#414 - Kaprekar Constant
\(6174\) is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits in decreasing order, we get \(7641-1467=6174\).
Even more remarkable is that if we start from any \(4\) digit number and repeat this process of sorting and subtracting, we'll eventually end up with \(6174\) or immediately with \(0\) if all digits are equal.
This also works with numbers that have less than \(4\) digits if we pad the number with leading zeroes until we have \(4\) digits.
E.g. let's start with the number \(0837\):
\(8730-0378=8352\)
\(8532-2358=6174\)
\(6174\) is called the Kaprekar constant. The process of sorting and subtracting and repeating this until either \(0\) or the Kaprekar constant is reached is called the Kaprekar routine.
We can consider the Kaprekar routine for other bases and number of digits.
Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers.
However, it can be shown that for \(5\) digits and a base \(b = 6t+3\neq 9\), a Kaprekar constant exists.
E.g. base \(15\): \((10,4,14,9,5)_{15}\)
base \(21\): \((14,6,20,13,7)_{21}\)
Define \(C_b\) to be the Kaprekar constant in base \(b\) for \(5\) digits. Define the function \(sb(i)\) to be
- \(0\) if \(i = C_b\) or if \(i\) written in base \(b\) consists of \(5\) identical digits
- the number of iterations it takes the Kaprekar routine in base \(b\) to arrive at \(C_b\), otherwise
Define \(S(b)\) as the sum of \(sb(i)\) for \(0 \lt i \lt b^5\).
E.g. \(S(15) = 5274369\)
\(S(111) = 400668930299\)
Find the sum of \(S(6k+3)\) for \(2 \leq k \leq 300\).
Give the last \(18\) digits as your answer.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=414. Published Sunday, 10th February 2013, 07:00 am. Solved by 342 members at time of mirroring.
Why this is useful
Algorithmic Development. Graph/state-space search transfers to routing, dependency resolution, and execution-path optimisation (Phases 16, 17).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.6 Recurrence Relations and Generating Functions · 2.7 Sequences, Series, Convergence, and Power Series
Recommended stepping-stone problems: #790 · #606 · #617
Concepts: sequences-series brute-force-reduction
Likely techniques: bfs-dfs digit-dp
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 414? Is it a bound (18), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single residue (the answer is reduced modulo a given number, so keep everything in modular arithmetic from the start).
- Write out, in your own words, the definition of kaprekar constant, kaprekar routine as the statement gives it. Which integers/objects are excluded by that definition?
- What do the arguments of S(b), S(15) 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 18?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 18 and the cost of testing one.
- Which sequences-series 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 sequences-series / digit-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 18, 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: "we get 7641-1467=6174.") - then run it. A surprise here is worth more than an hour of debugging later.
Scratchpad
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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 sequences-series structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the digit-dp 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 18 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 'digit-dp' 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: #790 · #606 · #617
- 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.