math

HOW TO MANAGE A HOTEL WITH AN INFINITE NUMBER OF ROOMS

By | archive, creative, math

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 1012 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 1023. 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 10100 (“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.

About Daniel Hudon

Daniel Hudon, originally from Canada, teaches writing, math, physics and astronomy in Boston. He has published a chapbook, Evidence for Rainfall (Pen and Anvil Press), a popular nonfiction book, The Bluffer’s Guide to the Cosmos (Oval Books) and has a travel manuscript, Traveling into Now, that is looking for a home. He has work coming up or appearing in The Chatttahoochee Review, {Ex}tinguished and {Ex}tinct: An Anthology of Things that No Longer {Ex}ist, Written River and The Little Patuxent Review. He blogs about environmental topics at econowblog.blogspot.com and some of his writing links can be found at people.bu.edu/hudon.

THE ABRAMS’ STORMTROOPER AXIOM

By | archive, creative, humour, math

It (hypothetically*) goes like this:

stormtrooperequation

* Like all good science, this needs some testing…

- – -

When news hit that Disney bought the rights to Star Wars, and that J.J. Abrams would be manning the first movie of a new trilogy, my inner geek went into giddy overdrive. This was because it gave me a chance to revisited my bucket list, which had previously scratched off “be an extra in a Star Wars movie” as something that was unattainable having presumed the prequels were my last chance. But now, there is (literally), A NEW HOPE. Even better, is the fact that my kids are old enough to also want this.

And so, being a science-y sort and all, I figured the first step would be to actually try and come up with a way to calculate the odds of such a thing happening, and hence you see the above – or what I have termed the Abrams’ Stormtrooper Axiom. In effect, this is an equation that aims to calculates the odds of you (or anyone) being cast as a stormtrooper in one of these new movies1.

Here’s how it works. We’ll first look at (1) which expresses the equation in its most obvious form.

stormtrooperequation01

When you look at this equation, there are three main components: two in the numerator: WkSblaster and bmiopthopt

And one in the denominator: 5.4^(1+bop+bow).

The denominator is an expression designed to address the likelihood of being cast, as having a dependence on the individual’s chance of contact with J.J. Abrams. Specifically, bop refers to the degrees of personal separation the individual is from the Director, whereas bow refers to the degrees of internet separation the individual is from the Director. The base of the exponential relationship is, of course, the standard May The Force Be With You Constant (or 5.4).

All told, if you have very little connection to the director, your odds can dwindle significantly, about 5.4(1+6+6) times, or roughly one in 3.3 billion! It also infers that even if you know JJ Abrams personally, it does not guarantee being cast – mathematically, the closest association would still work out to 5.4(1+1+1), or roughly a chance of one in 158. This is because there are other factors that need to come into play when determining whether an individual is right for a stormtrooper part.

Which is where the numerator expressions exert their influence. We can first begin with the bmiopthopt element, which essentially considers the physicality of the individual vying for a stormtrooper part. The bmi portion considers body shape, whereas the h portion considers height.

Each element can be further derived as:

stormtrooperequation02

Where (2) calculates divergence from an average body type (as expressed by an individual’s body mass index with m equals to the individual’s weight in kilograms, and h is equal to the individual’s height in metres). You’ll note that the more you veer away from an “average” body type, the greater the modification of the bmiopt number to a number less than one (and therefore further lowering your odds).

In the same manner, (3) calculates divergence from an optimal height (deemed 1.8 metres as determined from casual examination of Star Wars’ trivia – i.e. calculating Mark Hamill‘s height and noting the “Aren’t you a little short to be a Stormtrooper?” comment). Like the BMI calculation, the more you deviate from the optimal height, the greater the modification of the hopt number to a number less than one (and therefore further lowering your odds).

Note that both (2) and (3) are included in the overall equation for pragmatic prop design reasons (not every extra can have a custom made set of armour, so it makes sense if casting aimed for similar body types). Then, of course, there is the whole clone army narrative which might also presume the troops having similar physical features. (Also note that in case you weren’t familiar with the symbol, the straight up and down lines enclose a value where you only use the absolute number – i.e. remove the plus or minus sign).

