#673 - Beds and Desks
At Euler University, each of the \(n\) students (numbered from 1 to \(n\)) occupies a bed in the dormitory and uses a desk in the classroom.
Some of the beds are in private rooms which a student occupies alone, while the others are in double rooms occupied by two students as roommates. Similarly, each desk is either a single desk for the sole use of one student, or a twin desk at which two students sit together as desk partners.
We represent the bed and desk sharing arrangements each by a list of pairs of student numbers. For example, with \(n=4\), if \((2,3)\) represents the bed pairing and \((1,3)(2,4)\) the desk pairing, then students 2 and 3 are roommates while 1 and 4 have single rooms, and students 1 and 3 are desk partners, as are students 2 and 4.
The new chancellor of the university decides to change the organisation of beds and desks: a permutation \(\sigma\) of the numbers \(1,2,\ldots,n\) will be chosen, and each student \(k\) will be given both the bed and the desk formerly occupied by student number \(\sigma(k)\).
The students agree to this change, under the conditions that:
- Any two students currently sharing a room will still be roommates.
- Any two students currently sharing a desk will still be desk partners.
In the example above, there are only two ways to satisfy these conditions: either take no action (\(\sigma\) is the identity permutation), or reverse the order of the students.
With \(n=6\), for the bed pairing \((1,2)(3,4)(5,6)\) and the desk pairing \((3,6)(4,5)\), there are 8 permutations which satisfy the conditions. One example is the mapping \((1, 2, 3, 4, 5, 6) \mapsto (1, 2, 5, 6, 3, 4)\).
With \(n=36\), if we have bed pairing:
\((2,13)(4,30)(5,27)(6,16)(10,18)(12,35)(14,19)(15,20)(17,26)(21,32)(22,33)(24,34)(25,28)\)
and desk pairing
\((1,35)(2,22)(3,36)(4,28)(5,25)(7,18)(9,23)(13,19)(14,33)(15,34)(20,24)(26,29)(27,30)\)
then among the \(36!\) possible permutations (including the identity permutation), 663552 of them satisfy the conditions stipulated by the students.
The downloadable text files beds.txt and desks.txt contain pairings for \(n=500\). Each pairing is written on its own line, with the student numbers of the two roommates (or desk partners) separated with a comma. For example, the desk pairing in the \(n=4\) example above would be represented in this file format as:
1,3 2,4
With these pairings, find the number of permutations that satisfy the students' conditions. Give your answer modulo \(999\,999\,937\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=673. Published Sunday, 2nd June 2019, 07:00 am. Solved by 404 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact counting underlies discrete pricing lattices, scenario enumeration, and combinatorial probability (Phase 7).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.2 Primes, Sieves, and Integer Factorization · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #944 · #860 · #628
Concepts: combinatorics number-theory string-processing
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- What exactly is the input to problem 673? Is it a bound (999999937), 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 modulo 999999937.
- 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 constraint does the bound 999999937 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 999999937?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 999999937 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 - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 999999937, 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, with n=4, if (2,3) represents the bed pairing and (1,3)(2,4) the desk pairing, then students 2 and 3 are roommates while 1 and 4 have single rooms, and students 1 and 3 are desk partners, as are students 2 ...") - then run it. A surprise here is worth more than an hour of debugging later.
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You have a correct answer. That is the start of the learning, not the end.
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- 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 combinatorics structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the combinatorics 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 999999937 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 'combinatorics' 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: #944 · #860 · #628
- 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.