Pangrams
A pangram is a phrase or sentence that contains every letter
of the alphabet at least once.
Which alphabet, you ask ? Good question !
In the following, we shall basically restrict ourselves to
the 26 letters in the English language.
But you will also see many examples in Dutch, not only
since that language has a rich history in recreational linguistics, but because it plays a major role in this story.
And in Dutch appears the special ligature `ij' which can
sometimes be considered as a separate letter on its own. (Such as in the three-letter word
`wijf', but not so in the
eight-letter word `minijurk'. You see, Dutch is recreational
by itself. But we digress already.) However, in most of the
Dutch examples we adhere to the English alphabet and count
the `ij' as two letters, unless explicitly stated otherwise.
For the sake of completeness, it must also be noted that in
general we don't differentiate between capital letters and
lowercase letters. The `A' and the `a' are then two instances
of the same letter.
The traditional pangram example is the well-known typewriter
test phrase
The quick brown fox jumps over the lazy dog
One of the pangram word games is to construct the shortest phrase
that is still a pangram. This leads often to amusing yet rather
bizarre solutions, but also to some really fine examples, such as
The five boxing wizards jump quickly
and in Dutch (using the `xyz' alphabet, the `xijz' alphabet, and
the `xyijz' alphabet respectively)
Sexy qua lijf doch bang voor 't zwempak
Och, zwak vormpje blijft exquis ding
Doch 'n exquis gympje zwerft vlakbij
but here we are not really interested in those ones,
and we will not elaborate on them any further.
Self-referencing sentences
A self-referencing sentence makes a true assertion about the
sentence itself.
The following examples illustrate what is meant by this.
This sentence has five words
Deze zin heeft vijf woorden
Here we have twenty-eight letters
Hier zijn achtentwintig letters
Note that the English and the Dutch sentence of the first
example are perfect word-by-word translations of each other. However, it is probably even more remarkable that in the second
example we are able to make a valid (i.e. self-referencing)
translation at all.
Things become a bit complicated when we consider the (supposedly
self-referencing) sentence
This English sentence is difficult to translate into Dutch.
The straightforward translation of this sentence would be
Deze Engelse zin is moeilijk te vertalen in het Nederlands.
Does this Dutch sentence refer to itself ? That cannot be true,
because it would then incorrectly claim to be an English sentence.
So it must refer to the above English sentence. But that would make
that sentence self-contradicting, because it was actually very easy
to make the translation. We are stuck. Therefore a better candidate
for a Dutch translation of the English sentence would be
Deze Nederlandse zin is moeilijk te vertalen in het Engels.
It is probably more appropriate to call this a `transcription', `counterpart', or
`equivalent', than a `translation'.
We will encounter this sort of difficulties again when we are
later dealing with translations of pangrams.
A wealth of information is available on this topic, including many
paradoxical assertions such as the famous Epimenides paradox. Again, we are not interested in them in
general, but only in special
cases about letters and words.
The beginning
In January 1982, a year after taking over Martin Gardner's column "Mathematical
Games" in Scientific American, Douglas Hofstadter
published in his column "Metamagical Themas" (note the anagram)
solutions to the "problem of the self-answering question" posed
in his maiden column.
Among them was the following elaborate self-referencing sentence (from Howard
Bergerson):
In this sentence the word AND occurs twice, the word EIGHT occurs twice,
the word FOUR occurs twice, the word FOURTEEN occurs four times, the word
IN occurs twice, the word OCCURS occurs fourteen times, the word SENTENCE
occurs twice, the word SEVEN occurs twice, the word THE occurs fourteen times, the word THIS occurs
twice, the word TIMES occurs seven times, the
word TWICE occurs eight times, and the word WORD occurs fourteen times.
This inspired the Dutch linguist Rudy Kousbroek to produce in
February 1983 a Dutch counterpart example in the article "Welke
vraag heeft vierendertig letters ?" (note: a self-answering question):
In deze zin komt het woord IN twee keer voor, het woord DEZE ook twee keer, het woord ZIN twee
keer, het woord KOMT twee keer, het woord HET
zestien keer, het woord WOORD ook zestien keer, het woord TWEE tien keer,
het woord KEER zestien keer, het woord VOOR twee keer, het woord OOK
drie keer, het woord ZESTIEN vier keer, het woord TIEN twee keer, het
woord DRIE twee keer, het woord VIER twee keer, en het woord EN twee keer.
