Symmetrical Single-Digit Representations of Numbers

Consider a mathematical series, f^n (10) = 10^n + 10^(n-1) + ... ... +10^2 + 10^1 + 10^0.

It looks terrible and confusing, but what that basically means is that every number we know is a combination of the powers of ten (thousands, hundreds, tens and ones being the first four among the natural numbers series).

Take 45,231 for example. That's [4 x 10^4] + [5 x 10^3] + [2 x 10^2] + [3 x 10^1] + [1 x 10^0]. Think of it as Primary School Mathematics written in a bizarre way.

40,000 [4 x 10^4] --> 4 x 10,000

+ 5,000 [5 x 10^3] --> 5 x 1,000

+ 200 [2 x 10^2] --> 2 x 100

+ 30 [3 x 10^1] --> 3 x 10

+ 1 [1 x 10^0] --> 1 x 1

_______

45, 231. Does that make it clear?

{A simplified term would be abcde = [a x 10^4] + [b x 10^3] + [c x 10^2] + [d x 10^1] +

[e x 10^0]. It's possible for any of the numbers to be the same, and just as possible otherwise.}

Now, let's throw a curve-ball. Let's create equations where all digits are the same.

3 x f^2 (10) = [3 x 10^2] + [3 x 10^1] + [3 x 10^0] = 300 + 30 + 3 = 333.

Pay attention to the answer: 3 x f^2 (10) = 333.

If the 3 was changed into a 7, the answer would still remain the same, 7 x f^2 (10) = 777.

Changed to 2, and the equation 2 x f^4 (10) = 22222.

Changed to 8, and the equation 8 x f^5 (10) = 888888.

Changed to 9, and the equation 9 x f^0 (10) = 9.

With that said, we can surmise that 'a' x f^n (10) = 'aaaa...a'. The number will be made of a singular digit, and only that one digit.

This is known as a Single Digit Representation. This is a relatively popular series used commonly in various number patterns, and with the trend of exam questions moving up higher and higher it becomes important to understand the underlying concept behind such patterns.

So, for a number pattern with [6 x f^n (10)], the pattern formed will be 66, 666, 6666, 66666, 666666, 6666666... ad infinitum.

But!

Did you know, that these single-digit representations have properties to them? How, for example, some expressions for 'a' have the same answers no matter which digit you use between 1 to 9? Take the following equation for example:

aa - a

a + a

The above example is a fascinating one, because no matter what value 'a' assumes, the answer will always be the same.

11 - 1 = 22 - 2 = 33 - 3 = 44 - 4 = 55 - 5 = 66 - 6 = 77 - 7 = 88 - 8 = 99 - 9

1 + 1 2 + 2 3 + 3 4 + 4 5 + 5 6 + 6 7 + 7 8 + 8 9 + 9

All of the equations above have the same answer. 10/2, 20/4, 30/6, 40/8, they're all equations with different scales that lead to the same answer: 5. These are what we call Symmetrical Relations of Single-Digit Representations.

Some other fascinating examples are as follows:

aa + a aaa - a aaaaa + a aaa

a + a a + a aa + a a + a + a

6 55 926 37

(Credits to Inder J. Taneja for the examples)

It's not a stretch to say that all numbers can be created with single digit representations (if you 1+1+1... enough times to the desired value you'll get it), although finding symmetrical ones are rare. Can you think of other symmetrical Single Digit Representations?

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