#90 - Cube Digit Pairs
Each of the six faces on a cube has a different digit (\(0\) to \(9\)) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of \(2\)-digit numbers.
For example, the square number \(64\) could be formed:
In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below one-hundred: \(01\), \(04\), \(09\), \(16\), \(25\), \(36\), \(49\), \(64\), and \(81\).
For example, one way this can be achieved is by placing \(\{0, 5, 6, 7, 8, 9\}\) on one cube and \(\{1, 2, 3, 4, 8, 9\}\) on the other cube.
However, for this problem we shall allow the \(6\) or \(9\) to be turned upside-down so that an arrangement like \(\{0, 5, 6, 7, 8, 9\}\) and \(\{1, 2, 3, 4, 6, 7\}\) allows for all nine square numbers to be displayed; otherwise it would be impossible to obtain \(09\).
In determining a distinct arrangement we are interested in the digits on each cube, not the order.
- \(\{1, 2, 3, 4, 5, 6\}\) is equivalent to \(\{3, 6, 4, 1, 2, 5\}\)
- \(\{1, 2, 3, 4, 5, 6\}\) is distinct from \(\{1, 2, 3, 4, 5, 9\}\)
But because we are allowing \(6\) and \(9\) to be reversed, the two distinct sets in the last example both represent the extended set \(\{1, 2, 3, 4, 5, 6, 9\}\) for the purpose of forming \(2\)-digit numbers.
How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=90. Published Friday, 4th March 2005, 06:00 pm. Solved by 13,367 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:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 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: #53 · #29 · #24
Concepts: combinatorics geometry
Likely techniques: hashing
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Understand the problem
- What exactly is the input to problem 90? Is it a bound (one-hundred), 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 one-hundred impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly one-hundred?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by one-hundred 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 / hashing - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = one-hundred, 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, the square number 64 could be formed: In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below one-hundred: 01, 04, 09, 16, 25, 36, 49, 64, and 81.") - then run it. A surprise here is worth more than an hour of debugging later.
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- 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 hashing 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 one-hundred 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 'hashing' 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: #53 · #29 · #24
- 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.