In this article we explore some properties of the number names: the results of looking at the names of order, or looking at the whole string ONE, TWO, THREE, ..

Keith, Michael

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175 SOME PROPERTIES OF ONE, TWO, THREE ... ~ICHAEl KEITH Salem. Oregon In this article we explore some properties of the number names : the results of looking at the names in o rder. or looking at the wh ole s triog OX£. TWO. THREE •. . . SO~E BASIC PROPERTIES As o ne proceeds through the string ONE TWO THREE ...• the order in \<I.' hieh the letters of the alphabet occur is O~ETWHRPUIVSXGLYDAXBQPC. with J , n, and Z never appearing . Eighteen of these 23 letters have ap-peared by the time we get to 0:\£ THOeSAXD, but it takes a long time, comparatively . to get to the last five (XILLIO~ . BILlIOX. QCADRILLIO): , S EPTIlLIOX. OCTIL LIOX). The smallest number whose name co ntain s (at least) k different let-ters, for k :: 1 to 20, is : 3 O~E (also first 1 and 2) 4 THREe , EIGHT 6 THIRtEeX 7 POURTEe~ 9 TWENtY-POUR (also first B) 10 SEVeNTY-POUR 11 Ot\E IIUnDRed TWeLVe 13 ONE HUnDRed TWentY-PIVe (al so first 12) 14 O~E HVnDRed SIXTY-PiVe 15 TWO HU)lDREd SIXtY-PiVe 16 o~e THoUSAnD tWo hundRed sIXtY- fiVe 11 TWO tHoUSAND FIVe hundRed siXtY-eiGht 18 TWELVe tHOUSAXD FouR hundred sIXtY-eiGht 19 aXE XILlion TWo tHoUSAnD FiVe hundRed siXtY-eiGht 20 ONE BILlion one ~illion TWo tHoUSAnD FiVe hundRed siXtY-eiGht The final three letters are added by incorporating QUADRILLIO~. SEP-TILLION and OCTILLIO~ into the last number name. Sometimes we wish to examine the whole string from ONE to some number n. A fundamental question that then arises is : how many total letters are there? Of course we ca n just count them up one number at a time, but we seek a more efficient method . This can be done in general, but for brevity we just give t h e result for n equal to a thousand minus 1 . a millio n mi n us 1. and so on . 176 The number of letters in the word s for 1 to 10"-1 15 lOn-3 t (n).",' here t(3) = 18440 ten) ::: t(n-3) + tel) + 999(number of lette r s in the word for 100 - 3 ) for 0>3 and a multiple of 3 Por example, take 0 =6. The number of lette rs in the words 1. .. 999999 15 : 1000 times t(6- 3) + tel) + 999(number of letters in THOt;'SA~D) .: 1000(18440 + 18440 + 999(8» = 44872000 which ca n also be verified b y brute- fo rce counting . This tells us that if we choose a number at random from this range it will , on ave rag e , use ahout 45 lett.ers. Between one and o ne billion it's about 70 . and b etween o ne and o ne trillion. ahout 96. A so mew hat differen t ques tion is : how big a set of letter tiles 15 needed to spell any number in the range from 1 to 01 Por example. 37 tiles with distdbutio n (0 0 0 2 4 3 2 2 2 0 0 1 0 4 2 0 0 2 2 3 1 2 2 2 1 0) are s ufficient for n =100. The answers for n equal to 10. 100. 1000 . etc . are: 16. 37. S9, 82 101. 12S, 148 : eac h additional power of 10 increases the count by about 23. Xany of these sa me questions can be explored u s ing the string ZERO . ONE. TW O, etc . One o dd discovery is th e fact that the number of letters in ZERO , O!,\E . TWO ... SEVE~TY-~~~E is equal to the number of the beast (666) . ~o initial seg ments of OXE TWO THREE ... have this property. GEO~ETRIC PACKI~G F~3 For some valu es of n . the words O~E , TWO, THREE .. . (n) can be packed into a rectangle o ne of whose sides is n . This is a notable property , as there are . for example , only 14 values of n less than a mi1li.on for which this is true: 1 !x3 2595 2595x25 2 2.3 3870 3870x27 8 8.4 5568 5568.28 9 9.4 32592 32592x33 31 31.7 55019 55019.34 283 283.15 264740 264 740.41 786 786xl8 426067 42606 7x43 Actually. this list just shows which ones might. be able to be rectangle-pack.ed. Le .• those for which the number of cells in the rectangle is equal to the number of letters in the string . One of these could. in theory. be unpack.able ; but we (somewhat boldly) conjecture t his is nev-er the case , based on being able to pack the first four , plus the fact that larger ones should be easier to pack. 177 Por 0=9, ""' e note that 9x4 is also 6x6, so this motivates the question: what sequences of ONE through (n) can be packed into a square? There are many of these for n less than ten thousand (first number 15 n, second number is the size of the square): 9 6 _ 16 9 21 11 56 21 107 31 332 72 353 7 5 389 80 494 92 765 117-1683 195 2600 255 2945 275 3026 279 4008 330 4365 345 4585 355 6235 1ol9 6683 435 7507 463 8243 487 8605 499 8605 499 9341 521 97(,2 533 Again, these are only values for which a square packing migJlt be pos-sible. The first packing is possible, as shown below: 0 T f 0 u r n W P I V E 0 • N e 5 > x T H R E E I 5 e v e n N EIGHTE The second one on the list (ONE through SIXTEE~ fitting in a 9x9 square) ranks as perhaps the most amazing discovery made in this in -vestigation, because both n (16) and the number of points i n the s quare (81) are fourth powers! This means that. as shown in the figure belo". these both fill (not pac k) four-dimensional hyper c ubes: e S t F n i n e t 1 E h 0 S I X e e • • • • 9 V r U 0 n e 1 n • • • • h E e R T W 0 e T t N f • w e 1 v e v • • • • S I X T E E N e E · · · · f i f t e e n n L T H I R T E E N V f 0 u r t e e n E (a) 16 points liUs a 2x2x2x2 bypen:ube (b) ONE ... SlXTEEN fills a JxJxJxJ bypen:ube The diagram also shows how to pack aXE .. . SIXTEE~ into a 9x9 thus proving that the second item on the list above can be Packing the larger squares is left as a puzzle for the reader . sq uare, packed . A probabilistic calculation suggests that the number 16 is unique--there are probably no other value5 of n to;'ith this do<;~ty-hypercube-f'..l­ling property. We checked all numbers up to one hundred million and did not nnd another example . 

Some Properties of One, Two, Three...

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Some Properties of One, Two, Three...

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