Autograms:

Self-Enumerating Sentences

This website lists self-enumerating sentences, also known as autograms, in various languages. Some I have found on the web, a few were sent to me, and some I created myself.

The autograms on this website are ordered chronologically within each section, with the exception of non-pangrammatic reflexicons, which are ordered by length (pure and non-pure ones separately).

For a slightly longer overview, see the Wikipedia article on autograms. The essay “ Self-enumerating pangrams: A logological history ” provides detailed information on the history of English and Dutch autograms. Further links can be found throughout this page: all autograms that are not by me are linked to their source.

A chain is a sequence of autograms where each sentence enumerates its successor, with the last sentence enumerating the first one, closing the chain. Depending on the length of the chain, the sentences form pairs, triplets, quartets etc.

A reflexicon is an autogram which consists solely of letters and their counts, omitting any other text such as “This sentence contains”. Like conventional autograms, reflexicons are either pangrammatic or non-pangrammatic. A reflexicon is called pure if it does not contain dummy text in the form of “one #”, which is preferable because such text usually entails multiple solutions (any letter can be used that does not already occur), and that “detracts from their logological elegance” (Lee Sallows).

A pangram is a sentence that contains each letter of the alphabet at least once. A sentence that is both an autogram and a pangram is called pangrammatic autogram or self-enumerating pangram.

An autogram , also called self-enumerating or self-documenting sentence, is a sentence that lists the frequencies of its own letters. The numbers are spelled out so that they contribute to the letter count, and that makes creating an autogram quite difficult: The numbers influence the very frequencies they are documenting.

English

The English alphabet consists of the 26 letters of the basic Latin alphabet.

Non-Pangrammatic Autogram

The world’s first autogram, created by Lee Sallows in 1982:

Only the fool would take trouble to verify that his sentence was composed of ten a’s, three b’s, four c’s, four d’s, forty-six e’s, sixteen f’s, four g’s, thirteen h’s, fifteen i’s, two k’s, nine l’s, four m’s, twenty-five n’s, twenty-four o’s, five p’s, sixteen r’s, forty-one s’s, thirty-seven t’s, ten u’s, eight v’s, eight w’s, four x’s, eleven y’s, twenty-seven commas, twenty-three apostrophes, seven hyphens and, last but not least, a single !

Pangrammatic Autograms

In order to find self-enumerating pangrams, Lee Sallows built an electronic “pangram machine”. This is the first pangram it found:

This pangram lists four a’s, one b, one c, two d’s, twenty-nine e’s, eight f’s, three g’s, five h’s, eleven i’s, one j, one k, three l’s, two m’s, twenty-two n’s, fifteen o’s, two p’s, one q, seven r’s, twenty-six s’s, nineteen t’s, four u’s, five v’s, nine w’s, two x’s, four y’s, and one z.

The following pangram lists the total letter count as well:

This pangram contains two hundred nineteen letters: five a’s, one b, two c’s, four d’s, thirty-one e’s, eight f’s, three g’s, six h’s, fourteen i’s, one j, one k, two l’s, two m’s, twenty-six n’s, seventeen o’s, two p’s, one q, ten r’s, twenty-nine s’s, twenty-four t’s, six u’s, five v’s, nine w’s, four x’s, five y’s, and one z.

Non-Pangrammatic Reflexicons

139 characters, 14 distinct letters, not pure

Sixteen e’s, six f’s, one g, three h’s, nine i’s, nine n’s, five o’s, five r’s, sixteen s’s, five t’s, three u’s, four v’s, one w, four x’s

147 characters, 14 distinct letters, pure

Fifteen e’s, seven f’s, four g’s, six h’s, eight i’s, four n’s, five o’s, six r’s, eighteen s’s, eight t’s, four u’s, three v’s, two w’s, three x’s

