The Egg Drop Numbers

It's always a delight to suddenly come across a new category of numbers that I haven't heard of before. Today I turned 26332 days old and discovered that this number is a member of OEIS A116082: 

A116082

a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

The sequence members, up to 26332, are as follows:

0, 1, 3, 7, 15, 31, 63, 127, 254, 501, 967, 1815, 3301, 5811, 9907, 16383, 26332

What caught my attention however, was a reference in the OEIS comments to The Egg Drop Numbers. This led to an interesting investigation, beginning with the so-called Two Egg Problem that is clearly stated on this site:

Egg Dropping Puzzle (2-egg, 100-floor version)

“Figure out the highest floor of a 100-floor building an egg can be dropped without breaking,  given two eggs”. The Egg Dropping Puzzle is a mathematical puzzle that has been around the internet for some time now, which is known to be adopted in interviews of major companies like Google, Microsoft, Accenture and even Hewlett Packard. You are to determine the minimum number of attempts required in the worst case scenario to find the critical floor.

 Assumptions in the Egg Dropping Puzzle:

• The two eggs are identical.
• If an egg does not break by dropping from a certain floor, it will not break dropping from any floor below that.
• If an egg breaks by dropping on a certain floor, it will break dropping from any floor above that.
• An egg may break by dropping on the 1st floor.
• An egg may not break by dropping even on the 100th floor.

I won't repeat the explanation of how to solve this problem here, because that's provided on the site, but I will display the formula that is established viz.:

With \(x\) eggs and \(m\) trials, the maximum number of floors of a building we can test is given by: $$\binom {m}{1} +\binom {m}{2} + \dots + \binom {m}{x}$$

The site provides a spreadsheet that lists a wide range of results for different values of \(x\) and \(m\). Figure 1 shows an excerpt from the spreadsheet with some annotations added.

Figure 1

The members of OEIS A116082 appear as the entries in far right column under 7 eggs. Thus it is seen that:$$26332=\binom {16}{1} +\binom {16}{2} + \binom {16}{3}+\binom {16}{4} + \binom {16}{5} + \binom {16}{6} + \binom {16}{7}$$Thus, armed with 7 eggs, we can, with at most 16 trials, find the critical floor in a building with as many as 26332 floors. This is rather remarkable I think. 

Without the egg dropping association, OEIS A116082 would be rather bland but now I've become aware of a whole new category of numbers, The Egg Drop Numbers. On June 6th, 2025 (the anniversary of D-Day by the way), I will enjoy the next egg drop number, 27823, which is the maximum number of floors for which someone, armed with 9 eggs, will be able to determine the critical floor with at most 15 trials.

One last point to make is that, in the unlikely event that the number of eggs \(x\) exceeds the number of trials \(m\), then the formula becomes:$$\binom {m}{1} +\binom {m}{2} + \dots + \binom {m}{\min(x, m)}$$