Let x be the answer Aalia wants, the number of drops required.
So if the first egg breaks maximum she can have x-1 drops and so she must always put the first egg from height x. So she has determined that for a given x she must drop the first ball from x height. And now if the first drop of the first egg doesn't breaks she can have x-2 drops for the second egg if the first egg breaks in the second drop.
Taking an example, if 16 is her answer. She needs 16 drops to find out the answer First She drops from height 16,and if it breaks she tries all floors from 1 to 15. If the egg doesn't break then she has left 15 drops, so she will drop it from 16+15+1 =32nd floor. The reason being if it breaks at 32nd floor she can try all the floors from 17 to 31 in 14 drops (total of 16 drops). Now if it did not break then she is left with 13 drops. and she can figure out whether she can find out whether she can figure out the floor in 16 drops.
Lets take the case with 16 as the answer
1 + 15 16 if breaks at 16 checks from 1 to 15 in 15 drops
1 + 14 31 if breaks at 31 checks from 17 to 30 in 14 drops
1 + 13 45 .....
1 + 12 58
1 + 11 70
1 + 10 81
1 + 9 91
1 + 8 100 We can easily do in the end as we have enough drops to accomplish the task
Now finding out the optimal one she can see that she could have done it in either 15 or 14 drops only but how can she find the optimal one. The optimal one will be needing 0 linear trials in the last step.
So she could write it as
(1+p) + (1+(p-1))+ (1+(p-2)) + .........+ (1+0) >= 100.
Let 1+p=q which is the answer she is looking for
q (q+1)/2 >=100
Solving for 100 she gets q=14.
So the answer is: 14
Drop first orb from floors 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100... (i.e. move up 14 then 13, then 12 floors, etc) until it breaks (or doesn't at 100)