[ September 15, 2008 ]
Hopping Numerals
B.S. Rangaswamy (email)
I got the following astonishing figures from my 1993 diary. I am not
quite aware as to how I got them or how I arrived at these numbers.
Four fold product 
Hopping of last numeral to first place
A = 0 2 5 6 4 1
4A = 1 0 2 5 6 4
Eight fold product Hopping of last numeral to first place
E = 0 1 2 6 5 8 2 2 7 8 4 8 1
8E = 1 0 1 2 6 5 8 2 2 7 8 4 8
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Now, I have developed these further and arrived at the following hops :
Four fold products Hopping of last numeral to first place
[ A ]
A = 0 2 5 6 4 1 A = start
4A = 1 0 2 5 6 4
16A = 4 1 0 2 5 6
[ B ]
B = 0 5 1 2 8 2 B = 2A
4B = 2 0 5 1 2 8
16B = 8 2 0 5 1 2
[ C ]
C = 0 7 6 9 2 3 C = 3A
4C = 3 0 7 6 9 2
[ D ]
D = 2 3 0 7 6 9 D = 9A
4D = 9 2 3 0 7 6
Eight fold products Hopping of last numeral to first place
[ E ]
E = 0 1 2 6 5 8 2 2 7 8 4 8 1 E = start
8E = 1 0 1 2 6 5 8 2 2 7 8 4 8
[ F ]
F = 0 2 5 3 1 6 4 5 5 6 9 6 2 F = 2E
8F = 2 0 2 5 3 1 6 4 5 5 6 9 6
[ G ]
G = 0 5 0 6 3 2 9 1 1 3 9 2 4 G = 4E
8G = 4 0 5 0 6 3 2 9 1 1 3 9 2
[ H ]
H = 0 7 5 9 4 9 3 6 7 0 8 8 6 H = 6E
8H = 6 0 7 5 9 4 9 3 6 7 0 8 8
[ J ]
J = 1 0 1 2 6 5 8 2 2 7 8 4 8 J = 8E
8J = 8 1 0 1 2 6 5 8 2 2 7 8 4
[ K ]
K = 1 1 3 9 2 4 0 5 0 6 3 2 9 K = 9E
8K = 9 1 1 3 9 2 4 0 5 0 6 3 2
This phenomenon can also be seen in 2/3 digit numbers,
as illustrated below :
[ L ]
L = 0 2 7
26L = 7 0 2
[ M ]
M = 0 3 7
19M = 7 0 3
This can be seen in 3/4 digits numbers also, resulting
in ten fold products, though this is obvious but obscure.
[ N ]
N = 0 1 0 1 0 2 0 2 - - 0 9 0 9
10N = 1 0 1 0 2 0 2 0 - - 9 0 9 0
This can be observed in 4/5 digits numbers also,
resulting in 37 fold products.
[ P ]
P = 0 1 0 8 4
37P = 4 0 1 0 8
P = 0 1 3 5 5
37P = 5 0 1 3 5
P = 0 1 6 2 6
37P = 6 0 1 6 2
P = 0 1 8 9 7
37P = 7 0 1 8 9
P = 0 2 1 6 8
37P = 8 0 2 1 6
P = 0 2 4 3 9
37P = 9 0 2 4 3
It is seen that, all these 4/5 digit numbers are multiples of 271.
Apart from the fourfold products illustrated in 5/6 digit series,
there exist many more multifold products as in :
[ Q ]
Q = 0 1 0 1 0 1 - - - 0 9 0 9 0 9
10Q = 1 0 1 0 1 0 - - - 9 0 9 0 9 0
Q = 0 1 5 8 7 3 0 2 1 1 6 4
19Q = 3 0 1 5 8 7 4 0 2 1 1 6
Q = 0 2 6 4 5 5 0 3 1 7 4 6
19Q = 5 0 2 6 4 5 6 0 3 1 7 4
Q = 0 3 7 0 3 7 0 4 2 3 2 8
19Q = 7 0 3 7 0 3 8 0 4 2 3 2
Q = 0 4 7 6 1 9
19Q = 9 0 4 7 6 1
Q = 0 2 7 0 2 7
26Q = 7 0 2 7 0 2
Q = 0 1 1 6 5 5 0 1 3 9 8 6
43Q = 5 0 1 1 6 5 6 0 1 3 9 8
Q = 0 1 6 3 1 7 0 1 8 6 4 8
43Q = 7 0 1 6 3 1 8 0 1 8 6 4
Q = 0 2 0 9 7 9
43Q = 9 0 2 0 9 7
Q = 0 1 2 9 8 7
54Q = 7 0 1 2 9 8
It is a wonder that all (except 26Q) these 5/6 digit numbers
are multiples of 37.
