#265 - Binary Circles
\(2^N\) binary digits can be placed in a circle so that all the \(N\)-digit clockwise subsequences are distinct.
For \(N=3\), two such circular arrangements are possible, ignoring rotations:
For the first arrangement, the \(3\)-digit subsequences, in clockwise order, are:
\(000\), \(001\), \(010\), \(101\), \(011\), \(111\), \(110\) and \(100\).
Each circular arrangement can be encoded as a number by concatenating the binary digits starting with the subsequence of all zeros as the most significant bits and proceeding clockwise. The two arrangements for \(N=3\) are thus represented as \(23\) and \(29\):
\[\begin{align} 00010111_2 &= 23\\ 00011101_2 &= 29 \end{align}\]Calling \(S(N)\) the sum of the unique numeric representations, we can see that \(S(3) = 23 + 29 = 52\).
Find \(S(5)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=265. Published Saturday, 21st November 2009, 09:00 am. Solved by 4,748 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact counting underlies discrete pricing lattices, scenario enumeration, and combinatorial probability (Phase 7).
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Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.7 Dynamic Programming: Memoization and Tabulation · 19.11 Integer Partitions and Counting Structures · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 3.1 Vectors, Multivariable Functions, and Level Sets · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #29 · #24 · #68
Concepts: combinatorics geometry
Likely techniques: digit-dp hashing
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Understand the problem
- What exactly is the input to problem 265? Is it a bound (52), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- 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 S(N), S(3) 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 52?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 52 and the cost of testing one.
- Which combinatorics 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 combinatorics / digit-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 52, 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?
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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 combinatorics structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the digit-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 52 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 'digit-dp' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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- 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: #29 · #24 · #68
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