#88 - Product-sum Numbers
A natural number, \(N\), that can be written as the sum and product of a given set of at least two natural numbers, \(\{a_1, a_2, \dots, a_k\}\) is called a product-sum number: \(N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k\).
For example, \(6 = 1 + 2 + 3 = 1 \times 2 \times 3\).
For a given set of size, \(k\), we shall call the smallest \(N\) with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, \(k = 2, 3, 4, 5\), and \(6\) are as follows.
- \(k=2\): \(4 = 2 \times 2 = 2 + 2\)
- \(k=3\): \(6 = 1 \times 2 \times 3 = 1 + 2 + 3\)
- \(k=4\): \(8 = 1 \times 1 \times 2 \times 4 = 1 + 1 + 2 + 4\)
- \(k=5\): \(8 = 1 \times 1 \times 2 \times 2 \times 2 = 1 + 1 + 2 + 2 + 2\)
- \(k=6\): \(12 = 1 \times 1 \times 1 \times 1 \times 2 \times 6 = 1 + 1 + 1 + 1 + 2 + 6\)
Hence for \(2 \le k \le 6\), the sum of all the minimal product-sum numbers is \(4+6+8+12 = 30\); note that \(8\) is only counted once in the sum.
In fact, as the complete set of minimal product-sum numbers for \(2 \le k \le 12\) is \(\{4, 6, 8, 12, 15, 16\}\), the sum is \(61\).
What is the sum of all the minimal product-sum numbers for \(2 \le k \le 12000\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=88. Published Friday, 4th February 2005, 06:00 pm. Solved by 12,102 members at time of mirroring.
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Recommended stepping-stone problems: #52 · #33 · #27
Concepts: algebra
Likely techniques: hashing
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- What exactly is the input to problem 88? Is it a bound (12000), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- 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 12000 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 12000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 12000 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?
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- Predict the strategy: in one sentence, what will your solution do? (The classification says algebra / hashing - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 12000, 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, 6 = 1 + 2 + 3 = 1 x 2 x 3.") - then run it. A surprise here is worth more than an hour of debugging later.
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Which step of your solution were you least confident about, and what evidence would settle it?
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