#497 - Drunken Tower of Hanoi
Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods and disks of different sizes that can slide onto any of the rods. The game begins with a stack of \(n\) disks placed on the leftmost rod in descending order by size. The objective of the game is to move all of the disks from the leftmost rod to the rightmost rod, given the following restrictions:
- Only one disk can be moved at a time.
- A valid move consists of taking the top disk from one stack and placing it onto another stack (or an empty rod).
- No disk can be placed on top of a smaller disk.
Moving on to a variant of this game, consider a long room \(k\) units (square tiles) wide, labeled from \(1\) to \(k\) in ascending order. Three rods are placed at squares \(a\), \(b\), and \(c\), and a stack of \(n\) disks is placed on the rod at square \(a\).
Bob begins the game standing at square \(b\). His objective is to play the Tower of Hanoi game by moving all of the disks to the rod at square \(c\). However, Bob can only pick up or set down a disk if he is on the same square as the rod/stack in question.
Unfortunately, Bob is also drunk. On a given move, Bob will either stumble one square to the left or one square to the right with equal probability, unless Bob is at either end of the room, in which case he can only move in one direction. Despite Bob's inebriated state, he is still capable of following the rules of the game itself, as well as choosing when to pick up or put down a disk.
The following animation depicts a side-view of a sample game for \(n = 3\), \(k = 7\), \(a = 2\), \(b = 4\), and \(c = 6\):
Let \(E(n, k, a, b, c)\) be the expected number of squares that Bob travels during a single optimally-played game. A game is played optimally if the number of disk-pickups is minimized.
Interestingly enough, the result is always an integer. For example, \(E(2,5,1,3,5) = 60\) and \(E(3,20,4,9,17) = 2358\).
Find the last nine digits of \(\sum_{1\le n \le 10000} E(n,10^n,3^n,6^n,9^n)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=497. Published Sunday, 11th January 2015, 04:00 am. Solved by 669 members at time of mirroring.
Why this is useful
Probability Statistics. Expectation and state-based probability reasoning underpin pricing, risk, and statistical inference (Phases 7, 11, 13).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
10.4 Algorithms: Gradient Descent and Newton's Method · 10.1 Convex Sets and Convex Functions · 13.2 Monte Carlo Estimation and Error Analysis · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.8 Bit Manipulation and State Compression · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 3.1 Vectors, Multivariable Functions, and Level Sets · 7.3 Independence, Conditional Probability, and Conditional Expectation · 7.2 Expectation, Moments, and Key Inequalities · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #493 · #523 · #978
Concepts: game-theory geometry optimization probability
Likely techniques: bitmask-dp sorting
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 497? Is it a bound (10000), 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).
- 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 10000 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10000 and the cost of testing one.
- Which probability 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 probability / bitmask-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10000, 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, E(2,5,1,3,5) = 60 and E(3,20,4,9,17) = 2358.") - 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 probability structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bitmask-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 10000 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 'bitmask-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: #493 · #523 · #978
- 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.