A small group of logologists have long been intrigued by the use of all the alphabetical letters in a single sentence. No, I don\u27t mean such examples as PACK MY BOX WITH FIVE DOZEN LIQUOR JUGS, but an uncontrived and natural use of all the letters

Beaman, Ralph G.

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41 160 for an ev,THE PANGRAMMATIC WINDOW It! s a another t found for 6: 2/ RALPH G. BEAMAN Boothwyn, Pennsylvania 2! 2'IA small group of 10gologists have long been intrigued by the use of all the alphabetical letter s in a single sentence. No, I don I t mean " Each gro' such examples as PACK MY BOX WITH FIVE DOZEN LIQUOR JUG·S, York Tin but an uncontrived and natural use of all the letter s. 1972) , (: Saturday These are exceedingly rare. In fact, so much so that one delights ( Februar in finding a sentence having all but one of the letters. This A recent example that caught my eye was the sentence by Lowell send us b Thomas in The Saturday Evening Post for Winter, 1971: The r I was in an Assamese tea garden on a very hot day at about graITlmat three in the afternoon, relaxing after ,a jungle patrol, when I posed to noticed a quick ITlovement in the bushes across the compound. underuse ally call This quote misses only the letter Z. the windc quently d How rare are the s e? How many letter s must one check to find as large ITlost if not all of the letters? ing a Z) . that a pal FrOITl data obtained by Bell Telephone Laboratorie s, I note the even chal percent frequencie s of the least COITlITlon letter s are Z = 0.06, Q = 0.10, J = 0.10, X = 0.16, and K =0.60. These frequencies are ex­ Howr pressed so that the SUITl of all the letter frequencies is 100. fer the nl ston ChUl We see now why Lowell Thomas ITlis sed a Z. On the average, it occurs only 6 tiITles in 10,000 letters, or once in 1667 letters. To COITlpound the prob1eITl, one must also ensure that all the other let­ pen ters are included. I clOE The odds for finding all 26 letters in a string of letters selected } p1aining 1 at randoITl from English text ITlay be calculated by fundaITlental fre­ nUITlber c quency statistics. I letter wi: that the v For a 50- 50 chance of finding theITl all, I calculate a PangraITlITla­ equal to tic Window at about 2000 letter s. Actually, I needed only the letter s using log z, Q, J, and X, for the chance of finding K in a window of 2000 letter s exp( a} b is 99.994 per cent. All other letter s are even more probable; for raised to exaITlp1e, there should be about 250 E' s in a 2000-1etter string. The saITlp1e 0 fairly rare letter X is itself only of ITlinor consequence; the three Q, X and most uncommon letters (Z, Q and J) suggest a window of 1930 letter s obtain by f)y the use of 1 1 t mean 'J !UOR JUG,S, one delights by Lowell at about >1, when I compound. k to find note the • 06, 0 = es are ex­). ~verage, it :ers. To )ther let­9 .selected ~ntal fre­langramma­the letters f 2000 letter s 3-ble; for tring. The le three 1930 lette r s 161 for an even chance of catching them all. It! s one thing to calculate a mathematical probability, and yet another thing to verify it in practice. Here are the windows I have found for the whole alphabet in recent literature: 6300 2600 2500 1000 7000 2600 4000 5250 5500 8000 + 4800 2100 750 1850 1500 2580 750 2500 1100 1500 2430 7500 1475 1550 8400 Each group of five is from article s selected at random from ( 1) New York Times Book Review (January 30, 1972), (2) Fortune (January 1972), (3) New York Time s Magazine (February 6, 1972), (4) The Saturday Evening Post (Winter, 1971), and ( 5) Reader 1 s Digest ( February 1972) • This is a very small sample. Will each reader of Word Ways send us her or his own analysis of the True Pangrammatic Window? The randomness assumptions underlying the calculation of a Pan­grammatic Window can be questioned. Some authors are more dis­posed to using one letter than another (British authors, for example, underuse Z with spellings such as civilisation). Some subjects natur­ally call for more of one letter than another. Even more important, the window width is probably understated because the letter Z is fre­quently doubled (if all Z' s were in pair s in English text, then twice as large a window WOuld be required to have a 50- 50 chance of catch­ing a Z). Looking at the median of the 25 observations, I conclude that a pangrammatic window of 2500, not 2000, is required for an even chance of catching all letter s . How narrow can a specific example be? As a quote to beat, I of­fer the near-pangrammatic excerpt from The Birth of Britain by Win­ston Churchill, which use s all letter s but X in only 84 letter s: ... have enjoyed the romantic tragedy of Queen Eleanor penetrating the protecting maze at Woodstock by •.. I close with a brief note to the mathematically-inclined reader ex­plaining how pangrammatic windows are calculated. Let n denote the number of letter s in the window, and p( j) the probability that the jth letter will be drawn at random from English text. The probability that the window will contain one or more example s of the jth letter is equal to P{ j) = 1 - ( 1 - p( j) ) n. The nth power can be easily obtained using logarithms, or it can be approximated by exp{ -np( J) ) , where exp(a) is defined as the numerical constant e (equal to 2.71828 .. ) raised to the ath power. Finally, one obtains the probability that a s ample of n lette r s will contain one or more of each of the lette r s Z, 0, X and J by the formula P = P( z) P( q) P( x) P( j). Note that one must obtain by trial and error the n corre sponding to a P of O. 50. 

The Pangrammatic Window

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The Pangrammatic Window

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