#256 - Tatami-Free Rooms
Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
Assuming that the only type of available tatami has dimensions \(1 \times 2\), there are obviously some limitations for the shape and size of the rooms that can be covered.
For this problem, we consider only rectangular rooms with integer dimensions \(a, b\) and even size \(s = a \cdot b\).
We use the term 'size' to denote the floor surface area of the room, and - without loss of generality - we add the condition \(a \le b\).
There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet.
For example, consider the two arrangements below for a \(4 \times 4\) room:
The arrangement on the left is acceptable, whereas the one on the right is not: a red "X" in the middle, marks the point where four tatami meet.
Because of this rule, certain even-sized rooms cannot be covered with tatami: we call them tatami-free rooms.
Further, we define \(T(s)\) as the number of tatami-free rooms of size \(s\).
The smallest tatami-free room has size \(s = 70\) and dimensions \(7 \times 10\).
All the other rooms of size \(s = 70\) can be covered with tatami; they are: \(1 \times 70\), \(2 \times 35\) and \(5 \times 14\).
Hence, \(T(70) = 1\).
Similarly, we can verify that \(T(1320) = 5\) because there are exactly \(5\) tatami-free rooms of size \(s = 1320\):
\(20 \times 66\), \(22 \times 60\), \(24 \times 55\), \(30 \times 44\) and \(33 \times 40\).
In fact, \(s = 1320\) is the smallest room-size \(s\) for which \(T(s) = 5\).
Find the smallest room-size \(s\) for which \(T(s) = 200\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=256. Published Saturday, 19th September 2009, 01:00 am. Solved by 895 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:
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: #814 · #202 · #240
Concepts: combinatorics geometry brute-force-reduction
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Understand the problem
- What exactly is the input to problem 256? Is it a bound (200), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer.
- 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 T(s), T(70) 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 200?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 200 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 = 200, 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, consider the two arrangements below for a 4 x 4 room: The arrangement on the left is acceptable, whereas the one on the right is not: a red " X " in the middle, marks the point where four tatami meet.") - then run it. A surprise here is worth more than an hour of debugging later.
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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 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 200 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: #814 · #202 · #240
- 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.