#376 - Nontransitive Sets of Dice
Consider the following set of dice with nonstandard pips:
Die \(A\): \(1\) \(4\) \(4\) \(4\) \(4\) \(4\)
Die \(B\): \(2\) \(2\) \(2\) \(5\) \(5\) \(5\)
Die \(C\): \(3\) \(3\) \(3\) \(3\) \(3\) \(6\)
A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
If the first player picks die \(A\) and the second player picks die \(B\) we get
\(P(\text{second player wins}) = 7/12 \gt 1/2\).
If the first player picks die \(B\) and the second player picks die \(C\) we get
\(P(\text{second player wins}) = 7/12 \gt 1/2\).
If the first player picks die \(C\) and the second player picks die \(A\) we get
\(P(\text{second player wins}) = 25/36 \gt 1/2\).
So whatever die the first player picks, the second player can pick another die and have a larger than \(50\%\) chance of winning.
A set of dice having this property is called a nontransitive set of dice.
We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:
- There are three six-sided dice with each side having between \(1\) and \(N\) pips, inclusive.
- Dice with the same set of pips are equal, regardless of which side on the die the pips are located.
- The same pip value may appear on multiple dice; if both players roll the same value neither player wins.
- The sets of dice \(\{A,B,C\}\), \(\{B,C,A\}\) and \(\{C,A,B\}\) are the same set.
For \(N = 7\) we find there are \(9780\) such sets.
How many are there for \(N = 30\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=376. Published Sunday, 18th March 2012, 01:00 am. Solved by 338 members at time of mirroring.
Why this is useful
Probability Statistics. Expectation and state-based probability reasoning underpin pricing, risk, and statistical inference (Phases 7, 11, 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 · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 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: #406 · #560 · #783
Concepts: game-theory probability brute-force-reduction
Likely techniques: hashing
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 376? Is it a bound (30), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- Write out, in your own words, the definition of nontransitive set of dice as the statement gives it. Which integers/objects are excluded by that definition?
- What constraint does the bound 30 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 30?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 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 / hashing - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 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 (the statement says: "we get P(second player wins) = 7/12 > 1/2.") - 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 hashing 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 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 'hashing' 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: #406 · #560 · #783
- 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.