There can be more bewildering features in 6/7, 7/8 --- digit series,
which Time and Conscious Effort shall reveal.
❄ ❄
Here is a lone Gem Hop in 6/7 digit series :
R = 0 2 9 2 8 8 7
24R = 7 0 2 9 2 8 8
Obscure lot in 7/8 digit series :
S = 0 1 0 1 0 1 0 1 - - - 0 9 0 9 0 9 0 9
10S = 1 0 1 0 1 0 1 0 - - - 9 0 9 0 9 0 9 0
Rare Gem Hops in 7/8 digit series :
S = 0 1 3 6 9 8 6 3 S = start
22S = 3 0 1 3 6 9 8 6
S1 = 0 2 7 3 9 7 2 6 S1 = 2S
22S = 6 0 2 7 3 9 7 2
S2 = 0 4 1 0 9 5 8 9 S2 = 3S
22S = 9 0 4 1 0 9 5 8
S3 = 0 3 1 9 6 3 4 7 S3 = 2,333...S
22S = 7 0 3 1 9 6 3 4
S4 = 0 3 6 5 2 9 6 8 S4 = 2,666...S
22S = 8 0 3 6 5 2 9 6
Only three Hops in the 8/9 digit series :
T = 0 1 2 3 4 5 6 7 9
73T = 9 0 1 2 3 4 5 6 7
T = 0 ? ? ? ? ? ? ? ?
..T = ? ? ? ? ? ? ? ? ?
T = 0 ? ? ? ? ? ? ? ?
..T = ? ? ? ? ? ? ? ? ?
The first one is a Pearl !
Last two of these hops are intended for the involved reader to try and
arrive at these numerals and experience the thrill and joy of discovery.
Six Hopping Numerals are found in the 9/10 digit series :
U = 0 1 0 8 4 0 1 0 8 4
37U = 4 0 1 0 8 4 0 1 0 8
U = 0 1 3 5 5 0 1 3 5 5
37U = 5 0 1 3 5 5 0 1 3 5
U = 0 1 6 2 6 0 1 6 2 6
37U = 6 0 1 6 2 6 0 1 6 2
U = 0 1 8 9 7 0 1 8 9 7
37U = 7 0 1 8 9 7 0 1 8 9
U = 0 2 1 6 8 0 2 1 6 8
37U = 8 0 2 1 6 8 0 2 1 6
U = 0 2 4 3 9 0 2 4 3 9
37U = 9 0 2 4 3 9 0 2 4 3
plus an obscure lot also in these 9/10 digit series :
U = 0 1 0 1 0 1 0 1 0 1 - - - 0 9 0 9 0 9 0 9 0 9
10U = 1 0 1 0 1 0 1 0 1 0 - - - 9 0 9 0 9 0 9 0 9 0
When Zero hops in any number, result is only ten percent of its prior value.
Excluding numbers ending with zero, there are 8100 million 10 digit numbers.
I was unable to find even a single Hop Number in this galaxy of ten digit numbers.
Details of a few numbers which have missed the requirement just by a whisker are
illustrated below:
| Number | 1026957638 | 5504587156 | 4697986577 |
| Multiple | 78 | 11 | 15 |
| Product | 80102695764 | 60550458716 | 70469798655 |
| Hop Number | 80102695763 | 60550458715 | 70469798657 |
| Variance | 1 | 1 | 2 |
I was able to arrive at thirteen Hop Numbers apart from obscure lot
in 11/12 digit series as below :
V = 0 1 0 1 0 1 0 1 0 1 0 1 - - - 0 9 0 9 0 9 0 9 0 9 0 9
10V = 1 0 1 0 1 0 1 0 1 0 1 0 - - - 9 0 9 0 9 0 9 0 9 0 9 0
V = 0 1 0 5 8 2 0 1 0 5 8 2
19V = 2 0 1 0 5 8 2 0 1 0 5 8
V = 0 1 1 5 8 3 0 1 1 5 8 3
26V = 3 0 1 1 5 8 3 0 1 1 5 8
V = 0 1 1 6 5 5 0 1 1 6 5 5
43V = 5 0 1 1 6 5 5 0 1 1 6 5
V = 0 1 5 8 7 3 0 1 5 8 7 3
19V = 3 0 1 5 8 7 3 0 1 5 8 7
V = 0 1 5 4 4 4 0 1 5 4 4 4
26V = 4 0 1 5 4 4 4 0 1 5 4 4
V = 0 1 9 3 0 5 0 1 9 3 0 5
26V = 5 0 1 9 3 0 5 0 1 9 3 0
V = 0 2 3 1 6 6 0 2 3 1 6 6
26V = 6 0 2 3 1 6 6 0 2 3 1 6
V = 0 2 6 4 5 5 0 2 6 4 5 5
19V = 5 0 2 6 4 5 5 0 2 6 4 5
V = 0 2 7 0 2 7 0 2 7 0 2 7
26V = 7 0 2 7 0 2 7 0 2 7 0 2
V = 0 3 7 0 3 7 0 3 7 0 3 7
19V = 7 0 3 7 0 3 7 0 3 7 0 3
V = 0 3 0 8 8 8 0 3 0 8 8 8
26V = 8 0 3 0 8 8 8 0 3 0 8 8
V = 0 3 4 7 4 9 0 3 4 7 4 9
26V = 9 0 3 4 7 4 9 0 3 4 7 4
V = 0 4 7 6 1 9 0 4 7 6 1 9
19V = 9 0 4 7 6 1 9 0 4 7 6 1
❄ ❄
I admire and owe my deep gratitude to the 'Anamika' (unknown person)
who coined the Hopping Numeral and the 12/13 digit numbers,
as illustrated at E above.
