STEP 3 · ARRAYS · HASHING · EASY

Two Sum

WHAT IT SAYS

Return indices of two numbers that add up to a target.

WHAT IT'S REALLY ASKING

"As you walk the array, have you already seen the number that completes me?"

THE INSIGHT LADDER — FROM BRUTE FORCE TO OPTIMAL

Brute force: try every pair

O(n²) time · O(1) space

Two nested loops — for each element, scan the rest of the array hoping to find its partner.

WHERE THE WORK IS WASTED — For every element you re-walk territory you've already covered.

KEY OBSERVATION — THE UNLOCKlink

You are not searching. You are remembering.

The partner of nums[i] is fully determined: target − nums[i]. A question whose answer you can compute is a lookup, not a search. Lookups want a hash map.

One pass with a ledger

O(n) time · O(n) space

Before recording each number, ask the map for target − x. The map looks back so you never have to.

WHAT YOU TRADED — You bought O(1) recall with O(n) memory — space for the right to never re-scan.

WATCH THE IDEA RUN

need 7

found no

Need 7. Never seen it — record 5, move on.

step 1 / 5

THE PATTERN — SO YOU RECOGNIZE IT NEXT TIME

Search → Lookup

YOU'LL SEE IT AGAIN WHEN

You're repeatedly scanning for a value
That value is computable in O(1) from the current element

SAME BLUEPRINT, DIFFERENT PROBLEM

Subarray Sum Equals KLongest Consecutive Sequence

The bar isn't "solved it once." It's "could rebuild it from the observation."