Try the problem
See what you can do with just a blank slate.
ABOUT
About Markov Labs
Curated problems grounded in foundational ideas. Markov Labs is a free educational project.
PROBLEMS
Build intuition by working through it
Practice probability, expectation, games, logic, and strategy to learn structure, not recall. Start with the problem, use hints only when you need them, and review the path after you solve.
Check your answer
With support for both numerical and algebraic formats.
Hints and solution
Expose one thought at a time so you can get past a block without revealing the whole path.
Review the structure
Read the solution to connect the problem to the underlying idea.
* Revealing a hint or solution step can still leave the problem marked correct, but not flawless; the solution links to the reference section behind the work.
For example:
Matching dice
You roll a fair die twice, what is the probability that both rolls show the same number?
Row of Coins
Five fair coins are initially all heads. Every second, one coin is chosen uniformly at random and tossed. What is the expected time until all five coins are heads again?
Reference
Patterns worth knowing.
Notes, not a textbook
Intuition doesn’t always require derivation. Editorial notes separate the quick idea from the full argument.
Linked problems
Each reference page points back to related practice, making the map from technique to problem explicit.
Build an instinct
Focus on patterns, heuristics, shortcuts, and approximations that help you recognize structure on the fly.
A useful snippet
* In an undirected Markov Chain, the stationary mass of a state is proportional to its weighted degree. As seen below, each state in this chain can transition to exactly five other states, each differing in a single coin, and has the same self-loop probability. As such, every state has the same weighted degree, and the stationary distribution is uniform. Given the 25 total states, the expected return time to any given state, including all heads, is 32.
TOPICS Β· EXPLORE Β· DAILY
Find a place to start.
Browse by structure, daily rhythm, or what the catalog is surfacing now.
Topic pages
Open organized sections and the problems that belong to each idea.
Daily problem
A clean problem of the day when you want one sharp place to start.
Trending problems
Rotates toward recent activity, low solve rates, and other problems with unusual signal.
CHAIN GENERATOR
Build and inspect Markov chains visually.
Build chains visually or describe them with the custom Chain Language. Switch between Cartesian and polar layouts, validate transitions, inspect matrices, save models, and analyze stationary behavior.
Working from a larger state model? Share the Chain Language guide and your transition rules with an AI assistant to generate a first-pass input. Paste it into the editor, validate the chain, and refine it visually.
CONTRIBUTE
Add a question. Flag a mistake.
If you think we made a mistake or have a question you would like us to add, head over to Contribute. Send us a problem worth including, or tell us when something isnβt right.