146 characters, 14 distinct letters, pure

Sixteen e’s, five f’s, three g’s, six h’s, nine i’s, five n’s, four o’s, six r’s, eighteen s’s, eight t’s, three u’s, three v’s, two w’s, four x’s

Pangrammatic Reflexicon

The following reflexicon is not quite satisfactory since it does not use the correct plurals:

One a, one b, one c, one d, twenty-eight e, seven f, five g, five h, eight i, one j, one k, one l, one m, eighteen n, eighteen o, one p, one q, four r, two s, ten t, four u, five v, four w, one x, two y, one z

Chains

The right-hand sentence contains four a’s, one b, three c’s, three d’s, thirty-nine e’s, ten f’s, one g, eight h’s, eight i’s, one j, one k, four l’s, one m, twenty-three n’s, fifteen o’s, one p, one q, nine r’s, twenty-three s’s, twenty-one t’s, four u’s, seven v’s, six w’s, two x’s, five y’s, and one z. The left-hand sentence contains four a’s, one b, three c’s, three d’s, thirty-five e’s, seven f’s, four g’s, eleven h’s, eleven i’s, one j, one k, one l, one m, twenty-six n’s, fifteen o’s, one p, one q, ten r’s, twenty-three s’s, twenty-two t’s, four u’s, three v’s, five w’s, two x’s, five y’s, and one z.

This is a chain of three linked pangrams:

The second sentence contains three a’s, one b, three c’s, three d’s, thirty-four e’s, four f’s, one g, nine h’s, eleven i’s, one j, one k, one l, one m, twenty-four n’s, fourteen o’s, one p, one q, nine r’s, twenty-six s’s, twenty-four t’s, two u’s, five v’s, seven w’s, three x’s, five y’s and one z.

The third sentence contains three a’s, one b, three c’s, two d’s, thirty-nine e’s, six f’s, five g’s, ten h’s, thirteen i’s, one j, one k, two l’s, one m, twenty-one n’s, thirteen o’s, one p, one q, seven r’s, twenty-six s’s, twenty t’s, two u’s, seven v’s, five w’s, three x’s, four y’s and one z.

The first sentence contains three a’s, one b, four c’s, three d’s, thirty-seven e’s, eight f’s, one g, seven h’s, eight i’s, one j, one k, two l’s, one m, twenty-six n’s, eighteen o’s, one p, one q, eleven r’s, twenty-three s’s, eighteen t’s, six u’s, five v’s, five w’s, two x’s, five y’s and one z.

Matthias Belz, 2014

And here are four pangrams, linked in clockwise direction:

The sentence to the right contains three a’s, two b’s, three c’s, two d’s, thirty-eight e’s, seven f’s, three g’s, nine h’s, eleven i’s, one j, one k, two l’s, one m, twenty-three n’s, fifteen o’s, one p, one q, eight r’s, twenty-four s’s, twenty-two t’s, three u’s, seven v’s, eight w’s, one x, five y’s and one z. The sentence below contains three a’s, one b, three c’s, two d’s, thirty-nine e’s, four f’s, two g’s, nine h’s, eight i’s, one j, one k, three l’s, one m, twenty-four n’s, seventeen o’s, one p, one q, seven r’s, twenty-five s’s, twenty-five t’s, two u’s, five v’s, eight w’s, three x’s, five y’s and one z. The sentence above contains three a’s, one b, three c’s, two d’s, forty e’s, five f’s, five g’s, thirteen h’s, ten i’s, one j, one k, two l’s, one m, twenty-two n’s, fifteen o’s, one p, one q, nine r’s, twenty-four s’s, twenty-eight t’s, two u’s, five v’s, eight w’s, one x, five y’s and one z. The sentence to the left contains four a’s, two b’s, three c’s, two d’s, thirty-three e’s, nine f’s, three g’s, seven h’s, eleven i’s, one j, one k, one l, one m, twenty-two n’s, seventeen o’s, one p, one q, six r’s, twenty-one s’s, twenty-three t’s, two u’s, six v’s, eight w’s, one x, five y’s and one z.

