#301 - Nim
Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.
We'll consider the three-heap normal-play version of Nim, which works as follows:
- At the start of the game there are three heaps of stones.
- On each player's turn, the player may remove any positive number of stones from any single heap.
- The first player unable to move (because no stones remain) loses.
If \((n_1,n_2,n_3)\) indicates a Nim position consisting of heaps of size \(n_1\), \(n_2\), and \(n_3\), then there is a simple function, which you may look up or attempt to deduce for yourself, \(X(n_1,n_2,n_3)\) that returns:
- zero if, with perfect strategy, the player about to move will eventually lose; or
- non-zero if, with perfect strategy, the player about to move will eventually win.
For example \(X(1,2,3) = 0\) because, no matter what the current player does, the opponent can respond with a move that leaves two heaps of equal size, at which point every move by the current player can be mirrored by the opponent until no stones remain; so the current player loses. To illustrate:
- current player moves to \((1,2,1)\)
- opponent moves to \((1,0,1)\)
- current player moves to \((0,0,1)\)
- opponent moves to \((0,0,0)\), and so wins.
For how many positive integers \(n \le 2^{30}\) does \(X(n,2n,3n) = 0\) ?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=301. Published Saturday, 11th September 2010, 04:00 pm. Solved by 7,429 members at time of mirroring.
Why this is useful
General Problem Solving. Builds computational thinking, decomposition, and debugging discipline - transferable, without a specific financial application.
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
19.7 Dynamic Programming: Memoization and Tabulation
Concepts: game-theory
Likely techniques: grundy
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 301? Is it a bound (2^30), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- 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 X(1,2,3), X(n,2n,3n) 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 2^30?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 2^30 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 / grundy - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 2^30, 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 - then run it. A surprise here is worth more than an hour of debugging later.
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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 grundy 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 2^30 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 'grundy' 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: explore the same concept filter in the Lab
- 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.