#72 - Counting Fractions
Consider the fraction, \(\dfrac n d\), where \(n\) and \(d\) are positive integers. If \(n \lt d\) and \(\operatorname{HCF}(n,d)=1\), it is called a reduced proper fraction.
If we list the set of reduced proper fractions for \(d \le 8\) in ascending order of size, we get: \[\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \frac 2 7, \frac 1 3, \frac 3 8, \frac 2 5, \frac 3 7, \frac 1 2, \frac 4 7, \frac 3 5, \frac 5 8, \frac 2 3, \frac 5 7, \frac 3 4, \frac 4 5, \frac 5 6, \frac 6 7, \frac 7 8\]
It can be seen that there are \(21\) elements in this set.
How many elements would be contained in the set of reduced proper fractions for \(d \le 1\,000\,000\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=72. Published Friday, 18th June 2004, 06:00 pm. Solved by 25,661 members at time of mirroring.
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Prerequisites
Lessons that prepare you:
1.1 Sets, Functions, and Relations · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 2.1 Functions, Limits, and Continuity · 4.2 Linear Maps, Matrices, Rank, and the Null Space
Recommended stepping-stone problems: #16
Concepts: algebra
Likely techniques: exact-rational hashing sorting
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Understand the problem
- What exactly is the input to problem 72? Is it a bound (1000000), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- Write out, in your own words, the definition of reduced proper fraction as the statement gives it. Which integers/objects are excluded by that definition?
- What constraint does the bound 1000000 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1000000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000000 and the cost of testing one.
- Which algebra 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 algebra / exact-rational - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000000, 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 seen that there are 21 elements in this set.") - 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 algebra structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the exact-rational 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 1000000 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 'exact-rational' 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?
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- Related problem: #16
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