(A little side note: whereas in the English version the capital
words are listed in alphabetical order, the capital words in the
Dutch version are listed in the order in which they appear in the
whole sentence. In a sense the sentence repeats itself in the
capital words, at least partly.)
The big question is, can something similar be accomplished by counting
the individual letters in a sentence rather than the words ?
Self-enumerating sentences
A self-enumerating sentence is a self-referencing sentence whose text
consists solely of the enumeration of its letter content.
It is also called an autogram.
The answer to the question whether such sentence exists was given by
Lee Sallows, and was published by Hofstadter in the same January 1982
column in Scientific American mentioned above. Here it is:
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 !
Note that this beautiful sentence not only counts its own letters,
but also its punctuation marks!
Rudy Kousbroek reacted to this one as well, and constructed a
Dutch equivalent which he published also in his aforementioned
February 1983 article. He made the astute observation that in
Lee Sallows' sentence the letters `j', `q', and `z' were missing,
and considered that a minor flaw. He included all 26 letters in
his own translation, making it a pangram.
" Alleen 'n ezel gelooft dat ik de moeite heb genomen na te tellen dat deze
zin bestaat uit zestien a's, drie b's, vier c's, vijftien d's, negenenzeventig
e's, zes f's, veertien g's, zes h's, vierendertig i's, vier j's, zeven k's,
acht l's, vijf m's, achtendertig n's, zes o's, vier p's, een q, achttien r's,
vijfendertig s's, drieendertig t's, drie u's, dertien v's, drie w's, een x,
een y, tien z's, plus dertig komma's, vierentwintig afkappingstekens,
twee aanhalingstekens en, niet te vergeten, een enkel ! "
Unfortunately, the truth which this sentence expresses turned out
to be different from what he expected. Lee Sallows noted three (hmm, I see only two!)
miscalculations in the counts, making the sentence no longer self-enumerating.
So, ironically, it is indeed true that "only a
fool believes that I took the trouble to verify this sentence ...".
Rudy Kousbroek took his revenge only a month later. In March 1983
he came up with a revised version in the article "Instructies voor
het demonteren van een bom". This time he omitted the three letters
`q`, `x`, and `y' that appeared only once in the original version,
so the new one is no longer a pangram, but at least it is correctly
self-enumerating.
" Alleen 'n gek gelooft dat ik de moeite heb genomen om na te tellen dat deze
zin bestaat uit vijftien a's, drie b's, drie c's, veertien d's, vierenzeventig
e's, acht f's, zestien g's, vijf h's, vijfendertig i's, zes j's, acht k's,
negen l's, zes m's, negenendertig n's, zeven o's, vier p's, vijftien r's,
zesendertig s's, drieendertig t's, drie u's, veertien v's, vier w's,
tien z's, 'n vijfentwintig komma's, zesentwintig afkappingstekens,
de twee aanhalingstekens plus niet te vergeten een enkel ! "
But his revenge went much further.
Self-enumerating pangrams
At the bottom of his March 1983 article, Rudy Kousbroek included
the following masterpiece:
Dit pangram bevat vijf a's, twee b's, twee c's, drie d's, zesenveertig e's,
vijf f's, vier g's, twee h's, vijftien i's, vier j's, een k, twee l's,
twee m's, zeventien n's, een o, twee p's, een q, zeven r's, vierentwintig s's,
zestien t's, een u, elf v's, acht w's, een x, een y, en zes z's.
This sentence just enumerates its letter content and no longer
the punctuation marks, mentions every letter of the alphabet,
and consists of a minimum of words. The first clean example of
a real self-enumerating pangram!
The article ended with the devilish remark that without doubt
Lee Sallows wouldn't have trouble in finding a divine English
translation.
Lee Sallows set off to write a computer program (in Lisp) to search
for possible solutions to this `standard' problem, but realized soon
that the number of combinations to be investigated was prohibitively
overwhelming and too time consuming for his program.
He then took a radically different approach. Being an electronics engineer, he constructed a
`pangram machine' in special-purpose
hardware with circuitry designed to solve just this one problem.
The pangram machine was unable to find a solution starting with the
phrase ``This pangram contains ...'', being the proper translation
of the Dutch ``Dit pangram bevat ...''.
Other verbs were substituted instead of `contains', and after many
unsuccessful attempts the machine eventually produced an Eureka!
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.