Matthias Belz, 2014

In his memorable 1985 essay, “ In Quest of a Pangram ”, Lee Sallows presented a series of 25 numbered self-contained autograms , each one using a different verb. With today’s computing power (and a smart algorithm) it is possible to create a series of 25 linked autograms. The following chain uses the same verbs as Sallows in the exact same order:

The second pangram has five a’s, one b, one c, three d’s, twenty-eight e’s, six f’s, four g’s, nine h’s, eleven i’s, one j, one k, two l’s, two m’s, eighteen n’s, fifteen o’s, two p’s, one q, ten r’s, twenty-six s’s, twenty-four t’s, four u’s, four v’s, eight w’s, three x’s, four y’s and one z.

The third pangram totals five a’s, one b, two c’s, two d’s, thirty-six e’s, five f’s, four g’s, seven h’s, ten i’s, one j, one k, two l’s, two m’s, twenty-four n’s, fourteen o’s, two p’s, one q, six r’s, twenty-nine s’s, nineteen t’s, three u’s, eight v’s, eight w’s, three x’s, three y’s and one z.

The fourth pangram contains four a’s, two b’s, one c, two d’s, twenty-seven e’s, nine f’s, two g’s, six h’s, seven i’s, one j, one k, two l’s, three m’s, nineteen n’s, sixteen o’s, two p’s, one q, eleven r’s, twenty-seven s’s, eighteen t’s, seven u’s, five v’s, eight w’s, three x’s, five y’s and one z.

The fifth pangram numbers five a’s, two b’s, two c’s, two d’s, thirty e’s, six f’s, four g’s, seven h’s, eleven i’s, one j, one k, one l, three m’s, twenty n’s, fourteen o’s, two p’s, one q, seven r’s, twenty-six s’s, twenty-one t’s, three u’s, four v’s, nine w’s, four x’s, four y’s and one z.

The sixth pangram embraces five a’s, two b’s, one c, two d’s, twenty-three e’s, six f’s, four g’s, eight h’s, nine i’s, one j, one k, two l’s, two m’s, eighteen n’s, nineteen o’s, two p’s, one q, ten r’s, twenty-five s’s, twenty-one t’s, six u’s, four v’s, ten w’s, three x’s, five y’s and one z.

The seventh pangram harbours four a’s, one b, two c’s, two d’s, thirty-two e’s, eight f’s, four g’s, nine h’s, eleven i’s, one j, one k, one l, two m’s, twenty-one n’s, thirteen o’s, two p’s, one q, eight r’s, twenty-five s’s, twenty t’s, four u’s, six v’s, six w’s, two x’s, four y’s and one z.

The eighth pangram counts five a’s, one b, one c, two d’s, thirty e’s, eight f’s, three g’s, five h’s, ten i’s, one j, one k, four l’s, two m’s, twenty-three n’s, fifteen o’s, two p’s, one q, seven r’s, twenty-seven s’s, nineteen t’s, five u’s, six v’s, nine w’s, three x’s, four y’s and one z.

The ninth pangram tallies four a’s, one b, one c, two d’s, twenty-nine e’s, eight f’s, six g’s, nine h’s, twelve i’s, one j, one k, two l’s, two m’s, seventeen n’s, fifteen o’s, three p’s, one q, seven r’s, twenty-six s’s, twenty-two t’s, four u’s, five v’s, seven w’s, four x’s, four y’s and one z.

The tenth pangram exploits five a’s, one b, one c, two d’s, thirty-five e’s, eight f’s, four g’s, six h’s, seven i’s, one j, one k, four l’s, two m’s, eighteen n’s, fifteen o’s, two p’s, one q, eight r’s, twenty-seven s’s, twenty-two t’s, six u’s, eight v’s, ten w’s, three x’s, four y’s and one z.

