Numbers may be communicated in the form of
expressions -- written or spoken
-- using numerals
('1,' '3.1416...,' '4 + 4,' etc.) and in
the form of
words ('one,' 'three point one four one six,' four plus
four,' etc.).
The following observations are made with reference
to expressions in the form of words.
Among whole numbers from zero through twelve,
four of them may be
expressed as three-letter words:
’one,’ ‘two,’ ‘six’ and ‘ten.'
In addition,
‘and,’ a so-called function word, is also
a three-letter word.
Thus
there are a total of five three-letter words.
Also among the set of whole
numbers expressed as
words are four four-letter words:
’four,’ ’five,’ ‘nine’
and ‘zero.'
In addition, ‘plus,’ a function word, is a four-letter
word.
Thus there are a total of five four-letter words.
(We disregard the symbol
'+' as being, strictly
speaking, neither a word nor a numeral.)
Further,
there are among that set three five-letter
words:
‘three,’ ‘seven,’
and ‘eight.'
Moreover, ‘minus,’ a function word, is a five-letter
word.
Thus
there are four five-letter words. (The symbol '-'
is disregarded
similarly as per the symbol '+' above.)
Finally, among that set there are two six-letter words:
‘eleven’ and ‘twelve.’
To summarize:
five
three-letter words:
'one,' 'two,' 'six,' 'ten,' (plus 'and' )
five four-letter words:
five four-letter words:
'four,' 'five,' 'nine,'
('plus' and 'zero')
four five-letter words:
four five-letter words:
'three,' 'seven,'
'eight,' (plus 'minus')
two six-letter words:
two six-letter words:
'eleven,' (plus
'twelve')
Note that several of these 16 words appear between
Note that several of these 16 words appear between
parentheses
in the above list. This is because none of
the expressions observed below
is constructed using
any of the words so appearing. It is with reference
to
expressions formed from combinations of only 11 words,
therefore (i.e., those words not in parentheses) -- four
three-letter words, three four-letter words, three
five-letter words
and one six-letter word -- that the
observations are made.
These 11 words, i.e., the words designating the whole
numbers between 0 and 12
-- excepting 0 and 12 --
plus 'and,' can be combined in various
ways to yield a
series of 12-letter expressions (not counting spaces
between words or implicit punctuation) of which the
numerical value is 12.
E.g., ‘one, one and ten’ is such an expression, the
words of which contain a total of 12 letters and the
value of
which, expressed in numerals, is 12.
So is the expression ‘eight and four,' and so forth.
So far, 22 such expressions have been constructed.
11 of them – precisely
one half of the number of
expressions constructed and a total identical with
the count of number words drawn upon -- employ
the word 'and'; precisely 11 of them do not.
Here they appear in a
simple array (fig. 1):
O N E O N E AN D T E N
O N E T E N A N D ON E
O N E F O U R S E V E N
O N E S E V E N F O U R
O N E A N D E L E V E N
TWO TWO TWO S I X
T WO T WO S I X T WO
T WO S I X T WO T WO
T H R E E A N D N I N E
F O U R O N E S E V E N
F O U R FO U R F O U R
F O U R S E V E N O N E
F O U R A N D E I G H T
F I V E A N D S E V E N
S I X T WO T WO T WO
S E V E N A N D F I V E
S E V E N O N E F O U R
S E V E N F O U R O N E
E I G H T A N D F O U R
N I N E A N D T H R E E
T E N O N E AN D O N E
E L E V E N A N D O N E
F O U R O N E S E V E N
F O U R FO U R F O U R
F O U R S E V E N O N E
F O U R A N D E I G H T
F I V E A N D S E V E N
S I X T WO T WO T WO
S E V E N A N D F I V E
S E V E N O N E F O U R
S E V E N F O U R O N E
E I G H T A N D F O U R
N I N E A N D T H R E E
T E N O N E AN D O N E
E L E V E N A N D O N E
(fig. 1)
* * * * * * * * * * * * * * * * * *
E I G H T A N D F O U R
F O U R A N D E I G H T
T E N O N E A N D O N E
O N E T E N A N D O N E
O N E O N E A N D T E N
S E V E N F O U R O N E
S E V E N O N E F O U R
S E V E N A N D F I V E
O N E S E V E N F O U R
F O U R S E V E N O N E
F O U R O N E S E V E N
F I V E A N D S E V E N
O N E F O U R S E V E N
E L E V E N A N D O N E
O N E A N D E L E V E N
N I N E A N D T H R E E
T H R E E A N D N I N E
T W O T W O T W O S I X
S I X T W O T W O T W O
T W O S I X T W O T W O
T W O T W O S I X T W O
F O U R F O U R F O U R
(fig. 2)
E I G H T A N D F O U R
F O U R A N D E I G H T
T E N O N E A N D O N E
O N E T E N A N D O N E
S E V E N O N E F O U R
O N E S E V E N F O U R
F O U R O N E S E V E N
O N E F O U R S E V E N
O N E A N D E L E V E N
T W O T WO T W O S I X
S I X T W O T W O T W O
T W O S I X T W O T W O
T W O T W O S I X T W O
F O U R F O U R F O U R
T H R E E A N D N I N E
F I V E A N D S E V E N
S E V E N A N D F I V E
(fig. 3)
All but five expressions include 'e,'; all but five include
'n,' and all but four include 'e,' as
the figs above show.
Not the least interesting aspect of this phenomenon,
of course, regardless of
the manner in which its elements
are arrayed, is the fact of its random
character. That
these word combinations can be assembled to yield the
value of
12 is based largely on the chance circumstances
of their containing the
number of letters they do and the
way they are spelled. Obviously the same condition would
not obtain, or
would do so in some different manner, using
number words from languages other
than American English.
* * * * * * * * * * * * * * * * * * * *
To create 3x4-letter blocks, the 12 letters
of each expression
above may be ordered in various ways -- left-to-right,
right-
to-left, alternate (beginning from either end), top-to-bottom,
inward- or
outward-turning spirals beginning at different loci,
and so forth.
Inserting the letters in order left to right into four-letter rows
and stacking these
rows three-deep, top to bottom, results
in blocks as in the following
figures:
> O N E O > E I GH
> N E A N > T A ND
> D T E
N >
F OUR
etc.
The 22 blocks so constructed may be variously arrayed,
these
different arrays possessing various kinds of
symmetry.
One possible array,
shown below...