Still not completely satisfied, Lee Sallows got the inspiration to
replace the word `and' at the very end of the standard sentence with
`&', and on 22 November 1983 the pangram machine finally delivered
the answer
This pangram contains four a's, one b, two c's, one d, thirty e's,
six f's, five g's, seven 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, five r's, twenty-seven s's,
eighteen t's, two u's, seven v's, eight w's, two x's, three y's, & one z.
The pangram machine discovered many other solutions since then,
with different starting phrases. Some interesting ones will be
reproduced below.
In October 1984 Ed Miller proved that there are just two solutions
with the standard phrase ``This pangram contains ...'', and disclosed
the second one:
This pangram contains four a's, one b, two c's, one d, twenty-six e's,
six f's, three g's, six h's, eleven i's, one j, one k, two l's, two m's,
seventeen n's, fifteen o's, two p's, one q, eight r's, thirty s's,
seventeen t's, four u's, four v's, six w's, six x's, three y's, & one z.
The challenge of Lee Sallows
The results of the above exercises, including details about the
pangram machine, were reported in October 1984 by A. K. Dewdney
in his column "Computer Recreations" in Scientific American.
It was also in this article that Lee Sallows made the following wager:
"I bet 10 guilders nobody can come up with a self-enumerating solution
(or proof of its nonexistence) to the sentence beginning
``This computer-generated pangram contains ... and ...''
within the next 10 years."
How mistaken he was.
This computer-generated pangram contains six a's, one b, three c's,
three d's, thirty-seven e's, six f's, three g's, nine h's, twelve i's,
one j, one k, two l's, three m's, twenty-two n's, thirteen o's,
three p's, one q, fourteen r's, twenty-nine s's, twenty-four t's,
five u's, six v's, seven w's, four x's, five y's, and one z.
Already in January 1985 A. K. Dewdney reported in his column
"Computer Recreations" in Scientific American that this solution
was found by four different competitors, independent of each other,
and that it indeed was computer-generated in all four cases.
The four programmers were (in submission order) John Letaw,
Lawrence Tesler, Ed Miller, and William Lipp. Later Michael Gayle
and James Mittan submitted a fifth program, while Hans Buchwald
and Robert Wolfson contributed proposed algorithms.
Dewdney was surprised that all five solutions were exactly identical,
and astonished about the remarkable differences in computer time
used by the various programs. But there is an explanation.
Five overlooked pangrams
Only the program of Edward S. Miller employed an Exhaustive
Search Method, with ingeniously designed cutoffs and search
ranges to keep the execution time within reasonable bounds.
He could therefore prove that there is just one unique solution
to the `10-guilder' problem, and also that there is no solution
to the `standard' problem with the ``... contains ... and ...''
phrase, and exactly two solutions to the `standard' problem with
the ``... contains ... & ...'' modification.
In February 1985 Ed Miller came to the conclusion that not only
the other pangram programs employed too narrow search ranges and
boundary conditions (so that potential solutions could be skipped
too early), but that even the pangram machine suffered from this.
He examined seven variants of the `standard' problem (with the
verbs "contains", "comprises", "consists of", "is composed of",
"uses", "employs", "has") for which the pangram machine had found
no solution at all. Wrongly, because the exhaustive search method
revealed these overlooked pangrams:
This pangram uses four a's, one b, one c, two d's, thirty e's,
five f's, three g's, seven h's, eleven i's, one j, one k, three l's, two m's,
sixteen n's, twelve o's, two p's, one q, eight r's, twenty-nine s's,
sixteen t's, four u's, six v's, six w's, five x's, three y's, and one z.
This pangram uses four a's, one b, one c, two d's, twenty-six e's,
seven f's, three g's, six h's, eleven i's, one j, one k, two l's, two m's,
fifteen n's, fourteen o's, two p's, one q, eight r's, thirty s's,
sixteen t's, five u's, four v's, six w's, six x's, three y's, and one z.
This pangram employs four a's, one b, one c, two d's, thirty-six e's,
three f's, three g's, ten h's, seven i's, one j, one k, three l's, three m's,
seventeen n's, twelve o's, three p's, one q, ten r's, twenty-eight s's,
twenty-two t's, two u's, six v's, seven w's, three x's, five y's, and one z.
This pangram employs four a's, one b, one c, two d's, thirty-six e's,
six f's, three g's, eleven h's, ten i's, one j, one k, three l's, three m's,
eighteen n's, thirteen o's, three p's, one q, fourteen r's, twenty-six s's,
nineteen t's, five u's, three v's, three w's, four x's, four y's, and one z.
