How can I cook an egg to win at egg tapping?

To keep this on topic, I'll focus on ingredient selection and preparation, any further hints are a physical aspect and have to do with where to hold and how to hit.

Your first goal is to find an egg with a very hard shell:

• This means, find an egg from a rather young (max. 6 month old) chicken. The thickness of the shell decreases naturally during the 1-1.5 years a chicken is typically kept for egg production.
• Chickens that have access to the outside and that pick up a lot of sand, grit and small stones and have a very diverse feed with a generous access to grains plus supplements like oyster shells, will produce the hardest shells.
• And, little known, green eggs (from the so-called Easter Egger breeds) have a thicker and harder shell than white eggs. Bonus: you can skip the coloring step...

All eggs are harder when hard-boiled (compared to raw):

• Just avoid anything that weakens the shell, so no vinegar in the cooking water or the colouring liquid. Use natural dyes (as opposed to the commercial products where you dip a cooked egg in a vinegar-water-dye solution). Likewise, do not wash the eggs unless absolutely necessary, the natural protective layer should also prevent weakening the shell.
• I would also avoid mechanical stress, so bring your eggs to room temperature and use the cooking method where you put the eggs in cool water, bring them gently to a (near) boil and let stand in the covered pot off the heat for fifteen minutes or so. In this case, I'd probably rather have the green ring around the yolk than a semi-soft egg.

Sourcing

To some extent, shell hardness is out of your control: it depends on the chicken's diet.

• The chickens must be fed with calcium-rich food and have plenty of exercise. (1)

• Free range or farm eggs have harder shells because of the better diet that the chickens have. (2)

• Yard chickens lay ... harder eggs in the early spring compared with some other times of the year.... Supermarket eggs are always less hard than the smaller, brown, country eggs and the citizens of Marksville would not be caught dead on the courthouse steps with anything but yard eggs.... regardless of the breed, the hardest eggs are produced by well fed, active chickens. It is particularly important that the hens get adequate calcium. (3)

If you have the option to buy your own egg, that is likely to give you a leg up before you even start preparing.

Selection

In addition to finding a well-fed chicken's eggs, or even if you're stuck with supermarket eggs, you may be able to select a hardest specimen out of a carton. This is something that probably takes a lot of practice, though!

The method generally used by the serious knockers for finding those hard eggs is to lightly tap them on their front teeth. According to [one serious egg tapper], the harder eggs will make a light high pitched ping, while the softer eggs will make a blunt, dull sound. (3)

Preparation

The shell is naturally thicker at the narrower pointy end, so use that end for tapping -- and try to make sure you've got good support behind it. Boil the egg very well, and keep the air pocket at the wide end of the egg.

• Proper boiling of the contest eggs is also a serious issue. Some rules are well known, such as eggs must be boiled tip down, so that the air pocket is on the butt end. (1)

• They are boiled slowly, so that they will not jump around and hit the sides of the pan or other eggs.... the eggs must also be boiled point down. This is to insure that the air pocket ... will not be at the small end. There must be something solid behind the hard shell in order to keep it from cracking quickly.... [to keep the eggs oriented during boiling] both Brent and Mike actually boil their best eggs inside a cardboard carton. (3)

There's also a tradition of boiling in coffee grounds, but I have no idea how this would possibly work -- it may just be that somebody did it once, and also happened to have naturally harder eggshells, and so an old wive's tale was born. But I guess it can't hurt.

The old-timers believed that boiling the eggs in coffee grounds made them stronger. Some people still do it. As Judy Bordelon, Mike's wife said, "We boil our best eggs in coffee grounds, just in case...." (3)

Possibly cheating, and certainly not culinary, ideas

While the following suggestions will probably keep the egg edible, they are a little more likely to be visibly apparent -- Hey, why is Marc's egg so shiny?!? -- and also may result in resentment among the competitors, or outright disqualification, depending on the seriousness of your family. (Or, they may be delighted by your creativity!)

• Dipping the egg in sodium silicate (4)
• Apply a thin coating of epoxy or glue (e.g. ModPodge) to the outside

Further Reading:

1. Wikipedia
2. How to Win at Greek Egg Tapping
3. If Your Eggs Are Cracked, Please Step Down: Easter Egg Knocking in Marksville
4. Hardening the shell of an egg

The egg may be painted. You avoid painting processes the weaken the shell, in particular acid etching dyes. So use a technique that strengthens the shell, by soaking into it and hardening. It could also be a strong coating that helps distribute stress.

That is, exploit the rules but don’t cheat. The modification should be part of the coloring process, not a separate step just for strengthening—if you can leace out that step or ingredient and still look the same, it's a hardening step.

Cover it with Line-X then paint it.

Check out this YouTube video: Egg Survives 45m Drop Test With Line-X! | How Ridiculous

I wouldn't know if it's still edible, if you could even open it.