The eleventh pangram features four a’s, one b, one c, two d’s, twenty-six e’s, eight f’s, three g’s, eight h’s, eleven i’s, one j, one k, four l’s, two m’s, seventeen n’s, seventeen o’s, two p’s, one q, ten r’s, twenty-four s’s, twenty-five t’s, six u’s, four v’s, twelve w’s, two x’s, five y’s and two z’s.

The twelfth pangram utilizes five a’s, two b’s, one c, two d’s, thirty-one e’s, four f’s, four g’s, nine h’s, twelve i’s, one j, one k, two l’s, two m’s, twenty-three n’s, thirteen o’s, two p’s, one q, eight r’s, twenty-six s’s, twenty-five t’s, three u’s, two v’s, nine w’s, four x’s, four y’s and one z.

The thirteenth pangram tables four a’s, one b, two c’s, three d’s, twenty-six e’s, nine f’s, three g’s, six h’s, eight i’s, one j, one k, two l’s, two m’s, nineteen n’s, nineteen o’s, two p’s, one q, ten r’s, twenty-six s’s, twenty-two t’s, eight u’s, four v’s, nine w’s, three x’s, five y’s and one z.

The fourteenth pangram includes four a’s, one b, two c’s, two d’s, thirty-five e’s, six f’s, three g’s, eight h’s, ten i’s, one j, one k, two l’s, two m’s, twenty-one n’s, fourteen o’s, two p’s, one q, ten r’s, twenty-six s’s, twenty-one t’s, four u’s, five v’s, seven w’s, four x’s, four y’s and one z.

The fifteenth pangram recruits four a’s, one b, one c, two d’s, thirty-one e’s, seven f’s, three g’s, ten h’s, nine i’s, one j, one k, one l, two m’s, eighteen n’s, sixteen o’s, two p’s, one q, twelve r’s, twenty-six s’s, twenty-three t’s, six u’s, four v’s, seven w’s, three x’s, five y’s and one z.

The sixteenth pangram uses four a’s, two b’s, one c, two d’s, thirty-four e’s, five f’s, three g’s, eight h’s, nine i’s, one j, one k, one l, three m’s, twenty-two n’s, fourteen o’s, two p’s, one q, seven r’s, thirty-one s’s, twenty-three t’s, four u’s, six v’s, ten w’s, three x’s, five y’s and one z.

The seventeenth pangram subsumes six a’s, two b’s, one c, two d’s, thirty-three e’s, eight f’s, three g’s, seven h’s, nine i’s, one j, one k, two l’s, two m’s, twenty-one n’s, fifteen o’s, two p’s, one q, seven r’s, twenty-six s’s, twenty-three t’s, four u’s, seven v’s, nine w’s, two x’s, five y’s and one z.

The eighteenth pangram tabulates five a’s, one b, one c, two d’s, thirty-one e’s, seven f’s, two g’s, six h’s, ten i’s, one j, one k, two l’s, three m’s, twenty-two n’s, fifteen o’s, two p’s, one q, seven r’s, twenty-seven s’s, twenty-four t’s, three u’s, five v’s, nine w’s, four x’s, five y’s and one z.

The nineteenth pangram manifests five a’s, two b’s, one c, two d’s, thirty-one e’s, five f’s, four g’s, ten h’s, eleven i’s, one j, one k, two l’s, three m’s, twenty-one n’s, sixteen o’s, two p’s, one q, ten r’s, twenty-six s’s, twenty-six t’s, four u’s, three v’s, ten w’s, two x’s, five y’s and one z.

The twentieth pangram assembles four a’s, one b, one c, two d’s, thirty-two e’s, nine f’s, three g’s, seven h’s, thirteen i’s, one j, one k, two l’s, four m’s, twenty-two n’s, thirteen o’s, two p’s, one q, nine r’s, twenty-eight s’s, twenty-one t’s, four u’s, six v’s, eight w’s, three x’s, five y’s and one z.