Anyway, when you put it all together you get the expression (4).

stormtrooperequation04

Which only leaves Wk and Sblaster to be defined. Here, these two variables relate to two specific personality traits that are deemed important for the stormtrooper casting decision.

For instance, I don’t think I’m the only Star Wars fan who notices the incredibly poor marksmanship exhibited by the stormtroopers. There are many instances in the movies where there are many of them (with their weapons – presumably high tech in nature), in close proximity to the target, and yet, they still always fail to hit their target.

Given this observation, I’m left to assume that Stormtroopers, as a whole, have a deep distrust of guns, and with that discomfort tend to misfire (perhaps subconsciously). This also leads me to hypothesize that not only are they not very skilled, but that they are probably the sort that are not at all familiar with gun culture in their private lives.

Consequently, Sblaster is a number assigned to measure the individual’s relative experience wth guns, whereby a value of 1.0 represents full disconnect from the use of guns in their personal lives, and a number closer to zero represents an individual who is very familiar with gun culture.

Of course, perhaps the most important tangible characteristic (that could translate to a positive casting decision) is relative fandom itself. In other words, casting may be partly governed by how “into Star Wars” an individual is. Here, and in honor of Chewbacca’s reference of “pulling arms out of their sockets when they lose,” I’ve decided to use Wookie knowledge, or Wk as an indicator that can further increase casting chances. Essentially, this is a scale that ranges from 1 to 10, whereby 10 represents fanatical knowledge on all things Wookie, and 1 represents no knowledge at all. In effect, if you’re nuts about Star Wars (and wookies specifically), you can increase your chances of being cast by 10 fold.

In conclusion, I want to stress that this is the Abrams’ Stormtrooper Axiom, and by its very definition, an axiom is just a starting point. This means the equation will need more work, and it would be great suggestions to make it better. As it stands, it works as a general guideline using a number of test values2. As well, there is also the very real caveat of whether J.J. Abrams will even have stormtroopers in the new movies – never mind the fact that if he does, they may come in a different size, or be better at shooting, etc. In some respects, this reminds me a little of Schrödinger’s cat (we can call our version Abrams’ Stormtrooper): we won’t really know what he has in mind until he lets us open the box.

- – -

Footnotes

1. In general, I’ve used information from the original trilogy for points of reference.

2. For instance, an individual with no connection at all will result in a number that works against the backdrop of the total human population numbers. For J.J. Abrams, himself, where bop and bow are equal to zero, and his Wk is likely quite high, the equation would further calculate that he has practically perfect odds of being cast as a stormtrooper (which makes sense given his role in the movie). For the sake of comparison, I’ve calculated my own odds to be approximately: 0.00000519 or about one in 19,000.

About David Ng

David (@ng_dave) is Faculty at the Michael Smith Labs. His writing has appeared in places such as McSweeney's, The Walrus, and also as an occasional blogger at boingboing.net. If you're looking for a graphic for your next science talk, he encourages you to check out his blog, popperfont.net.

UP OR DOWN? AN EFFICIENCY-BASED ARGUMENT FOR OPTIMAL TOILET SEAT PLACEMENT

By | archive, journal club, math

- – -

UP OR DOWN?

As long as males and females have been living with indoor plumbing they have argued over the placement of the toilet seat. Most often, males leave the toilet seat up and females leave the toilet seat down. Males may not necessarily have a problem with the toilet seat down, but then females will suffer from wet bottom syndrome from time to time. If males leave the seat up, females may “fall” into the toilet particularly at night. A solution to this problem is to always leave the toilet seat in a particular position: the toilet seat remains down and males must lift the toilet seat to urinate, but then return it to its down position; alternatively, the toilet seat remains up and females must always place the seat down to use the toilet and return it to its upright position when done.

