#165 - Intersections
A segment is uniquely defined by its two endpoints.
By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.
Moreover when two segments have exactly one point in common it might be the case that that common point is an endpoint of either one of the segments or of both. If a common point of two segments is not an endpoint of either of the segments it is an interior point of both segments.
We will call a common point \(T\) of two segments \(L_1\) and \(L_2\) a true intersection point of \(L_1\) and \(L_2\) if \(T\) is the only common point of \(L_1\) and \(L_2\) and \(T\) is an interior point of both segments.
Consider the three segments \(L_1\), \(L_2\), and \(L_3\):
- \(L_1\): \((27, 44)\) to \((12, 32)\)
- \(L_2\): \((46, 53)\) to \((17, 62)\)
- \(L_3\): \((46, 70)\) to \((22, 40)\)
It can be verified that line segments \(L_2\) and \(L_3\) have a true intersection point. We note that as the one of the end points of \(L_3\): \((22,40)\) lies on \(L_1\) this is not considered to be a true point of intersection. \(L_1\) and \(L_2\) have no common point. So among the three line segments, we find one true intersection point.
Now let us do the same for \(5000\) line segments. To this end, we generate \(20000\) numbers using the so-called "Blum Blum Shub" pseudo-random number generator.
\[\begin{align} s_0 &= 290797\\ s_{n + 1} &= s_n \times s_n \pmod{50515093}\\ t_n &= s_n \pmod{500} \end{align}\]To create each line segment, we use four consecutive numbers \(t_n\). That is, the first line segment is given by:
\((t_1, t_2)\) to \((t_3, t_4)\).
The first four numbers computed according to the above generator should be: \(27\), \(144\), \(12\) and \(232\). The first segment would thus be \((27,144)\) to \((12,232)\).
How many distinct true intersection points are found among the \(5000\) line segments?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=165. Published Saturday, 27th October 2007, 10:00 am. Solved by 3,108 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:
13.2 Monte Carlo Estimation and Error Analysis · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 7.3 Independence, Conditional Probability, and Conditional Expectation · 7.2 Expectation, Moments, and Key Inequalities · 7.1 Probability Spaces, Random Variables, and Distributions
Concepts: computational-geometry probability brute-force-reduction
Likely techniques: hashing inclusion-exclusion memoization
Learning mode
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Understand the problem
- What exactly is the input to problem 165? Is it a bound (5000), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- 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 5000 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 5000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 5000 and the cost of testing one.
- Which computational-geometry 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 computational-geometry / inclusion-exclusion - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 5000, 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: "It can be verified that line segments L_2 and L_3 have a true intersection point.") - 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.
- 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 computational-geometry structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the inclusion-exclusion 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 5000 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 'inclusion-exclusion' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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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: explore the same concept filter in the Lab
- 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.