Unique Numbers By Shyam Sunder Gupta

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_nas 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 U₃= 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 U₃is always 198. Similarly for a number consisting of 4 consecutive digits, the Unique number U₄= 3087. Given below is a table of Unique numbers from U₂to U₁₀(U₁ = 0).

U₂	=	09
U₃	=	198
U₄	=	3087
U₅	=	41976
U₆	=	530865
U₇	=	6419754
U₈	=	75308643
U₉	=	864197532
U₁₀	=	9753086421

A glance at the table will reveal the following fascinating characteristics of Unique numbers:

The digital root, that is, the ultimate sum of the digits of any Unique number is 9.
From U₄to U_10,the first digit of any Unique number U_nis n-1 and the sum of the first and last digits is 10. On removing the first and last digits, the remaining number is U_n-2 – 1. For example, in U₅= 41976, the sum of the first and last digits is 4 + 6 = 10, the first digit is n-1 = 5 – 1 = 4. On removing the first and last digits, the remaining number is 197 which is U₃– 1 = 198 –1 = 197. Thus, knowing U₂and U₃, we can get subsequent Unique numbers from U₄to U₁₀.
On removing the top digit in a column, the remaining digits are in the same sequence as in subsequent columns. For example, in the seventh column (from left), if the top digit 3 is removed, the remaining digits 1, 0, 9, 8, 7 and 6 are in the same sequence as in the eighth column.
The middle digit of U_2n+1(where n = 1, 2, 3 and 4) is always 9, while the two middle digits of U_2n(where n = 2, 3, 4 and 5) are always 08).
U₉contains all the digits from 1 to 9 and U₁₀contains all the digits from 0 to 9.
If we take the difference of two consecutive Unique numbers, that is, (U_n+1- U_n), we get the following interesting pattern.

U₂– U₁	=	09
U₃– U₂	=	189
U₄– U₃	=	2889
U₅– U₄	=	38889
U₆– U₅	=	488889
U₇– U₆	=	5888889
U₈– U₇	=	68888889
U₉– U₈	=	788888889
U₁₀– U₉	=	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 U₃= 198, so U₃'= 891. The following interesting pattern is obtained by summing U_nand U_n'.

U₃+ U₃'	=	1089
U₄+ U₄'	=	10890
U₅+ U₅'	=	109890
U₆+ U₆'	=	1098900
U₇+ U₇'	=	10998900
U₈+U₈'	=	109989000
U₉+ U₉'	=	1099989000
U₁₀+ U₁₀'	=	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.

(U₂– U₁)/9	=	1
(U₃– U₂)/9	=	21
(U₄– U₃)/9	=	321
(U₅– U₄)/9	=	4321
(U₆– U₅)/9	=	54321
(U₇– U₆)/9	=	654321
(U₈– U₇)/9	=	7654321
(U₉– U₈)/9	=	87654321
(U₁₀– U₉)/9	=	987654321

Relation of Unique numbers with Kaprekar Constant:

If 4-digit Kaprekar constant is denoted by K₄ i.e. 6174 and the reverse of K₄ by K₄' i.e. 4716 then it can be noted that U₄+ U₄' = K₄+ K₄' i.e.

3087 + 7803 = 10890 = 6174 + 4716

Similarly for 3-digit Kaprekar constant, we get K₃ = 495 and K₃' = 594, So

It can be noted that U₃+ U₃' = K₃+ K₃' 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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