#863 - Different Dice
Using only a six-sided fair dice and a five-sided fair dice, we would like to emulate an \(n\)-sided fair dice.
For example, one way to emulate a 28-sided dice is to follow this procedure:
- Roll both dice, obtaining integers \(1\le p\le 6\) and \(1\le q\le 5\).
- Combine them using \(r = 5(p-1) + q\) to obtain an integer \(1\le r\le 30\).
- If \(r\le 28\), return the value \(r\) and stop.
- Otherwise (\(r\) being 29 or 30), roll both dice again, obtaining integers \(1\le s\le 6\) and \(1\le t\le 5\).
- Compute \(u = 30(r-29) + 5(s-1) + t\) to obtain an integer \(1\le u\le 60\).
- If \(u>4\), return the value \(((u-5)\bmod 28) + 1\) and stop.
- Otherwise (with \(1\le u\le 4\)), roll the six-sided dice twice, obtaining integers \(1\le v\le 6\) and \(1\le w\le 6\).
- Compute \(x = 36(u-1) + 6(v-1) + w\) to obtain an integer \(1\le x\le 144\).
- If \(x>4\), return the value \(((x-5)\bmod 28) + 1\) and stop.
- Otherwise (with \(1\le x\le 4\)), assign \(u:=x\) and go back to step 7.
The expected number of dice rolls in following this procedure is 2.142476 (rounded to 6 decimal places). Note that rolling both dice at the same time is still counted as two dice rolls.
There exist other more complex procedures for emulating a 28-sided dice that entail a smaller average number of dice rolls. However, the above procedure has the attractive property that the sequence of dice rolled is predetermined: regardless of the outcome, it follows (D5,D6,D5,D6,D6,D6,D6,...), truncated wherever the process stops. In fact, amongst procedures for \(n=28\) with this restriction, this one is optimal in the sense of minimising the expected number of rolls needed.
Different values of \(n\) will in general use different predetermined sequences. For example, for \(n=8\), the sequence (D5,D5,D5,...) gives an optimal procedure, taking 2.083333... dice rolls on average.
Define \(R(n)\) to be the expected number of dice rolls for an optimal procedure for emulating an \(n\)-sided dice using only a five-sided and a six-sided dice, considering only those procedures where the sequence of dice rolled is predetermined. So, \(R(8) \approx 2.083333\) and \(R(28) \approx 2.142476\).
Let \(S(n) = \displaystyle\sum_{k=2}^n R(k)\). You are given that \(S(30) \approx 56.054622\).
Find \(S(1000)\). Give your answer rounded to 6 decimal places.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=863. Published Saturday, 11th November 2023, 07:00 pm. Solved by 281 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:
10.4 Algorithms: Gradient Descent and Newton's Method · 10.1 Convex Sets and Convex Functions · 13.2 Monte Carlo Estimation and Error Analysis · 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 · 19.6 Recurrence Relations and Generating Functions · 2.7 Sequences, Series, Convergence, and Power Series · 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: #352 · #380 · #394
Concepts: dynamic-programming numerical-methods optimization probability sequences-series brute-force-reduction
Likely techniques: precision-control
Learning mode
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Understand the problem
- What exactly is the input to problem 863? Is it a bound (1000), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer. Required format: Give your answer rounded to 6 decimal places.
- 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 R(n), R(8) 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 1000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000 and the cost of testing one.
- Which numerical-methods 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 numerical-methods / precision-control - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000, 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, one way to emulate a 28-sided dice is to follow this procedure: Roll both dice, obtaining integers 1<= p<= 6 and 1<= q<= 5.") - 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 numerical-methods structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the precision-control 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 1000 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 'precision-control' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
Self-assess (mastery is not a correct number)
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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: #352 · #380 · #394
- 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.