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The 196-Problem

The 196-Problem is one of the biggest unsolved mysteries as far as the mathematics is concerned. The reason behind this mystery is that it has not been proved yet whether 196 is a Lychrel number or non-Lychrel number; a number that produces Palindrome number after repeatedly performing the reverse-then-add operation.

As you can see, the terms – Lychrel number, non-Lychrel number, and Palindrome number have been used above. Before going into detail of The 196-Problem, it is important to know about these terms in order to understand this problem effectively.

Lychrel number

The name Lychrel was coined by computer scientist Wade Van Landingham. Lychrel number is a natural number that do not result in forming numeric palindromic sequence or Palindrome number even after pefroming multiple iterations [n] of the reverse-then-add operation.

Non-Lychrel number

If any natural number result in forming numeric palindromic sequence or Palindrome number after pefroming one or more iterations of the reverse-then-add operation, then it is called non-Lychrel number.

Palindrome number

Palindrome number is a number which is read the same from left to right and right to left. In other words, Palindrome number is a number which remains the same even after reversing the digits.

For example, number 101 is a Palindrome number as this number remains the same even after reversing the digits.

Forming Palindrome Number through Reverse-then-add Operation

For example, take the number 195

Reversing the digits and adding to the original number

195 + 591 => 786 [First Iteration]

Again, reversing the digits and adding to the number formed in First Iteration

786 + 687 => 1473 [Second Iteration]

Again, reversing the digits and adding to the number formed in Second Iteration

1473 + 3741 => 5214 [Third Iteration]

Again, reversing the digits and adding to the number formed in Third Iteration

5214 + 4125 => 9339 [Fourth Iteration]

The end-result 9339 is a Palindrome number formed in four iterations, but this procedure reverse-then-add is to be repeated until Palindrome number is formed.

Now you would have got a brief understanding of the terms – Lychrel number, non-Lychrel number, and Palindrome number.

More About The 196-Problem

A majority of the numbers result into a Palindrome number after one or more iterations of reverse-then-add operation. If any number does not form a Palindrome number after performing multiple iterations of reverse-then-add operation, then it is called Lychrel number, else non-Lychrel number.

For nearly all numbers, if reverse-then-add operation is carried out either one or multiple times, then the end-result is a Palindrome number.

Below are some examples:

Number 15

Reverse-then-add operation: 15 + 51 => 66

Number 25

Reverse-then-add operation: 25 + 52 => 77

Number 57

Reverse-then-add operation: 57 + 75 => 132

132 + 231 => 363

Number 85

Reverse-then-add operation: 85 + 58 => 143

143 + 341 => 484

The vast majority of the numbers form a Palindrome number in one or more iterations after performing reverse-then-add operation. However, when this operation is carried out with the number 196, then a Palindrome number is not formed no matter how many times reverse-then-add operation is performed.

The number 196 is the smallest number that does not lead to form a Palindrome number through reverse-then-add operation. Due to this reason, number 196 may be considered a Lychrel number till it is proved to be a non-Lychrel number, and this problem is known as The 196-Problem.

Hope you have understood The 196-Problem. In case of any issue, kindly provide your valuable comments regarding the same.

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