CONCEPT BLOGS
Deep concept tutorials — written the way we teach Multi-phase guides with mental models, worked examples, diagrams, and real edge cases — the public form of Learnisim AI's conceptual learning. Read here; train it interactively in the app.
HOW WE TEACH · SAME ARC AS THE APP
Why → Model → Worked → Practice → Edge cases → Apply
LEVEL All beginner intermediate advanced
TOPIC All advanced algorithm analysis algorithms amortized arrays backtracking beginner bfs binary search binary search trees binary-tree binary-trees bits bitwise bst bucket coding interview coding-interview complexity analysis computer science cycle data structures data-structures deque dfs dp dsa dynamic programming dynamic-programming floyd geometry graph graph-theory graphs greedy greedy algorithms hash map hash set hashmap heap in-place intermediate intervals interview prep interview-prep kadane kmp leetcode linked list matrix mental model monotonic stack optimization performance pointers prefix prefix sum problem solving problem-solving programming fundamentals queue recursion scheduling serialization simulation sliding window sorting stack streaming strings time complexity topo tree trees trie two pointers union-find worked example xor
intermediate 8 min · Oct 8, 2026
Master the n & (n-1) bit-clearing trick by tracing popcounts for 0..5 and Kernighan's algorithm on numbers 7 and 8 step by step.
dsa bits algorithms bitwise optimization
Read article → beginner 7 min · Oct 8, 2026
Learn how XOR bitwise cancellation effortlessly finds the single number in [4, 1, 2, 1, 2] with O(1) space complexity. Step-by-step trace included.
dsa bits xor
Read article → 
![Single number in [4, 1, 2, 1, 2] overview diagram](/blogs/single-number-xor-explained-tracing-4-1-2-1-2/diagrams/hero.svg)