It will help to know a little math. We don’t mean fancy math like differential calculus, the Mandelbrot set and Fermat’s Last Theorem. We mean the ability to juggle lists with an infinite number of items in your head without getting dizzy. An appreciation for very large numbers will also be beneficial. Most people have a concept of infinity that is far too small. You must disabuse them of their petty notions and expand their idea of infinity at every opportunity.

You can prime your guests in this direction by keeping a few large numbers at your disposal. For example, you should know that if you wanted to rebuild the Empire State Building so that it is completely filled with pennies, you would need about 1.8 trillion (written as 1.8 x 10^{12} in scientific notation). While that’s about three hundred times greater than the present world population, and an awful lot of pennies, it’s easy to find larger numbers. The largest number most of your guests will have met in their education is Avogadro’s number (named after the only scientist in history whose name rhymes with avocado), which is the number of atoms in twelve grams of carbon, 6.02 x 10^{23}. This formidable number is comparable to the number of grains of sand in the all the beaches and deserts of the Earth, or to the number of stars in the universe, take your pick. And you’re barely warming up.

Larger still is a googol, which is 10^{100} (“ten raised to the power 100” or ten multiplied by itself one hundred times). If you wanted to fill the observable universe with a googol of pennies, you would have enough left over to fill another thousand billion universes. Evidently, a googol is starting to get up there, size-wise, but it’s a mere drop of water in the ocean of the largest number so far used in a mathematical proof, known as Graham’s number. Graham himself wondered if we’d spent too much time thinking about small numbers and maybe all the exciting stuff happens at really big numbers. Graham’s number is almost unimaginably large. It is built in sixty-four stages as a tower of exponents and exponents of exponents. The third stage alone is a tower of 3’s (3 to the power 3 to the power 3 to the power 3 to the power 3…) seven trillion levels high. And that’s only the third stage – sixty-one more number-building stages follow. Converting a googol’s worth of atoms (more than are present in the visible universe) to pencil lead would allow you to write out only the first few digits of Graham’s number. In the interests of educating your guests about infinity, which is still larger than Graham’s number, we suggest making an announcement on a sandwich board outside your restaurant, below the daily special. By the way, you can say, have you pondered Graham’s number lately?

An infinite number of rooms, then, is clearly a lot. Never mind what goes on in these rooms (if your hotel is full, everything that *can* happen in a hotel room is happening), your task is to make sure your guests are happy. Due to the size of your hotel (by far the largest in the universe), you will have some particular challenges to ensure your guests want to return for another stay.

Your first task will be to hire more bell hops. However many you may have at the moment, we can assure you it won’t be enough. Consider the following scenario. A guest registers for a room and because of the enormously popular Amateur Mathematician’s convention, you assign him a room in the corridor whose room numbers are all in the billions. You dispatch a bell hop with your guest to help him find his room. Another guest shows up with two suitcases and you assign her a room in the same corridor. But due to the length of the corridor, the first bell hop isn’t due back for hours, if not days. You dispatch another bell hop with this guest and tend to the next who is now registering. Meanwhile, a guest whose room number is in the quadrillions calls the front desk and asks if it’s possible to send up an infinite number of rolls of typewriter tape, so you dispatch a bell hop to fetch some. Then a guest named Zeno calls to ask how he can ever hope to leave his room if each step he takes toward the door is half the size of his previous step. Again, you dispatch a bell hop to assist him. And just when you’re ready to take a break, a guest calls to complain about what sounds to be a *very* large number of monkeys banging away on typewriters next door. You should remind the complaining guest that this is the Hotel Infinity, after all, where everything can happen, and attempt to appease him or her by dispatching a couple of bell hops and a security guard to check out the situation. Clearly, in a hotel like this, you can never have too many bell hops. (We also suggest making a connection with a reliable banana supplier. If you really do have an infinite number of monkeys banging out the complete works of Shakespeare in one of your rooms, you can bet that when it’s time for their banana break they are going to be *very* hungry.)

One thing you should expect is that many guests will ask to check into the room with the highest number. Here they will think they have fooled you because if your hotel has a room with the highest number then how can it have an infinite number of rooms? They may drum their fingers on the counter, stand with arms akimbo or otherwise look askance at you while awaiting your reply. You could smile politely, inform your guest that the room is already occupied and offer them a room numbered for the largest known prime number, or you might remind your guest that infinity is a concept not a number (it may help to point out that there are an infinite number of real numbers between zero and one). Or you could simply frown and say, “Yes, it is available, but it’s a long, long, long way down an infinite corridor and you are liable to starve on your way there.”

In fact, starvation might be a problem for any guest trying to reach a room whose number is suitably large. Encourage your guests to stop at the all-you-can-eat buffet before trudging the ghastly lengths of your corridors to find their rooms. Tell them they can even get a plate to go – they will need it.

Another problem will be guests forgetting their room numbers. While there are hints to remember certain numbers (for example, 65536 is the sixteenth power of two and 65537 is a prime number) some of these tricks will become much more cumbersome as the room numbers get dozens or hundreds, not to mention billions of digits long. For this reason, we recommend that you tell your guests to use room service whenever possible and, except in the case of an emergency, to never leave their rooms.

Be sure to keep the housekeeping inventory up to date. It will be far easier to order new items on a regular basis than to wait until you run out – delivery of an infinite number of little bottles of shampoo could take forever.

Knowing a little bit about infinity will be useful on the occasions when all your rooms are full. Turning away a guest who has a reservation would be bad for business. As soon as people hear that the hotel is full they will sue you for false advertising and you will be ruined. One of the unique features of your hotel is that even when there is no vacancy, you can still accommodate more guests.

In this situation, rather than putting up the No Vacancy sign, which would also be bad for future business, kindly ask all your guests to shift to a room whose number is one higher than their present room. This will free up room number 1 and you can happily accommodate your just-arrived guest.

But suppose all your rooms are occupied and an infinite number of buses pulls up with an infinite number of passengers, each with a reservation. What to do, what to do? Don’t fret and by no means turn them away. Instead, invite your many arrivals to a drink at the bar while you arrange for your present guests to shift to a room twice as great – an act that will free up all the odd-numbered rooms, of which there are an infinite number. To avoid a real-time experiment in chaos theory, you will need your entire fleet of bell hops not just to assist with the Great Room Shift but also to assist with the math because not everyone can mentally multiply thousand-digit numbers by two.

The main thing in running a hotel like this is to be responsive – *anything* can happen and, as unpredictable situations arise, you will need to think creatively so solve them. You want to be known as the place where guests are truly taken care of – if that means chocolate baskets on the pillow upon arrival, two-for-one cosmo specials in the atrium and having bell hops who can recite epic poetry to entertain guests, so be it. Don’t forget rate cuts (with so many rooms, you probably don’t need to charge much more than a dollar per room in the first place) during the low season and promotions to draw tourists at particular times of year. Run a tight ship, keep costs down and who knows? Perhaps one day you can expand your operation and open up another, larger hotel across the street.