Twelve Hops in 12/13 series :
W = 0 1 2 5 7 8 6 1 6 3 5 2 2
16W = 2 0 1 2 5 7 8 6 1 6 3 5 2
W = 0 2 5 3 1 6 4 5 5 6 9 6 2 see also intro
8W = 2 0 2 5 3 1 6 4 5 5 6 9 6
W = 0 1 8 8 6 7 9 2 4 5 2 8 3
16W = 3 0 1 8 8 6 7 9 2 4 5 2 8
W = 0 3 7 9 7 4 6 8 3 5 4 4 3
8W = 3 0 3 7 9 7 4 6 8 3 5 4 4
W = 0 2 5 1 5 7 2 3 2 7 0 4 4
16W = 4 0 2 5 1 5 7 2 3 2 7 0 4
W = 0 5 0 6 3 2 9 1 1 3 9 2 4
8W = 4 0 5 0 6 3 2 9 1 1 3 9 2
W = 0 6 3 2 9 1 1 3 9 2 4 0 5 see also intro
8W = 5 0 6 3 2 9 1 1 3 9 2 4 0
W = 0 7 5 9 4 9 3 6 7 0 8 8 6 see also intro
8W = 6 0 7 5 9 4 9 3 6 7 0 8 8
W = 0 4 4 0 2 5 1 5 7 2 3 2 7
16W = 7 0 4 4 0 2 5 1 5 7 2 3 2
W = 0 8 8 6 0 7 5 9 4 9 3 6 7
8W = 7 0 8 8 6 0 7 5 9 4 9 3 6
W = 0 5 0 3 1 4 4 6 5 4 0 8 8
16W = 8 0 5 0 3 1 4 4 6 5 4 0 8
W = 0 5 6 6 0 3 7 7 3 5 8 4 9
16W = 9 0 5 6 6 0 3 7 7 3 5 8 4
Seven Hops in 13/14 series :
X = 0 1 2 5 5 2 3 0 1 2 5 5 2 3
24X = 3 0 1 2 5 5 2 3 0 1 2 5 5 2
X = 0 1 6 7 3 6 4 0 1 6 7 3 6 4
24X = 4 0 1 6 7 3 6 4 0 1 6 7 3 6
X = 0 2 0 9 2 0 5 0 2 0 9 2 0 5
24X = 5 0 2 0 9 2 0 5 0 2 0 9 2 0
X = 0 2 5 1 0 4 6 0 2 5 1 0 4 6
24X = 6 0 2 5 1 0 4 6 0 2 5 1 0 4
X = 0 2 9 2 8 8 7 0 2 9 2 8 8 7
24X = 7 0 2 9 2 8 8 7 0 2 9 2 8 8
X = 0 3 3 4 7 2 8 0 3 3 4 7 2 8
24X = 8 0 3 3 4 7 2 8 0 3 3 4 7 2
X = 0 3 7 6 5 6 9 0 3 7 6 5 6 9
24X = 9 0 3 7 6 5 6 9 0 3 7 6 5 6
Fourteen Hop Numbers in 14/15 series :
Y = 0 1 0 7 5 2 6 8 8 1 7 2 0 4 3
28Y = 3 0 1 0 7 5 2 6 8 8 1 7 2 0 4
Y = 0 1 0 8 4 0 1 0 8 4 0 1 0 8 4
37Y = 4 0 1 0 8 4 0 1 0 8 4 0 1 0 8
Y = 0 1 4 3 3 6 9 1 7 5 6 2 7 2 4
28Y = 4 0 1 4 3 3 6 9 1 7 5 6 2 7 2
Y = 0 1 3 5 5 0 1 3 5 5 0 1 3 5 5
37Y = 5 0 1 3 5 5 0 1 3 5 5 0 1 3 5
Y = 0 1 7 9 2 1 1 4 6 9 5 3 4 0 5
28Y = 5 0 1 7 9 2 1 1 4 6 9 5 3 4 0
Y = 0 1 6 2 6 0 1 6 2 6 0 1 6 2 6
37Y = 6 0 1 6 2 6 0 1 6 2 6 0 1 6 2
Y = 0 2 1 5 0 5 3 7 6 3 4 4 0 8 6
28Y = 6 0 2 1 5 0 5 3 7 6 3 4 4 0 8
Y = 0 1 8 9 7 0 1 8 9 7 0 1 8 9 7
37Y = 7 0 1 8 9 7 0 1 8 9 7 0 1 8 9
Y = 0 2 5 0 8 9 6 0 5 7 3 4 7 6 7
28Y = 7 0 2 5 0 8 9 6 0 5 7 3 4 7 6
Y = 0 2 7 0 2 7 0 2 7 0 2 7 0 2 7 Note the nice
26Y = 7 0 2 7 0 2 7 0 2 7 0 2 7 0 2 "702" pattern !
