Problems & Puzzles: Puzzles

Puzzle 33.- Find numbers like this: 15551= Fifteen Thousand Five Hundred Fifty One = 383

(Proposed by G.L. Honaker, Jr. to me and to Patrick De Geest at , 9/12/98)

In the original proposal of this puzzle Honaker was asking for a scheme of the following type:

N à E(N), English expression of N à V(E), value of E

Where:

1. N and E should be palprimes
2. E(N) is the expression of the number N in English language
3. V(E) must add the values of the letters employed in E(N), assigning the following values to the letters: A=1, B=2, etc.

Very soon Patrick obtained "the number of the beast" from a palprime:

1215121 (palprime) = One Million Two Hundred Fifteen Thousand One Hundred Twenty One = 666. (P. De Geest)

and a second example of the like found by Honaker:

1496941 = O+N+E+M+I+L+L+I+O+N+ F+O+U+R+H+U+N+D+R+E+D+N+I+N+E+T+Y+S+I+X+ T+H+O+U+S+A+N+D+ N+I+N+E+H+U+N+D+R+E+D+F+O+R+T+Y+O+N+E = 727 = palindromic prime (Patrick De Geest)

Then I simply made my own search in Spanish (a=1, ñ=15, z=27):

1380831 (pal not prime) = un millón trescientos ochenta mil ochocientos treinta y uno = 666 (C. Rivera)

106909601(palprime) = ciento seis millones novecientos nueve mil seiscientos uno = 666(C. Rivera)

1936391 (palprime) =un millón novecientos treinta y seis mil trescientos noventa y uno = 787(palprime) (C. Rivera)

Maybe you would like to find other primes and/or palprimes in English; or maybe you would like to puzzle this way in your own language.

As for sure you have imagined, the very core of this puzzle is to develop your own code to make "easy" the step N à E(N). Do it… you’ll enjoy…

(On request you can ask by e-mail for the following codes available at this moment: NUMESP.ub, by Carlos Rivera and HONAPUZZ.ub by Patrick De Geest)

As a continuation of this puzzle, yesterday (Saturday, 12/12/98) I proposed to cycle the scheme designed by Honaker to an arbitrary starting number just to see where ends the game… By example, In English I constructed the following sequence:

123456789, 964, 273, 298, 268, 278, 291, 253, 254, 258, 247, 281(prime), 240, 216, 228, 288, 255, 240 (a loop!)

Then I asked to Honaker, Patrick and Mike Keith (we were in touch that Saturday in an informal meeting by e-mails ) if every number always falls in a loop in every language.

The question did not last long in the air. Mike almost immediately answered this:

"This problem is discussed in Eckler's recent book "Making the Alphabet Dance." He cites a paper by Borgmann from 1967 that shows that every number in English converges to the 216-...-240 loop given by Carlos. Every number requires at most 18 steps to enter the cycle: an example of this is "4".

Isn't the following a remarkable fact?… Every number in English converges to a single loop?!!!…

Evidently I started trying to detect the loops available in Spanish and I found that same morning - without a very exhaustive search – only three (3) loops:

1) 196, 242, 286, 196...

2) 222, 249, 309, 222...

3) 251, 298, 291, 300, 251...

Well, now the following questions appear interesting:

Q1)Can you verify that in English there exists only one loop (as Borgmann says)?

Q2)Can you see if in Spanish are there other loops besides the 3 found by Carlos Rivera.

Q3)Would you like to find the loops formed in other languages (come in friends of France, Belgium, Korea, Poland, and other countries!)

Claudio Meller wrote (May 2011!!!)

Carlos, a mi en español también me dan tres bucles pero ligeramente distintos debe haber algo que entendí mal

el bucle de 196 me da : 196,  242,  280,  196 y a ti te da 196, 242, 286, 196
Yo calculo doscientos cuarenta y dos así :
4 + 16 + 20 + 3 + 9 + 5 + 14 + 21 + 16 + 20 + 3 + 22 + 1 + 19 + 5 + 14 + 21 + 1 + 26 + 4 + 16 + 20 = 280
Donde está el error?
A mi para el de trescientos me da 153
21 + 19 + 5 + 20 + 3 + 9 + 5 + 14 + 21 + 16 + 20 = 153
y a ti te da 251, creo que este te lo salteaste.

pero en lineas generales aparecen siempre estos tres bucles

196,  242,  280,  196
251,  298,  291,  300, 153, 251
222,  249,  309,  222
En la semana probaré en inglés...

En inglés  aparentemente caen siempre en alguno de los términos del siguiente bucle:

326, 327, 340, 273, 336, 320, 275, 322, 333, 324, 335, 310, 207, 235, 312, 255, 278, 329, 317, 277, 345, 315, 233, 326

***

Jan van Delden wrote (May, 2011)

There are 3 final periodic cycles. I used a small check, but that should suffice.

[280,189,200,121,213,196,218,201,145,220,223,278]
[219,214]
tweehonderdnegentien
tweehonderdveertien

[168,205].
honderdachtenzestig
tweehonderdvijf

[B.T.W. the ij is counted as i+j, to distinguish it from y].

There are some peculiarities in Dutch. On the left side of the = sign the value of the number and on the right the way we express this (normally):

Numbers of the form 10^k with a special word to denote 10^k have no initial 1 when counting:

However:

2000=2 1000  (the same as in English)

Multiples of 100 and 1000 are treated differently:

1099= 1000 99
1199= 11 100 99 instead of 1000 199 (children might do it this way)

However some people might express 1000000 as 1 1000000 especially if one has won the lottery. I wouldn’t mind.

I’m especially interested in French, since the way they count have some interesting properties.

For instance:

70 =60+10      soixante dix
80 = 4.20        quatre vingts
90 = 4.20+10:  quatre vingt dix

Where does this originate from?

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