#111 - Primes with Runs
Considering \(4\)-digit primes containing repeated digits it is clear that they cannot all be the same: \(1111\) is divisible by \(11\), \(2222\) is divisible by \(22\), and so on. But there are nine \(4\)-digit primes containing three ones: \[1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111.\]
We shall say that \(M(n, d)\) represents the maximum number of repeated digits for an \(n\)-digit prime where \(d\) is the repeated digit, \(N(n, d)\) represents the number of such primes, and \(S(n, d)\) represents the sum of these primes.
So \(M(4, 1) = 3\) is the maximum number of repeated digits for a \(4\)-digit prime where one is the repeated digit, there are \(N(4, 1) = 9\) such primes, and the sum of these primes is \(S(4, 1) = 22275\). It turns out that for \(d = 0\), it is only possible to have \(M(4, 0) = 2\) repeated digits, but there are \(N(4, 0) = 13\) such cases.
In the same way we obtain the following results for \(4\)-digit primes.
| Digit, d | M(4, d) | N(4, d) | S(4, d) |
| 0 | 2 | 13 | 67061 |
| 1 | 3 | 9 | 22275 |
| 2 | 3 | 1 | 2221 |
| 3 | 3 | 12 | 46214 |
| 4 | 3 | 2 | 8888 |
| 5 | 3 | 1 | 5557 |
| 6 | 3 | 1 | 6661 |
| 7 | 3 | 9 | 57863 |
| 8 | 3 | 1 | 8887 |
| 9 | 3 | 7 | 48073 |
For \(d = 0\) to \(9\), the sum of all \(S(4, d)\) is \(273700\).
Find the sum of all \(S(10, d)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=111. Published Friday, 16th December 2005, 06:00 pm. Solved by 8,572 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.2 Primes, Sieves, and Integer Factorization
Recommended stepping-stone problems: #231 · #347 · #387
Concepts: number-theory
Likely techniques: digit-dp prime-test sieve
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- What exactly is the input to problem 111? Is it a bound (10), 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 do the arguments of S(4,d), M(n,d) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10?
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- Predict the strategy: in one sentence, what will your solution do? (The classification says number-theory / digit-dp - do you agree, and why?)
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- Related problem: #231 · #347 · #387
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