Y = 0 2 1 6 8 0 2 1 6 8 0 2 1 6 8
37Y = 8 0 2 1 6 8 0 2 1 6 8 0 2 1 6
Y = 0 2 8 6 7 3 8 3 5 1 2 5 4 4 8
28Y = 8 0 2 8 6 7 3 8 3 5 1 2 5 4 4
Y = 0 2 4 3 9 0 2 4 3 9 0 2 4 3 9
37Y = 9 0 2 4 3 9 0 2 4 3 9 0 2 4 3
Y = 0 3 2 2 5 8 0 6 4 5 1 6 1 2 9
28Y = 9 0 3 2 2 5 8 0 6 4 5 1 6 1 2
Ten Hop Numbers in 15/16 digit series :
Z = 0 1 3 6 9 8 6 3 0 1 3 6 9 8 6 3
22Z = 3 0 1 3 6 9 8 6 3 0 1 3 6 9 8 6
Z = 0 1 8 2 6 4 8 4 0 1 8 2 6 4 8 4
22Z = 4 0 1 8 2 6 4 8 4 0 1 8 2 6 4 8
Z = 0 2 2 8 3 1 0 5 0 2 2 8 3 1 0 5
22Z = 5 0 2 2 8 3 1 0 5 0 2 2 8 3 1 0
Z = 0 1 3 0 7 1 8 9 5 4 2 4 8 3 6 6
46Z = 6 0 1 3 0 7 1 8 9 5 4 2 4 8 3 6
Z = 0 2 7 3 9 7 2 6 0 2 7 3 9 7 2 6
22Z = 6 0 2 7 3 9 7 2 6 0 2 7 3 9 7 2
Z = 0 3 1 9 6 3 4 7 0 3 1 9 6 3 4 7
22Z = 7 0 3 1 9 6 3 4 7 0 3 1 9 6 3 4
Z = 0 5 8 8 2 3 5 2 9 4 1 1 7 6 4 7
12Z = 7 0 5 8 8 2 3 5 2 9 4 1 1 7 6 4
Z = 0 3 6 5 2 9 6 8 0 3 6 5 2 9 6 8
22Z = 8 0 3 6 5 2 9 6 8 0 3 6 5 2 9 6
Z = 0 1 9 6 0 7 8 4 3 1 3 7 2 5 4 9
46Z = 9 0 1 9 6 0 7 8 4 3 1 3 7 2 5 4
Z = 0 4 1 0 9 5 8 9 0 4 1 0 9 5 8 9
22Z = 9 0 4 1 0 9 5 8 9 0 4 1 0 9 5 8
I do not intend to go beyond this series, in view of Jurjen N.E. Bos' notings below
( email from [ Nov 10, 2008 ] ).
Some theory :
Write the hopping equation as k*(a*10+b) = b*10^(n-1)+a
(b is one digit, a is the rest, n is the length and k is the factor).
E.g.: 4 * 25641 = 102564
k = 4, a = 2564, b = 1, and n = 6.
Now multiply both ends of the equation with 10, and subtract 10a+b :
10*k*(a*10+b)-10a-b = 10*(b*10^(n-1)+a)-10a-b
simplifying to
b*(10k-1)*(a*10+b) = 10^n-1
This implies that the number 10^n-1 has to be written as the product
of three factors :
b
10k-1
a*10+b
Furthermore, we know (a*10+b)*k>10^(n-1), so that b must be 1.
So the recipe is :
Factor the number 10^n-1.
Group the factors to produce a factor ending in 9, call this 10k-1.
Compute (10^n-1)/(10k-1) and call this 10a+b.
Here's the result up to n=24 (the first line is the program in PARI).
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I wish to share these astonishing features of Hopping Numerals with
you and the readers of your popular website World!Of Numbers.
A000174 