UNIQUE NUMBERS

If a number An consisting of n consecutive digits in ascending order is subtracted from the number An' obtained by reversing the digits of An, then the difference is always a constant. This constant is termed as the Unique number Un 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 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 U3 is 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 U2 to 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:

1. The digital root, that is, the ultimate sum of the digits of any Unique number is 9.

2. From U4 to U10, the first digit of any Unique number Un is n-1 and the sum of the first and last digits is 10. On removing the first and last digits, the remaining number is Un-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 U2 and U3 , we can get subsequent Unique numbers from U4 to U10 .

3. 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.

4. The middle digit of U2n+1 (where n = 1, 2, 3 and 4) is always 9, while the two middle digits of U2n (where n = 2, 3, 4 and 5) are always 08).

5. U9 contains all the digits from 1 to 9 and U10 contains all the digits from 0 to 9.

6. If we take the difference of two consecutive Unique numbers, that is, (Un+1 - Un ), 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 Un' denote the number obtained from a Unique number Un by writing its decimal digits in reverse order. For example U3 = 198, so U3' = 891. The following interesting pattern is obtained by summing Un and Un'.

 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, (Un+1 - Un )/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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