Let's analyze the number 111. This is a 3-digit repunit, meaning it consists of repeated units. 111 can be factored as 3 times 37. The sum of its digits is 3. In binary representation, 111 is written as 1101111. An interesting property is that when you square 111, you get 12,321, which is a palindrome - it reads the same forwards and backwards.
Let's explore the fascinating pattern when we square the number 111. The result is 12,321, which is a palindrome - reading the same forwards and backwards. This is part of a beautiful pattern with repunits. When we square 1, we get 1. Squaring 11 gives us 121. Squaring 111 results in 12,321. This pattern continues: 1111 squared equals 1,234,321, and 11,111 squared equals 123,454,321. Each result forms a perfect palindrome with an ascending-descending sequence of digits.
Let's understand why squaring 111 results in a palindrome. First, we can express 111 as the sum of powers of 10: 10 squared plus 10 to the first power plus 10 to the zeroth power. When we square this expression using the binomial theorem, we get a series of terms. Combining like terms, we end up with coefficients that form a symmetric pattern: 1, 2, 3, 2, 1. These coefficients multiply the corresponding powers of 10, giving us 1 times 10 to the fourth, 2 times 10 cubed, 3 times 10 squared, 2 times 10, and 1 times 1. This equals 10000 plus 2000 plus 300 plus 20 plus 1, which is 12,321 - a perfect palindrome.
Let's develop a general formula for repunits and their squares. A repunit is a number consisting of n repeated digit 1s. We can express a repunit with n digits using the formula: R_n equals (10^n minus 1) divided by 9. For example, with n equals 3, we get 111. When we square a repunit, we get (10^n minus 1)^2 divided by 81. This formula generates the palindromic pattern we've observed. Looking at our examples: when n equals 1, R_1 is 1, and its square is 1. When n equals 2, R_2 is 11, and its square is 121. For n equals 3, R_3 is 111, and its square is 12,321. This pattern continues with each repunit square forming a perfect palindrome.
Let's summarize what we've learned about the number 111. First, 111 is a 3-digit repunit, consisting of repeated digit 1s. When squared, 111 produces 12,321, which is a perfect palindrome reading the same forwards and backwards. This is part of a broader pattern where all repunit squares form palindromes. We can express any repunit with n digits using the formula R_n equals (10 to the power of n minus 1) divided by 9. The palindromic structure in the squares comes from the binomial expansion coefficients. This pattern demonstrates how even simple numbers like 111 can reveal beautiful and elegant mathematical structures.