Reaman Numerals Plural

Sequel to Reaman Numeral.

Copyright 1997 by Paul Niquette. All rights reserved.


 

The integers in the puzzle are all Reaman Numerals with the common multiplier digit of 4.

Each has a distinct do-si-do digit as indicated.


102,564
times 4 equals
410,256 
128,205
times 4 equals
512,820 
153,846
times 4 equals
615,384 
179,487
times 4 equals
717,948 
205,128
times 4 equals
820,512 
230,769
times 4 equals
923,076


As for that 57-digit number in the hint -- why, it is merely another Reaman Numeral.
 

1,016,949,152,542,372,881,355,932,203,389,830,508,474,576,271,186,440,677,966
times 6 equals
6,101,694,915,254,237,288,135,593,220,338,983,050,847,457,627,118,644,067,796
    It has one property in common with the first number in the puzzle; namely, the multiplier digit and the do-si-do digit are the same.

The following equation is a generalized version of the one we derived elsewhere.

      m x = (x - d) / 10 + 10n-1 d , where:
        d = the do-si-do digit (constrained)
        m = the multiplier digit (given)
        n = the number of digits in x (unknown)
        x = the Reaman Numeral (unknown)
Using a few algebraic steps we are able to rearrange our equation so that x winds up all by itself over on the left hand side.
      x = (10n d - d) / (10m - 1)
We will always know the value of the denominator of this fraction, inasmuch as, in finding a Reaman Numeral, m must be given. Here is a handy table of denominators.
 
m =
9
8
7
6
5
4
3
2
10m -1=
89
79
69
59
49
39
29
19

The numerator has the same form for any value of digit d. Thus, the fraction may be expressed as follows:
 
x =
d-1
9
9
9
...
9
10-d
/ (10m - 1)

The algorithm applies four steps...

Step 1. Pick a value for digit d,
Step 2. Put digit d - 1 at the leftmost end of a string of nines,
Step 3. Put digit 10 - d at the rightmost end of that string of nines, and
Step 4. Carry out the indicated long division by 10m - 1.
Here is a handy table of numerators.
 
d = 
9
8
7
6
5
4
3
2
Numerator
89~91
79~92
69~93
59~94
49~95
39~96
29~97
19~98
During the long division in Step 4, we have to keep inserting nines as we go along until the quotient comes out even. As we have seen, that can take quite a long time.

Finally, one correspondent, Don Lauria, has suggested that the Reaman Numerals should be given a symbolic representation as R(d,m), such that, for example, R(4,4) = 102,564, R(5,4) = 128,205, and R(5,5) = 102,040,816,326,530,612,244,897,959,183,673,469,387,755.  So, then, how many Reaman Numerals are there?


Epilog:

Few activities given to mankind carry more solemn responsibilities than those that arise in creating puzzles.  As a courtesy to solvers, both the statement and the solution must be precisely correct.

More than six years went by after the publication of Reaman Numerals Plural, and with no challenges received, one might reasonably assume that the material is error free.  So then several Reaman Numerals were appropriated for what was purported to be the world's first Cross-Number Puzzle, and zowie: a typographical errer was reported and reported and reported...  The mistrake has been corected now, thanks to the allertness of two many solvers to acknowlage.
 


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