The 100-story building and the two eggs

1. In the worst case it is not advisable to always advance with equal leaps.
2. If the first egg breaks, the remaining section with the second must fit exactly in the attempts you have left.
3. That is why the jumps of the first egg must be adjusted as you go up.

Answer: The worst case minimum is 14 throws. Explanation: Two risks must be balanced:

• if the first egg breaks too soon, you need many linear attempts with the second one;
• if you reserve it too much, you spend too many throws raising. The optimal strategy is to lower the remaining margin one by one:
• first try on the 14th floor,
• then on the 27th,
• then in 39,
• then in 50,
• and so on, each time adding a jump one smaller unit. Because? Because this way, if the egg breaks in one test, the number of floors left to check with the second one exactly matches the number of throws still available. You need a number $n$ such that

$$ 1+2+\cdots+n \ge 100. $$

The smallest is $n=14$, because

$$ 14\cdot 15/2 = 105. $$ That's why 14 is the guaranteed minimum. $$