can always find the highest floor an egg can be dropped from with a worst case of 14 total drops.
What if we only had one egg? How could we find the correct floor?
Because we can’t use the egg again if it breaks, we’d have to play it safe and drop the egg from every floor, starting at the bottom and working our way up. In the worst case, the egg won’t break until the top floor, so we’d drop the egg 100 times.
What does having two eggs allow us to do differently?
Since we have two eggs, we can skip multiple floors at a time until the first egg breaks, keeping track of which floors we dropped it from. Once that egg breaks we can use the second egg to try every floor, starting on the last floor where the egg didn’t break and ending on the floor below the one where it did break.
How should we choose how many floors to skip with the first egg?
What about trying a binary ↴ approach? We could drop the first egg halfway up the building at the 50th floor. If the egg doesn’t break, we can try the 75th floor next. We keep going like this, dividing the problem in half at each step. As soon as the first egg breaks, we can start using our second egg on our (now-hopefully narrow) range of possible floors.
If we do that, what’s the worst case number of total drops?
The worst case is that the highest floor an egg won’t break from is floor 48 or 49. We’d drop the first egg from the 50th floor, and then we’d have to drop the second egg from every floor from 1 to 49, for a total of 50 drops. (Even if the highest floor an egg won’t break from is floor 48, we still won’t know if it will break from floor 49 until we try.)
Can we do better than this binary approach?
50 is probably too many floors to skip for the first drop. In the worst case, if the first egg breaks after a small number of drops, the second egg will break after a large number of drops. And if we went the other way and skipped 1 floor every time, we’d have the opposite problem! What would the worst case floor be then?
The worst case would be floor 98 or 99—the first egg would drop a large number of times (at every floor from 2-100 skipping one floor each time) and the last egg would drop a small number of times (only on floor 99), for a total of 51 drops.
Can we balance this out? Is there some number between 50 and 1—the number of floors we’ll skip with each drop of the first egg—where the first and second eggs would drop close to the same number of times in the worst case?
Yes, we could skip 10 floors each time. The worst case would again be floor 98 or 99, but we’d only drop the first egg 10 times and the second egg 9 times, for a total of 19 drops!
Is that the best we can do?
Let’s look at what happens with this strategy each time the first egg doesn’t break. How does the worst case total number of drops change?
The worst case total number of drops increases by one each time the first egg doesn’t break
For example, if the egg breaks on our first drop from the 10th floor, we may have to drop the second egg at each floor between 1 and 9 for a worst case of 10 total drops. But if the egg breaks when we skip to the 20th floor we will have a worst case of 11 total drops (once for the 10th floor, once for the 20th, and all of the floors between 11 and 19)!
How can we keep the worst case number of drops from increasing each time the first egg doesn’t break?
Since the maximum number of drops increases by one each time we skip the same amount of floors, we could skip one fewer floor each time we drop the first egg!
But how do we choose how many floors to skip the first time?
Well, we know two things that can help us:
Now that we know this, can we figure out the ideal number of floors to skip the first time?
To be able to decrease the number of floors we skip by one every time we move up, and to minimize the number of floors we skip the first time, we want to end up skipping just one floor at the very top. Can we model this with an equation?
Let’s say n is the number of floors we’ll skip the first time, and we know 1 is the number of floors we’ll skip last. Our equation will be:
How can we solve for n? Notice that we’re summing every number from 1 to n.
The left side is a triangular series. ↴ Knowing this, can we simplify and solve the equation?
We know the left side reduces to:
so we can plug that in:
and we can rearrange to get a quadratic equation:
which gives us 13.651.
We can’t skip a fraction of a floor, so how many floors should we skip the first time? And what’s our final worst case total number of drops?
We’ll use the first egg to get a range of possible floors that contain the highest floor an egg can be dropped from without breaking. To do this, we’ll drop it from increasingly higher floors until it breaks, skipping some number of floors each time.
When the first egg breaks, we’ll use the second egg to find the exact highest floor an egg can be dropped from. We only have to drop the second egg starting from the last floor where the first egg didn’t break, up to the floor where the first egg did break. But we have to drop the second egg one floor at a time.
With the first egg, if we skip the same number of floors every time, the worst case number of drops will increase by one every time the first egg doesn’t break. To counter this and keep our worst case drops constant, we decrease the number of floors we skip by one each time we drop the first egg.
When we’re choosing how many floors to skip the first time we drop the first egg, we know:
Knowing this, we set up the following equation where n is the number of floors we skip the first time:
This triangular series ↴ reduces to n * (n+1) / 2 = 100 which solves to give n = 13.651. We round up to 14 to be safe. So our first drop will be from the 14th floor, our second will be 13 floors higher on the 27th floor and so on until the first egg breaks. Once it breaks, we’ll use the second egg to try every floor starting with the last floor where the first egg didn’t break. At worst, we’ll drop both eggs a combined total of 14 times.
Highest floor an egg won't break from
13 Floors we drop first egg from 14 Floors we drop second egg from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 Total number of drops 14
Highest floor an egg won't break from
98 Floors we drop first egg from 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99 Floors we drop second egg from 96, 97, 98 Total number of drops 14
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