UNIQUE NUMBERS |
If a number A_{n} consisting of n consecutive digits in ascending order is subtracted from the number A_{n}'_{ }obtained by reversing the digits of A_{n, }then the difference is always a constant. This constant is termed as the Unique number U_{n }as reported by me earlier in [1].
For example, a 3-digit number 345 if subtracted from its reverse 543, yields a difference of 198. Thus U_{3 }= 198. Another 3-digit number, say, 678 if subtracted from its reverse 876 will also yield the same difference, that is, 198. Thus for any number consisting of 3 consecutive digits, the Unique number U_{3 }is always 198. Similarly for a number consisting of 4 consecutive digits, the Unique number U_{4 }= 3087. Given below is a table of Unique numbers from U_{2 }to U_{10 }(U_{1} = 0).
U_{2} |
= |
09 |
U_{3} |
= |
198 |
U_{4} |
= |
3087 |
U_{5} |
= |
41976 |
U_{6} |
= |
530865 |
U_{7} |
= |
6419754 |
U_{8} |
= |
75308643 |
U_{9} |
= |
864197532 |
U_{10} |
= |
9753086421 |
A glance at the table will reveal the following fascinating characteristics of Unique numbers:
U_{2 }– U_{1} |
= |
09 |
U_{3}– U_{2} |
= |
189 |
U_{4}– U_{3} |
= |
2889 |
U_{5}– U_{4} |
= |
38889 |
U_{6}– U_{5} |
= |
488889 |
U_{7}– U_{6} |
= |
5888889 |
U_{8}– U_{7} |
= |
68888889 |
U_{9}– U_{8} |
= |
788888889 |
U_{10 }– U_{9} |
= |
8888888889 |
It can be seen that the first digit of all numbers gradually increases from 0 to 8, the last digit is 9 and the remaining digits are 8.
All the above properties were reported earlier in [1].
Let U_{n}' denote the number obtained from a Unique number U_{n} by writing its decimal digits in reverse order. For example U_{3 }= 198, so U_{3}'_{ }= 891. The following interesting pattern is obtained by summing U_{n }and U_{n}'.
U_{3}+ U_{3}' |
= |
1089 |
U_{4}+ U_{4}' |
= |
10890 |
U_{5}+ U_{5}' |
= |
109890 |
U_{6}+ U_{6}' |
= |
1098900 |
U_{7}+ U_{7}' |
= |
10998900 |
U_{8}+U_{8}' |
= |
109989000 |
U_{9}+ U_{9}' |
= |
1099989000 |
U_{10 }+ U_{10}' |
= |
10999890000 |
Abhinav Sharma vide his email dated 22-02-2015 informed that If we divide the difference of two consecutive Unique numbers by 9,
that is, (U_{n+1 }- U_{n })/9, we get the following interesting pattern.
(U_{2 }– U_{1})/9 |
= |
1 |
(U_{3}– U_{2})/9 |
= |
21 |
(U_{4}– U_{3})/9 |
= |
321 |
(U_{5}– U_{4})/9 |
= |
4321 |
(U_{6}– U_{5})/9 |
= |
54321 |
(U_{7}– U_{6})/9 |
= |
654321 |
(U_{8}– U_{7})/9 |
= |
7654321 |
(U_{9}– U_{8})/9 |
= |
87654321 |
(U_{10}– U_{9})/9 |
= |
987654321 |
Relation of Unique numbers with Kaprekar Constant:
If 4-digit Kaprekar constant is denoted by K_{4} i.e. 6174 and the reverse of K_{4} by K_{4}' i.e. 4716 then it can be noted that U_{4}+ U_{4}' = K_{4}+ K_{4}' i.e.
3087 + 7803 = 10890 = 6174 + 4716 |
Similarly for 3-digit Kaprekar constant, we get K_{3} = 495 and K_{3}' = 594, So
It can be noted that U_{3}+ U_{3}' = K_{3}+ K_{3}' i.e.
198 + 891 = 1089 = 495 + 594 |
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[1] Unique Numbers, S. S. Gupta, Science Today, January 1988, India.
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