#828 - Numbers Challenge
It is a common recreational problem to make a target number using a selection of other numbers. In this problem you will be given six numbers and a target number.
For example, given the six numbers \(2\), \(3\), \(4\), \(6\), \(7\), \(25\), and a target of \(211\), one possible solution is:
\[211 = (3+6)\times 25 − (4\times7)\div 2\]This uses all six numbers. However, it is not necessary to do so. Another solution that does not use the \(7\) is:
\[211 = (25−2)\times (6+3) + 4\]Define the score of a solution to be the sum of the numbers used. In the above example problem, the two given solutions have scores \(47\) and \(40\) respectively. It turns out that this problem has no solutions with score less than \(40\).
When combining numbers, the following rules must be observed:
- Each available number may be used at most once.
- Only the four basic arithmetic operations are permitted: \(+\), \(-\), \(\times\), \(\div\).
- All intermediate values must be positive integers, so for example \((3\div 2)\) is never permitted as a subexpression (even if the final answer is an integer).
The attached file number-challenges.txt contains 200 problems, one per line in the format:
where the number before the colon is the target and the remaining comma-separated numbers are those available to be used.
Numbering the problems 1, 2, ..., 200, we let \(s_n\) be the minimum score of the solution to the \(n\)th problem. For example, \(s_1=40\), as the first problem in the file is the example given above. Note that not all problems have a solution; in such cases we take \(s_n=0\).
Find \(\displaystyle\sum_{n=1}^{200} 3^n s_n\). Give your answer modulo \(1005075251\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=828. Published Saturday, 4th February 2023, 04:00 pm. Solved by 813 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:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.2 Primes, Sieves, and Integer Factorization · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #944 · #860 · #628
Concepts: combinatorics number-theory
Likely techniques: modular-exponentiation
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 828? 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. Required format: Give your answer modulo 1005075251.
- Write out, in your own words, the definition of score of a solution as the statement gives it. Which integers/objects are excluded by that definition?
- What constraint does the bound 200 impose, and is it inclusive or exclusive?
- 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 / modular-exponentiation - 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, given the six numbers 2, 3, 4, 6, 7, 25, and a target of 211, one possible solution is: 211 = (3+6)x 25 - (4x7)/ 2 This uses all six numbers.") - 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 modular-exponentiation 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 'modular-exponentiation' 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: #944 · #860 · #628
- 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.