The trouble with this solution is: in which position should the toilet seat be placed? This decision has, no doubt, been the source of many arguments in male-female households. Previous scientific research has been undertaken on this household problem. Choi (2002) uses an optimization method to identify the efficient placement of the toilet seat. He finds that unless the costs of changing the toilet seat position are asymmetric across the parties involved, the optimal toilet seat placement follows the selfish rule: do not change the toilet seat position when you are finished using the toilet. Harter (2005) and Siddiqi (2006) both use game theoretic models to show that optimal toilet seat placement is up. However, Harter (2005) does note that in order to avoid marital conflict the toilet seat may best be in the down position.

In this paper, an efficiency-based argument is used to show which position the toilet seat should be in, depending on the composition of the household. This is done through a mathematical modeling approach that extends previous research by considering households with more than one male and more than one female. Because it takes effort to raise and lower the toilet seat, the toilet seat should be left in the position that minimizes the number of times it must be moved. It is shown that the optimal toilet seat placement depends on the ratio of males to females.

METHODOLOGY

In order to determine the optimal toilet seat placement, a mathematical modeling approach is taken. In order to perform such modeling, a number of axioms must be made.

Axiom 1: Females always use the toilet with the seat in the down position.

Axiom 2: Males urinate with the toilet seat up in the up position and defecate with the toilet seat in the down position.

Axiom 3: Males and females defecate once per day and urinate 7 times per day.

Axiom 3 is clearly critical for the results, but in a sensitivity analysis the results presented below were shown to be robust. Considering these axioms, the toilet seat ratio is calculated as follows:

This ratio is bounded by zero and unity. If the TSR is greater than 0.50, the optimal toilet seat placement is up; if the TSR is less than 0.50, the optimal toilet seat placement is down; and if the TSR is equal to 0.50 the optimal placement is either up or down. The TSR is calculated for all combinations of 1 – 5 males and 0 – 6 females.

RESULTS

The results of the mathematical modeling are shown in Figure 1 and Table 1. Figure 1 also shows the 0.50 mark (grey line) and all TSR values greater than 0.50 in Table 1 are in bold.

Figure 1. Toilet Seat Ratio

Table 1. Toilet Seat Ratios, ad nauseam

The first point to notice in this analysis is that the claims of previous research have not been replicated here: when there is one female and one male in the household the optimal toilet seat placement is down. However, as evident in Table 1, all hope for having the toilet seat in the up position is not lost for males.

Overall, the general results clearly show that if the number of females is greater than or equal to the number of males the optimal placement of the toilet seat is down. Additionally, when males outnumber females, the optimal toilet seat placement is not always up: when there are four males in a household, the optimal toilet seat placement is only definitively up when there are two or fewer females; and when there are five males in a household, the optimal toilet seat placement is only definitively up when there are three or fewer females.

CONCLUSION

Through the use of mathematical modeling, the analysis in this paper has shown that the optimal placement of the toilet seat can be calculated based on the number of males relative to the number of females. The general result (that is not sensitive to reasonable changes in Axiom 3) is that when the number of females in a household is greater than or equal to the number of males the optimal placement of the toilet seat is down. Therefore, there is no longer any need for males and females to argue over the placement of their toilet seat as long as they are concerned with the efficient expenditure of household energy.

REFERENCES

Choi, J.P. (2002). Up or down? A male economist’s manifesto on the toilet seat etiquette. Department of Economics, Michigan State University Working Paper.

Harter, R. (2005). A game theoretic approach to the toilet seat problem. The Science Creative Quarterly.

Siddiqi, H. (2006). The social norm of leaving the toilet seat down: a game theoretic analysis. MPRA Paper No. 856.

About Martin A. Andresen

Martin A. Andresen is a Star Wars and Battlestar Galactica fan. Though at times confused for an academic, Prof. Andresen spends much of his time playing video games, watching television, and playing with trains and race cars. He still wants to be a Jedi when he grows up.

