#568 - Reciprocal Games II
Tom has built a random generator that is connected to a row of \(n\) light bulbs. Whenever the random generator is activated each of the \(n\) lights is turned on with the probability of \(\frac 1 2\), independently of its former state or the state of the other light bulbs.
While discussing with his friend Jerry how to use his generator, they invent two different games, they call the reciprocal games:
Both games consist of \(n\) turns. Each turn is started by choosing a number \(k\) randomly between (and including) \(1\) and \(n\), with equal probability of \(\frac 1 n\) for each number, while the possible win for that turn is the reciprocal of \(k\), that is \(\frac 1 k\).
In game A, Tom activates his random generator once in each turn. If the number of lights turned on is the same as the previously chosen number \(k\), Jerry wins and gets \(\frac 1 k\), otherwise he will receive nothing for that turn. Jerry's expected win after playing the total game A consisting of \(n\) turns is called \(J_A(n)\). For example \(J_A(6)=0.39505208\), rounded to \(8\) decimal places.
For each turn in game B, after \(k\) has been randomly selected, Tom keeps reactivating his random generator until exactly \(k\) lights are turned on. After that Jerry takes over and reactivates the random generator until he, too, has generated a pattern with exactly \(k\) lights turned on. If this pattern is identical to Tom's last pattern, Jerry wins and gets \(\frac 1 k\), otherwise he will receive nothing. Jerry's expected win after the total game B consisting of \(n\) turns is called \(J_B(n)\). For example \(J_B(6)=0.43333333\), rounded to \(8\) decimal places.
Let \(D(n)=J_B(n)−J_A(n)\). For example, \(D(6) = 0.03828125\).
Find the \(7\) most significant digits of \(D(123456789)\) after removing all leading zeros.
(If, for example, we had asked for the \(7\) most significant digits of \(D(6)\), the answer would have been 3828125.)
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=568. Published Saturday, 3rd September 2016, 04:00 pm. Solved by 347 members at time of mirroring.
Why this is useful
Direct Quant. Markov/absorbing-state and simulation reasoning is exactly the machinery behind pricing, risk, and execution models (Phases 7, 8, 13).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
13.2 Monte Carlo Estimation and Error Analysis · 13.1 Random-Number Generation and Sampling · 19.8 Bit Manipulation and State Compression · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 2.4 Taylor Series and Local Approximation · 5.1 Floating-Point Arithmetic, Conditioning, and Stability · 7.3 Independence, Conditional Probability, and Conditional Expectation · 7.2 Expectation, Moments, and Key Inequalities · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #481 · #567 · #849
Concepts: game-theory graph-theory numerical-methods probability brute-force-reduction
Likely techniques: bfs-dfs bitmask-dp monte-carlo precision-control
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 568? Is it a bound (123456789), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a real number quoted to a stated precision, so the whole computation must control rounding error.
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What do the arguments of D(6), D(n) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 123456789?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 123456789 and the cost of testing one.
- Which game-theory fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says game-theory / bitmask-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 123456789, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "For example J_A(6)=0.") - then run it. A surprise here is worth more than an hour of debugging later.
Scratchpad
Mathematical notes, formulas, pseudocode, hypotheses, complexity notes. Saved automatically with your progress.
Python workbench
Real Python (Pyodide) in a sandboxed Web Worker - no network, no filesystem, no DOM access. Ctrl/Cmd+Enter runs. Escape leaves the editor. Stop terminates the worker.
Check your answer
Answers are checked against a salted hash held in a separate file - not printed in this page. This prevents accidental spoilers; it is not cryptographic protection (see the build notes).
Progressive hints
Optimization
You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
Which step of your solution were you least confident about, and what evidence would settle it?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the game-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bitmask-dp idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 123456789 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'bitmask-dp' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
Self-assess (mastery is not a correct number)
You reach Mastered only when you have solved it, rated yourself at least Solid across the dimensions, and written a real explanation.
Confidence
Low confidence schedules this problem for spaced review, even if you solved it.
Mastery check
- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #481 · #567 · #849
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.