Playing with digits: supercells, palindromes, Kaprekar’s routine and the Collatz sequence, digit sums, and estimating.
Key ideas
A supercell holds a number bigger than all its neighbours.
A palindrome reads the same both ways: 121, 4334. Reversing a number and adding often leads to one.
Kaprekar’s routine: arrange a 4-digit number’s digits largest-first and smallest-first and subtract; repeating always reaches 6174 (if the digits are not all the same).
Collatz: if a number is even, halve it; if odd, multiply by 3 and add 1. Every number tried so far reaches 1.
Estimate by rounding, then check whether the exact answer is close.
Remember
Kaprekar step
largest arrangement − smallest arrangement
Kaprekar’s constant
6174 (7641 − 1467 = 6174)
Collatz rule
even → n ÷ 2; odd → 3n + 1
Watch out
Dropping a leading zero in Kaprekar’s routine: if a step gives 999, keep it as 0999 so it still has four digits.
Stopping a sequence one step too early.
Calling a number a palindrome when only its first and last digits match.
Worked example
Start Kaprekar's routine with 5320. Write each subtraction until you reach 6174.