This pangram has five a's, one b, one c, two d's, twenty-eight e's,
five f's, three g's, seven h's, ten i's, one j, one k, one l, two m's,
twenty n's, thirteen o's, two p's, one q, five r's, twenty-three s's,
twenty t's, one u, six v's, nine w's, two x's, five y's, and one z.
Variations on a theme
As said before, the pangram machine could be loaded with different
starting phrases. By extending the text a bit and carefully selecting
the verbs, it was possible to generate an ordered list:
This first pangram has five a's, one b, one c, two d's, twenty-nine e's,
six f's, four g's, eight h's, twelve i's, one j, one k, three l's, two m's,
nineteen n's, twelve o's, two p's, one q, eight r's, twenty-six s's,
twenty t's, three u's, five v's, nine w's, three x's, four y's, and one z.
This second pangram totals five a's, one b, two c's, three d's, twenty-nine e's,
six f's, four g's, seven h's, ten i's, one j, one k, two l's, two m's,
twenty-one n's, sixteen o's, two p's, one q, eight r's, twenty-eight s's,
twenty-three t's, four u's, four v's, nine w's, three x's, five y's, and one z.
This third pangram contains five a's, one b, two c's, three d's, twenty-six e's,
six f's, two g's, four h's, ten i's, one j, one k, two l's, two m's,
twenty-two n's, seventeen o's, two p's, one q, seven r's, twenty-nine s's,
twenty-one t's, four u's, six v's, eleven w's, four x's, five y's, and one z.
After a while it becomes almost more difficult to choose the verb
than to construct the solution.
The entire list of the first twenty-five
numbered pangrams
The standard problem revisited
The solution to the `standard' problem had the flaw that the Dutch
word `en' was translated into `&' instead of the desired `and'.
By trying variations for the noun in the starting phrase instead
of the verb, the following came out of the pangram machine (again with two alternatives):
This autogram contains five a's, one b, two c's, two d's, thirty-one e's,
five f's, five g's, eight h's, twelve i's, one j, one k, two l's, two m's,
eighteen n's, sixteen o's, one p, one q, six r's, twenty-seven s's,
twenty-one t's, three u's, seven v's, eight w's, three x's, four y's, and one z.
This autogram contains five a's, one b, two c's, two d's, twenty-six e's,
six f's, two g's, four h's, thirteen i's, one j, one k, one l, two m's,
twenty-one n's, sixteen o's, one p, one q, five r's, twenty-seven s's,
twenty t's, three u's, six v's, nine w's, five x's, five y's, and one z.
which translates into the now perfectly equivalent Dutch version
Dit autogram bevat vijf a's, twee b's, twee c's, drie d's, zevenenveertig e's,
zes f's, vijf g's, twee h's, veertien i's, vijf j's, een k, twee l's, twee m's,
zeventien n's, twee o's, een p, een q, zes r's, vierentwintig s's,
achttien t's, twee u's, elf v's, negen w's, een x, een y, en vijf z's.
The three Dutch alphabets
The Dutch examples are mostly written using the English alphabet,
i.e. the special ligature `ij' counts as two separate letters, and
the 25th letter of the alphabet is the `y'. The same classification
is used by the standard Dutch reference dictionary "van Dale".
Here is another example:
Dit pangram gebruikt vijf a's, twee b's, drie c's, zeven d's, vijfenveertig
e's, drie f's, zeven g's, drie h's, zeventien i's, drie j's, twee k's, een l,
twee m's, twintig n's, een o, twee p's, een q, negen r's, drieentwintig s's,
zeventien t's, twee u's, acht v's, acht w's, een x, een y, en zes z's.
If the ligature `ij' is considered as a separate letter (which is
strictly speaking more conforming to the nature of the Dutch language)
the alphabet is extended to 27 letters, "tolerating" both `y' and `ij'.
This also has a solution:
Dit pangram uit zich in zes a's, een b, vijf c's, drie d's, vierenveertig e's,
vier f's, vijf g's, vijf h's, achttien i's, een j, een k, een l, twee m's,
tweeentwintig n's, een o, twee p's, een q, acht r's, eenentwintig s's, achttien
t's, twee u's, tien v's, zeven w's, een x, een y, vier ij's, en vier z's.
Purists may claim that the `y' is a actually foreign letter and
does not formally belong to the Dutch language. The 25th letter
is the `ij' ligature, and there is no `y' at all.
