99 Bottles of Beer

"99 Bottles of Beer" or "100 Bottles of Pop on the Wall" is a traditional reverse counting song from the United States and Canada. It is popular to sing on road trips, as it has a very repetitive format which is easy to memorize and can take a long time when sung in full. In particular, the song is often sung by children on long school bus trips, such as class field trips, or on Scout or Girl Guide outings.

Lyrics
The song's lyrics are as follows, beginning with n=99: (n) bottles of beer on the wall. (n) bottles of beer. Take one down, pass it around, (n-1) bottles of beer on the wall.

The same verse is repeated, each time with one bottle fewer, until there is none left. Variations on the last verse following the last bottle going down include lines such as: No more bottles of beer on the wall, no more bottles of beer. Go to the store and buy some more, 99 bottles of beer on the wall...

Or: No more bottles of beer on the wall, no more bottles of beer. We've taken them down and passed them around; now we're drunk and passed out!

Other alternate lines read: If that one bottle should happen to fall, what a waste of alcohol!

Or: No more bottles of beer on the wall, no more bottles of beer. There's nothing else to fall, because there's no more bottles of beer on the wall.

Or the song does not stop at the last "1" or "0" bottles of beer but continues counting with &minus;1 (Negative one) Bottles of beer on the wall Take one down, pass it around, &minus;2 (negative 2) bottles of beer on the wall... continuing onward through the negative numbers.

Full-length recitals
Singing all verses takes an extraordinarily long time. The American comedian Andy Kaufman used this for comedic effect early in his career when he actually sang all 100 verses.

Atticus, a band from Knoxville, Tennessee, recorded a thirteen and a half minute live version of the song in its entirety at the Glasgow Cathouse in Scotland. It was included in the 2001 album Figment. Rich Stewart aka Homebrew Stew listed it as the number one drinking song out of 86 in an article for Modern Drunkard Magazine the following year.

Mathematically inspired variants
Donald Byrd has collected dozens of variants inspired by mathematical concepts and written by himself and others. (A subset of his collection has been published. ) Byrd argues that the collection has pedagogic as well as amusement value. Among his variants are:
 * "Infinity bottles of beer on the wall". If one bottle is taken down, there are still infinite bottles of beer on the wall (thus creating an unending sequence much like "The Song That Never Ends").
 * "Aleph-null bottles of beer on the wall". Aleph-null is the size of the set of all natural numbers, and is the smallest infinity and the only countable one; therefore, even if an infinite aleph-null of bottles fall, the same amount remains.
 * "Aleph-one/two/three/etc. bottles of beer on the wall". Aleph-one, two, three, etc. are uncountable infinite sets, which are larger than countable ones; therefore, if only a countable infinity of bottles fall, an uncountable number remains.

Other versions in Byrd's collection involve concepts including geometric progressions, differentials, Euler's identity, complex numbers, summation notation, the Cantor set, the Fibonacci sequence, and the continuum hypothesis, among others.

References in computer science
The computer scientist Donald Knuth proved that the song has a complexity of $$O(\log N)$$ in his in-joke-article "The Complexity of Songs".

Numerous computer programs exist to output the lyrics to the song. This is analogous to "Hello, World!" programs, with the addition of a loop. As with "Hello, World!", this can be a practice exercise for those studying computer programming, and a demonstration of different programming paradigms dealing with looping constructs and syntactic differences between programming languages within a paradigm.

The program has been written in over 1,500 different programming languages.