#534 - Weak Queens
The classical eight queens puzzle is the well known problem of placing eight chess queens on an \(8 \times 8\) chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of \(92\) distinct configurations can be found for eight queens. The general case asks for the number of distinct ways of placing \(n\) queens on an \(n \times n\) board, e.g. you can find \(2\) distinct configurations for \(n=4\).
Let's define a weak queen on an \(n \times n\) board to be a piece which can move any number of squares if moved horizontally, but a maximum of \(n - 1 - w\) squares if moved vertically or diagonally, \(0 \le w \lt n\) being the "weakness factor". For example, a weak queen on an \(n \times n\) board with a weakness factor of \(w=1\) located in the bottom row will not be able to threaten any square in the top row as the weak queen would need to move \(n - 1\) squares vertically or diagonally to get there, but may only move \(n - 2\) squares in these directions. In contrast, the weak queen is not handicapped horizontally, thus threatening every square in its own row, independently from its current position in that row.
Let \(Q(n,w)\) be the number of ways \(n\) weak queens with weakness factor \(w\) can be placed on an \(n \times n\) board so that no two queens threaten each other. It can be shown, for example, that \(Q(4,0)=2\), \(Q(4,2)=16\) and \(Q(4,3)=256\).
Let \(S(n)=\displaystyle\sum_{w=0}^{n-1} Q(n,w)\).
You are given that \(S(4)=276\) and \(S(5)=3347\).
Find \(S(14)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=534. Published Saturday, 14th November 2015, 01:00 pm. Solved by 381 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.8 Bit Manipulation and State Compression · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 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: #256 · #324 · #762
Concepts: combinatorics geometry brute-force-reduction
Likely techniques: backtracking bitmask-dp 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 534? Is it a bound (3347), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: one extremal object (or the value attached to it), so a search-with-pruning shape is natural.
- Write out, in your own words, the definition of weak queen on an n x n board as the statement gives it. Which integers/objects are excluded by that definition?
- What do the arguments of Q(n,w), Q(4,0) 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 3347?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 3347 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 / bitmask-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 3347, 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: "You are given that S(4)=276 and S(5)=3347.") - 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 combinatorics structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bitmask-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 3347 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 'bitmask-dp' 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: #256 · #324 · #762
- 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.