Like Kaprekar's constant from before, there are more numbers where repeating a calculation using only the digits reveals a surprising rule. Meet palindromes, which read the same forwards and backwards; happy numbers, which eventually reach 1 when you keep squaring and adding their digits; and narcissistic numbers, whose digits raised to a power sum back to the number itself.
A palindrome is a number like 121 that reads the same from front to back as it does from back to front. A happy number is a number that eventually reaches 1 when you repeatedly square each digit and add them up; if it doesn't, it gets stuck in a loop cycling through 4-16-37-58-89-145-42-20. A narcissistic number is a number like 153 = 1³+5³+3³, where raising each digit to the power of the digit count and adding them together gives back the number itself.
Numbers like 121 and 3553, which read the same from front to back as from back to front, are called palindromes. Some, like 727, are even both a palindrome and prime.
Try repeatedly squaring each digit and adding them up. If you eventually reach 1, it's a happy number; if you get stuck in the loop 4-16-37-58-89-145-42-20, it's an unhappy number.
153 has 3 digits, so raise each digit to the 3rd power and add them: 1³+5³+3³ = 1+125+27 = 153 — exactly itself!