#315 - Digital Root Clocks
Sam and Max are asked to transform two digital clocks into two "digital root" clocks.
A digital root clock is a digital clock that calculates digital roots step by step.
When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.
For example, if the clock is fed the number 137, it will show: "137" → "11" → "2" and then it will go black, waiting for the next number.
Every digital number consists of some light segments: three horizontal (top, middle, bottom) and four vertical (top-left, top-right, bottom-left, bottom-right).
Number "1" is made of vertical top-right and bottom-right, number "4" is made by middle horizontal and vertical top-left, top-right and bottom-right. Number "8" lights them all.
The clocks consume energy only when segments are turned on/off.
To turn on a "2" will cost 5 transitions, while a "7" will cost only 4 transitions.
Sam and Max built two different clocks.
Sam's clock is fed e.g. number 137: the clock shows "137", then the panel is turned off, then the next number ("11") is turned on, then the panel is turned off again and finally the last number ("2") is turned on and, after some time, off.
For the example, with number 137, Sam's clock requires:
| "137" | : | (2 + 5 + 4) × 2 = 22 transitions ("137" on/off). |
| "11" | : | (2 + 2) × 2 = 8 transitions ("11" on/off). |
| "2" | : | (5) × 2 = 10 transitions ("2" on/off). |
Max's clock works differently. Instead of turning off the whole panel, it is smart enough to turn off only those segments that won't be needed for the next number.
For number 137, Max's clock requires:
| "137" |
: |
2 + 5 + 4 = 11 transitions ("137" on) 7 transitions (to turn off the segments that are not needed for number "11"). |
| "11" |
: |
0 transitions (number "11" is already turned on correctly) 3 transitions (to turn off the first "1" and the bottom part of the second "1"; the top part is common with number "2"). |
| "2" |
: |
4 transitions (to turn on the remaining segments in order to get a "2") 5 transitions (to turn off number "2"). |
Of course, Max's clock consumes less power than Sam's one.
The two clocks are fed all the prime numbers between A = 107 and B = 2×107.
Find the difference between the total number of transitions needed by Sam's clock and that needed by Max's one.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=315. Published Sunday, 19th December 2010, 10:00 am. Solved by 3,878 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.
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 · 1.1 Sets, Functions, and Relations · 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 · 2.1 Functions, Limits, and Continuity · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 8.1 Markov Chains
Recommended stepping-stone problems: #72 · #76 · #80
Concepts: algebra game-theory number-theory
Likely techniques: markov-chain prime-test
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 315? Is it a bound (137), 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 137 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 137?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 137 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?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says algebra / markov-chain - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 137, 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, if the clock is fed the number 137, it will show: " 137 " → " 11 " → " 2 " and then it will go black, waiting for the next number.") - then run it. A surprise here is worth more than an hour of debugging later.
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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 algebra structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the markov-chain 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 137 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 'markov-chain' 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: #72 · #76 · #80
- 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.