This "formal" standpoint yields even a double solution:
Dit pangram uit zich in zes a's, een b, vijf c's, vier d's, tweeenveertig
e's, drie f's, vier g's, vijf h's, zestien i's, een j, een k, een l,
twee m's, achttien n's, een o, twee p's, een q, acht r's, vierentwintig s's,
zestien t's, twee u's, acht v's, zes w's, een x, drie ij's, en zeven z's.
Dit pangram uit zich in vijf a's, een b, vier c's, twee d's, vierenveertig
e's, vijf f's, vijf g's, vier h's, zestien i's, een j, een k, twee l's,
twee m's, negentien n's, een o, twee p's, een q, acht r's, tweeentwintig s's,
zeventien t's, twee u's, elf v's, acht w's, een x, vier ij's, en vier z's.
There is even a fourth alphabet, which is only used by the
Dutch PTT in telephone books. The ligature `ij' is here
discarded completely, and replaced everywhere by `y'.
This is rather absurd. For the sake of pangrams, this "PTT"
alphabet is equivalent to the formal alphabet.
Magic pairs
A certain starting phrase may yield more than one solution.
We have seen this already for the results of some variants
of the `standard' problem. But if some of the initial words
can be reshuffled, a rather magic effect can be reached.
Loaded with Dutch words and phrases, the pangram machine
produced the beautiful magic pair
Dit pangram bevat maar acht a's, twee b's, drie c's, vijf d's, eenenveertig
e's, vijf f's, zes g's, drie h's, zeventien i's, vijf j's, een k, een l,
drie m's, twintig n's, een o, twee p's, een q, acht r's, tweeentwintig s's,
vijftien t's, een u, negen v's, zes w's, een x, een y, and vier z's.
Maar dit pangram bevat negen a's, twee b's, vier c's, vier d's, drieenveertig
e's, twee f's, zeven g's, vier h's, vijftien i's, twee j's, een k, een l,
drie m's, twintig n's, een o, twee p's, een q, negen r's, twintig s's,
achttien t's, een u, acht v's, acht w's, een x, een y, and twee z's.
for which there exists the similar English translation
This pangram tables but five a's, three b's, one c, two d's, twenty-eight e's,
six f's, four g's, six h's, ten i's, one j, one k, three l's, two m's,
seventeen n's, twelve o's, two p's, one q, seven r's, twenty-nine s's,
twenty t's, five u's, six v's, eight w's, four x's, four y's, and one z.
But this pangram tables five a's, three b's, one c, two d's, twenty-nine e's,
six f's, six g's, eight h's, eleven i's, one j, one k, three l's, two m's,
seventeen n's, fourteen o's, two p's, one q, eight r's, twenty-eight s's,
twenty-two t's, six u's, four v's, eight w's, four x's, four y's, and one z.
Mutually descriptive pairs
There are other, even more magic, pairs. On 22 March 1985 Ed Miller
printed (on two separate sheets, lacking a double-sided printer)
the magnificent
The sentence on the reverse side contains three a's, one b, three c's,
three d's, forty-three e's, seven f's, two g's, nine h's, eight i's,
one j, one k, two l's, one m, twenty-four n's, sixteen o's, one p,
one q, eleven r's, twenty-seven s's, twenty-three t's, three u's,
six v's, seven w's, two x's, five y's, and one z.
The sentence on the reverse side contains three a's, one b, three c's,
three d's, forty-six e's, four f's, two g's, ten h's, eight i's,
one j, one k, two l's, one m, twenty-four n's, fifteen o's, one p,
one q, eleven r's, twenty-nine s's, twenty-three t's, two u's,
seven v's, seven w's, three x's, five y's, and one z.
However, it turns out after all that something similar can
in fact be printed one-sided.
-
The adjacent text utilizes four a's, one b, two c's, three d's, thirty-six e's,
five f's, three g's, nine h's, eleven i's, two j's, one k, four l's, one m,
eighteen n's, thirteen o's, one p, one q, eight r's, twenty-seven s's,
twenty-four t's, four u's, four v's, seven w's, five x's, four y's, and two z's.
|
-
The adjacent text utilizes four a's, one b, two c's, three d's, thirty-two e's,
nine f's, three g's, eight h's, eleven i's, two j's, one k, three l's, one m,
seventeen n's, fifteen o's, one p, one q, eleven r's, twenty-six s's,
twenty-one t's, eight u's, six v's, six w's, three x's, four y's, and two z's.
|
And the next pair belongs to the same category, but is even
more mutually descriptive.