The twenty-first pangram summons four a’s, one b, two c’s, three d’s, thirty-six e’s, six f’s, three g’s, nine h’s, nine i’s, one j, one k, two l’s, two m’s, twenty-four n’s, fifteen o’s, two p’s, one q, eleven r’s, twenty-eight s’s, nineteen t’s, five u’s, five v’s, seven w’s, three x’s, five y’s and one z.

The twenty-second pangram shows five a’s, one b, one c, four d’s, thirty-four e’s, six f’s, four g’s, nine h’s, eleven i’s, one j, one k, three l’s, two m’s, nineteen n’s, fourteen o’s, three p’s, one q, nine r’s, twenty-six s’s, twenty-two t’s, three u’s, seven v’s, eight w’s, three x’s, seven y’s and one z.

The twenty-third pangram displays four a’s, one b, two c’s, three d’s, thirty-one e’s, seven f’s, three g’s, seven h’s, eight i’s, one j, one k, one l, two m’s, twenty-two n’s, sixteen o’s, three p’s, one q, eight r’s, twenty-five s’s, twenty-six t’s, four u’s, five v’s, eleven w’s, one x, five y’s and one z.

The twenty-fourth pangram produces four a’s, one b, two c’s, two d’s, thirty-three e’s, ten f’s, three g’s, eight h’s, nine i’s, one j, one k, two l’s, two m’s, nineteen n’s, fifteen o’s, two p’s, one q, ten r’s, twenty-seven s’s, twenty-two t’s, five u’s, seven v’s, ten w’s, two x’s, five y’s and one z.

The twenty-fifth pangram evinces four a’s, one b, two c’s, three d’s, thirty-four e’s, six f’s, four g’s, seven h’s, fifteen i’s, one j, one k, two l’s, two m’s, twenty-four n’s, twelve o’s, two p’s, one q, seven r’s, twenty-nine s’s, eighteen t’s, two u’s, five v’s, seven w’s, three x’s, three y’s and one z.

The first pangram discloses five a’s, one b, two c’s, three d’s, twenty-nine e’s, nine f’s, five g’s, eight h’s, nine i’s, one j, one k, two l’s, two m’s, nineteen n’s, sixteen o’s, two p’s, one q, nine r’s, twenty-five s’s, eighteen t’s, six u’s, three v’s, seven w’s, three x’s, four y’s and one z.

Matthias Belz, 2015

Percentages

A variant suggested by Lee Sallows and first publicly mentioned by Chris Patuzzo is to use percentages instead of absolute counts. When I read that this problem had never been solved, I gave it a try and found the following pangram:

Rounded to one decimal place, four point three percent of the letters in this sentence are a’s, zero point one percent are b’s, four point three percent are c’s, zero point seven percent are d’s, twenty point six percent are e’s, one point one percent are f’s, zero point six percent are g’s, one point four percent are h’s, five point seven percent are i’s, zero point one percent are j’s, zero point one percent are k’s, zero point seven percent are l’s, zero point three percent are m’s, eleven point six percent are n’s, eight point two percent are o’s, seven point seven percent are p’s, zero point one percent are q’s, ten point seven percent are r’s, five point eight percent are s’s, ten point one percent are t’s, zero point nine percent are u’s, one point six percent are v’s, zero point four percent are w’s, zero point seven percent are x’s, zero point three percent are y’s and one point eight percent are z’s.

