Definitions

pangrammatic

Pangrammatic window

A pangrammatic window is a stretch of naturally occurring text that contains all the letters in the alphabet.

Shortest Examples

Until recently, the shortest known window was found in Lillie de Hegermann-Lindencrone's 1912 book In the Courts of Memory: "I sang, and thought I sang very well; but he just looked up into my face with a very quizzical expression, and said, 'How long have been singing, Mademoiselle?'" at 56 letters. A 47-letter example has, however, been found on the internet: "JoBlo's movie review of The Yards: Mark Wahlberg, Joaquin Phoenix, Charlize Theron...".

Probability of Occurrence

Generally, according to the law of probability, the shorter the work, the longer the pangrammatic window will be.

Using the frequencies of the letters, it is easy to show this. For a sequence length m, the probability it will contain all 26 letters is P(a)P(b)...P(y)P(z) where P(n)=1-(1-p(n))m. Inputing the letter frequencies, the probability that a 1,700-letter sequence will contain all 26 letters is about 50%. At 1000, there is about a 19.5% chance, and at 2,500, there is about a 73% chance. Technically, the probability of a perfect pangrammatic window, i.e. one 26 letters long, is about 556 billion to one.

For example, the shortest pangrammatic window in Around the World in Eighty Days, by Jules Verne, is 155 letters: "Once a man approached and spoke to him, and the detective merely replied by shaking his head. Thus the night passed. At dawn, the half-extinguished disc of the sun rose above a misty horizon; but it was now possible to recognise objects two miles off. Phileas Fogg and the squad had gone southward; in the south all was still vacancy."

The shortest pangrammatic window in the United States Declaration of Independence, a significantly shorter work, is 592 letters: "We hold these truths to be self-evident, that all men are created equal, that they are endowed, by their Creator, with certain unalienable Rights, that among these are Life, Liberty, and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or abolish it, and to institute new Government, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed."

References

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