-
The righthand 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 lefthand 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.
|
A bimagic angram
A special case of a double solution is worth mentioning separately.
It has magic of its own.
-
This angram contains four a's, two b's, two c's, one d, twenty-seven e's,
eight f's, four g's, five h's, ten i's, one j, one k, one l, two m's,
twenty n's, fifteen o's, one q, six r's, twenty-seven s's, eighteen t's,
five u's, six v's, seven w's, three x's, four y's, one z, but no _.
-
This angram contains four a's, two b's, two c's, one d, twenty-seven e's,
eight f's, four g's, five h's, eleven i's, one j, one k, two l's, two m's,
twenty n's, fifteen o's, one q, six r's, twenty-seven s's, nineteen t's,
five u's, six v's, eight w's, three x's, four y's, one z, but no _.
We have now entered the realm of autograms that are no pangrams.
The solitary z
In case a certain letter does not appear in the text of the
starting phrase, nor in any of the counting words, it has to
be included in the enumeration with a count of `one' to make
the sentence a pangram. So far we have seen this happen in
all above examples. Would it be possible to get rid of these
rather redundant standalone additions ?
Almost, so it seems, as we can produce the autogram
This sentence employs two a's, two c's, two d's, twenty-eight e's,
five f's, three g's, eight h's, eleven i's, three l's, two m's,
thirteen n's, nine o's, two p's, five r's, twenty-five s's,
twenty-three t's, six v's, ten w's, two x's, five y's, and one z.
which has several trivial variations since `one z' could have
been just as well `one q', but from an esthetic standpoint
the `z' is the obvious choice. Nevertheless, it turns out to
be possible to construct a pure solution after all.
This sentence employs two a's, two c's, two d's, twenty-six e's,
four f's, two g's, seven h's, nine i's, three l's, two m's,
thirteen n's, ten o's, two p's, six r's, twenty-eight s's,
twenty-three t's, two u's, five v's, eleven w's, three x's, and five y's.
To emphasize this solitary occurrence of the `z', consider
the following surprising triplet:
This sentence contains three a's, three c's, two d's, twenty-six e's,
five f's, three g's, eight h's, thirteen i's, two l's,
sixteen n's, nine o's, six r's, twenty-seven s's, twenty-two t's,
two u's, five v's, eight w's, four x's, five y's, and only one z.
Only this sentence contains three a's, three c's, two d's, twenty-nine e's,
five f's, three g's, eight h's, twelve i's, three l's,
nineteen n's, nine o's, six r's, twenty-four s's, twenty-one t's,
two u's, five v's, eight w's, two x's, five y's, and one z.
This sentence contains only three a's, three c's, two d's, twenty-five e's,
nine f's, four g's, eight h's, twelve i's, three l's,
fifteen n's, nine o's, eight r's, twenty-four s's, eighteen t's,
five u's, four v's, six w's, two x's, and four y's.
A self-enumerating paradox
This sentence doesn't contain two a's, three c's, two d's, twenty-six e's,
six f's, three g's, seven h's, eight i's, fifteen n's, ten o's, seven r's,
twenty-eight s's, twenty-one t's, four u's, four v's, seven w's, three x's,
four y's, & two &'s.
Self-enumerating palingrams
A palingram is a phrase that has exactly the same syllable, word, or letter sequence when you read it
backwards.
A letter palingram is normally called a palindrome.
As such they are obviously a bit off-topic here, but Lee Sallows
donated a special palingram that is also self-enumerating, so
it should definitely be mentioned.
It deserves a page of its own, and goes without further words.
The art of
self-enumerating palingrams
Counting letters and letters
As an extension to the `standard' problem and the `10-guilder' problem, here are two pangrams which not only enumerate their
individual letter content, but also mention their total letter count.
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.
This computer-generated sentence contains two hundred forty-seven letters:
four a's, one b, four c's, five d's, forty-four e's, nine f's, three g's,
seven h's, eleven i's, one j, one k, three l's, two m's, twenty-nine n's,
nineteen o's, two p's, one q, fourteen r's, thirty-one s's, twenty-five t's,
seven u's, eight v's, seven w's, two x's, six y's, and one z.
After eliminating some of the one-time letters, there is also
a non-pangram version.