WRITE PRIME

By | archive, creative, math

999999999999999999999999999999999999999999999999999999
900000000000000000000000000000000000000000000000000009
900000000000000000000000000000000000000000000000000009
900088000088008888880000888888008888888800888888880009
900088000088008800088800008800000008800000880000000009
900088000088008800088800008800000008800000880000000009
900088000088008800880000008800000008800000888888800009
900088088088008888800000008800000008800000880000000009
900088088088008800880000008800000008800000880000000009
900008800880008800088000008800000008800000880000000009
900008800880008800008800888888000008800000888888880009
900000000000000000000000000000000000000000000000000009
900000000000000000000000000000000000000000000000000009
999999999999999999999999999999999999999999999999999999
*10^375+1

- – -

I obtain much pleasure from writing.

Also from reading books.

Literature is the greatest art form in the world.

And so I imagine a writer living in a small efficiency apartment in the southern part of the United States.

A third-floor apartment, 33 steps up black cast iron stairs in a back alley.

The apartment has a main bed/living-room and a chair, a small kitchen, a tiny refrigerator, a stove, a small bathroom.

No television, no cell phone, no computer.

But the writer has a typewriter and an old radio setting on a small table with a lamp next to it.

He has a few tattered books, some clothes, an abstract painting on the wall, not much else.

The man has loved literature since he was a small boy.

Escaping into stories and novels and picture books.

But he is also a man of action, not a wimp.

He lifts weights regularly and occasionally gets into brutal fist fights in crappy bars.

The man has been writing ever since the age of 11; and found his “literary voice” at a young age.

Every day the man sits at his typewriter and writes for hours – brilliant short stories and minimal poems come reeling off his typewriter.

There is a primitive beauty to the prose in his stories, a roughness to the visage of his piercing minimal poems, all of it fresh and compelling and packed with genuine emotion, which showcases his unique perspective of the world.
He sends his work out to magazines and journals once a week by regular mail.

And soon he finds that the literary game is NOT RIGGED.

Editors and other writers actually RECOGNIZE and appreciate his literary genius.

They publish his poems and stories in their magazines and quickly he gains the respect and attraction of literary agents and five major NY publishers.

Three small books come out in rapid succession: a novella, a book of short stories, a book of minimal poems. They gain mostly positive reviews in the best periodicals in spite of the revolutionary way the young writer has re-evaluated and almost “attacked” modern literature.

The writer receives large advances and generous royalty checks and moves out of his tiny efficiency apartment.

He gets married and buys a large house and an expensive sports car and continues writing every single day.

And the writer does not become an alcoholic, he does not develop a horrible drug habit, he does not commit suicide.

He only works harder at his writing and fully nurtures his devastating talent.

One summer he decides to buy an old laptop computer equipped with only a primitive word processor and a spell check – no other programs.

He writes a 40,000-word brutalist novel in two months that becomes a true classic of southern gothic literature.

And the writer goes on to become one of the greatest authors America has ever produced and wins numerous major awards and becomes a multi-millionaire.

The End.

About Jason Earls

Jason Earls is the author of Mathematical Bliss, the Underground Guitar Handbook, Concrete Primes, Red Zen, How to Become a Guitar Player from Hell, I Sin Every Number, Cocoon of Terror, Heartless Bastard In Ecstasy, and other books.

A SOCIAL IMPOSSIBILITY THEOREM

By | archive, math

For centuries, generations of philosophers and social scientists have studied the sources of income inequality in human society. Finally, a major breakthrough in the form of a mathematical proof of a fundamental truth, long suspected by millions of laboring individuals, and now shown rigorously.

Theorem: It is impossible for engineers, scientists and the professoriate to earn as much money as business executives.

Proof:

Postulate 1: Knowledge is Power.
Postulate 2: Time is Money.

As every engineer knows,

powerequation

Since philosophers and economists have amply demonstrated that Knowledge = Power, and Time = Money, we have via substitution,

eq2

Solving for Money, one obtains:

eq3

and taking limits,

eq4

Result: The less you know, the more you make.

Quod Erat Demonstrandum

About Haynes Goddard

Haynes Goddard is an environmental economist and Professor of Economics at the University of Cincinnati who finds humor in most of life's dimensions. This little theorem has been around for some time, but cried out for more intellectual respectability, i.e., a more formal title and mathematical form.