The O(n) Club: Integer Break — How to Multiply Your Happiness (and Numbers)
⚡ TL;DR
Split your number n into at least two positive integers. Maximize the product? Think dynamic programming or just spam 3s if you want to show off. Quick Java for the curious:🧠 How Devs Usually Mess This Up
We’ve all been that developer: break everything into 1s (hope is free, right?)—then watch your product stubbornly stay at 1. Or you return n itself for n=3, even though the rules say ‘actually break it, please.’ Interviewers sigh, recruiters cry, nobody’s happy. Bonus points for panicking and hardcoding some random cases at midnight.
🔍 What’s Actually Going On
This is the playground of greedy meets DP. Splitting into too many pieces (like all 2s or worse, all 1s) is like trying to cut a pizza into atomic slices—sad and hopeless. The sweet spot? Threes. Chop off as many 3s as you can, unless you’re stuck with leftovers that are, well, socially awkward—like 2 or 4. The magical math behind it? Multiplying 3s gives you quicker exponential growth than lower numbers. It’s not magic, it’s just algebra’s slightly drunk cousin, optimization.
🛠️ PseudoCode
- Initialize: Set
dp[1]=1(it can’t break but we’re polite). - For each i from 2 to n:
- For every
jin1 to (i-1)(try every first break):- Option one: don’t break further:
j * (i-j) - Option two: break the rest:
j * dp[i-j] - Set
dp[i]to the maximum you get.
- Option one: don’t break further:
- For every
- Return
dp[n]and take a victory lap.