This sentence contains one hundred and ninety-seven letters:
four a's, one b, three c's, five d's, thirty-four e's, seven f's, one g,
six h's, twelve i's, three l's, twenty-six n's,
ten o's, ten r's, twenty-nine s's, nineteen t's,
six u's, seven v's, four w's, four x's, five y's, and one z.
So far, only the redundant one-time letters have been disposed
of sometimes. But for constructing full sentences a starting
phrase was needed, with a rather arbitrary text. Would it be
possible to get rid of all such arbitrary elements altogether ?
The shortest self-enumerating phrase
In June 1985 the Dutch linguist Hugo Brandt Corstius
wrote under his pseudonym (one of many) Piet Grijs the article "Humor and
wiskunde". He cited one of the "Stellingen" appended
to the PhD thesis of Jan Moors. This proposition states that
Vijf v's, vijf ij's, vijf f's, vijf s's.
is the shortest self-enumerating phrase in Dutch.
(Note that this time the `xijz' alphabet is used.)
Subsequently Eric Wassenaar challenged Ed Miller to construct the
shortest English counterpart. In July 1985 Ed Miller came up with
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.
which has many trivial variations (since `one g, one w' could be
just as well `one a, one b' etc.). Without the arbitrary one-time
letters the phrase becomes slightly longer.
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.
Very close after that one comes with just one letter more
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.
The last two phrases are examples of what Lee Sallows calls
a "reflexicon". This opens a whole new aspect in the field
of recreational linguistics. But that goes beyond the scope
of this document.
The longest self-enumerating pangram
By extending the leading phrase, one can try to create very long
self-enumerating pangrammatic stories. How long was demonstrated
by John Letaw. It needs its own page. All further comments seem
superfluous and unnecessary.
Abraham Lincoln's Gettysburg
Abragram
On hindsight, it contains one amusing error. The abragram claims
70 d's whereas there are actually 86. The pangram program probably
didn't take into account the 2 d's in the 8 words `hundred' showing
up in the enumeration. The letter `d' does not appear in number-words
between 1 and 99. That fact is used by pangram programs to achieve
considerable speedup.
Epilogue
This is not a static document. It will be updated as more information
becomes available. It reflects what is known to me at the date that
is mentioned in the header. All mistakes and errors are mine.
This epilogue contains three a's, one b, two c's, two d's, thirty e's,
four f's, two g's, six h's, ten i's, one j, one k, two l's, one m,
twenty-one n's, seventeen o's, two p's, one q, six r's, twenty-seven s's,
twenty-one t's, three u's, five v's, nine w's, three x's, five y's, and one z.
References
- Douglas R. Hofstadter, Metamagical themas
Scientific American,
Vol. 246,
No. 1,
pp. 12-17
- Rudy Kousbroek, Welke vraag heeft vierendertig letters ?
Nieuwe Rotterdamse Courant,
Cultureel Supplement 640,
p. 3
- Rudy Kousbroek, Instructies voor het demonteren van een bom
Nieuwe Rotterdamse Courant,
Cultureel Supplement 644,
p. 9
- Rudy Kousbroek, De logologische ruimte
Meulenhoff,
Amsterdam,
pp. 135-153
- A. K. Dewdney, Computer recreations
Scientific American,
Vol. 251,
No. 4,
pp. 18-22
- Edward S. Miller, Pangram-solving program
Private correspondence to Lee Sallows
- Lee C. F. Sallows, In quest of a pangram
Private correspondence to Edward S. Miller
- A. K. Dewdney, Pangram programs
- A. K. Dewdney, Computer recreations
Scientific American,
Vol. 252,
No. 1,
pp. 10-13
- Edward S. Miller, Five overlooked pangrams
Private correspondence to A. K. Dewdney
- Lee C. F. Sallows, In quest of a pangram
Abacus,
Vol. 2,
No. 3,
pp. 22-40
- John R. Letaw, Pangrams: a nondeterministic approach
Abacus,
Vol. 2,
No. 3,
pp. 42-47
- Piet Grijs, Humor en wiskunde
de Volkskrant,
Kunst bijlage,
p. 19
- Edward S. Miller, Shortest self-enumerating sentences
Private correspondence to Eric Wassenaar
- Lee C. F. Sallows, Reflexicons
Word Ways,
Vol. 25,
No. 3,
pp. 131-141
- Lee C. F. Sallows, Pangrams with webbed feet
Private correspondence to Eric Wassenaar
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