Matthias Belz, 2015

At around the same time (in late 2015), Chris Patuzzo found a similar pangram , also using one decimal place. Shortly after that pangram was made public, Josh Bevan published one that uses two decimal places, followed by a pangram using three:

This sentence is dedicated to Chris Patuzzo/Lee Sallows and to within three decimal places three point five three eight percent of the letters in this sentence are a’s, zero point one zero four percent are b’s, three point four three four percent are c’s, zero point seven two eight percent are d’s, twenty point one eight seven percent are e’s, two point zero eight one percent are f’s, zero point nine three seven percent are g’s, two point four nine seven percent are h’s, six point five five six percent are i’s, zero point one zero four percent are j’s, zero point one zero four percent are k’s, zero point seven two eight percent are l’s, zero point two zero eight percent are m’s, ten point one nine eight percent are n’s, nine point zero five three percent are o’s, five point seven two three percent are p’s, zero point one zero four percent are q’s, ten point zero nine four percent are r’s, five point two zero three percent are s’s, ten point three zero two percent are t’s, one point one four five percent are u’s, one point nine seven seven percent are v’s, one point five six one percent are w’s, zero point five two zero percent are x’s, zero point two zero eight percent are y’s, and two point seven zero six percent are z’s.

I only learned about this about two years later, and it rekindled my interest in this kind of autogram. After devising a new approach, I managed to find pangrams with up to six decimal places. Here’s an example:

Rounded to five decimal places, two point six five two five two percent of the letters of this sentence are a’s, zero point zero eight eight four two percent are b’s, two point six five two five two percent are c’s, zero point four four two zero nine percent are d’s, nineteen point eight zero five four eight percent are e’s, three point four four eight two eight percent are f’s, one point seven six eight three five percent are g’s, two point nine one seven seven seven percent are h’s, seven point eight six nine one four percent are i’s, zero point zero eight eight four two percent are j’s, zero point zero eight eight four two percent are k’s, zero point three five three six seven percent are l’s, zero point one seven six eight three percent are m’s, ten point two five six four one percent are n’s, eight point nine three zero one five percent are o’s, four point seven seven four five four percent are p’s, zero point zero eight eight four two percent are q’s, nine point five four nine zero seven percent are r’s, four point nine five one three seven percent are s’s, nine point six three seven four nine percent are t’s, two point zero three three six zero percent are u’s, two point seven four zero nine four percent are v’s, one point six seven nine nine three percent are w’s, zero point nine seven two five nine percent are x’s, zero point zero eight eight four two percent are y’s and one point nine four five one eight percent are z’s.

Matthias Belz, 2017

Nevertheless, I had the feeling that using rounded values instead of exact values is a flaw that makes these autograms less perfect than the regular ones. I wondered if it were possible to create autograms that use exact percentages, and it turned out that it is:

Exactly three point eight seven five percent of the letters of this autogram are a’s, zero point one two five percent are b’s, three point five percent are c’s, zero point two five percent are d’s, twenty-one point two five percent are e’s, three point seven five percent are f’s, zero point three seven five percent are g’s, one point five percent are h’s, seven point two five percent are i’s, zero point one two five percent are j’s, zero point one two five percent are k’s, zero point three seven five percent are l’s, zero point two five percent are m’s, nine point seven five percent are n’s, seven point five percent are o’s, six point five percent are p’s, zero point one two five percent are q’s, nine point three seven five percent are r’s, five point one two five percent are s’s, ten percent are t’s, zero point three seven five percent are u’s, four point six two five percent are v’s, one point five percent are w’s, zero point five percent are x’s, zero point three seven five percent are y’s and one point five percent are z’s.

Matthias Belz, 2017

The following is an autogram that is not a pangram and thus shorter, and it also uses exact percentages:

This self-enumerating sentence is composed of exactly zero point eight percent a’s, five point two percent c’s, zero point six percent d’s, seventeen percent e’s, one point eight percent f’s, one point two percent g’s, one point two percent h’s, seven point two percent i’s, one percent l’s, zero point six percent m’s, twelve point six percent n’s, nine point two percent o’s, eight point six percent p’s, six point six percent r’s, seven point six percent s’s, eleven point four percent t’s, one point four percent u’s, one point four percent v’s, one point four percent w’s, one point eight percent x’s, zero point four percent y’s and one percent z’s.

Matthias Belz, 2017