Suppose a land with a finite number of castles. Each castle is connected (via roads) with exactly 3 other castles.

One knight leaves from his castle and starts travelling. He moves in the following way. 1) When he begins from his home castle (first step) he randomly chooses a road to take. 2)For the second step he randomly chooses the left or right road, but can't go back. 3)Every other step is deterministic. If he came to this castle by taking the left road next he takes the right and vice versa and can never just go back.

Prove that he will eventually return to his original castle.

  • $\begingroup$ Can he visit castles multiple times? Does "every other step" mean all steps after the second, or every odd numbered step except 1, (3,5,7...n)? $\endgroup$
    – CG.
    Jan 14, 2020 at 14:54
  • $\begingroup$ yes he can visit the same castle multiple times and it means every step after the second $\endgroup$ Jan 14, 2020 at 15:05
  • $\begingroup$ Suppose the starting castle is connected to castles 1, 2 and 3. If the left road leads to castle 1 and the right road leads to castle 2 and those are the only choices available, how does the traveller get to castle 3? $\endgroup$
    – Lawrence
    Mar 20, 2023 at 16:42

1 Answer 1


First of all,

since there are only finitely many castles and finitely many roads, there will necessarily be a repetition where the knight arrives at castle C by road R1 and leaves it by road R2 on two occasions. (After which, clearly everything between those two occurrences will repeat for ever.)

And now the point is that

the process is reversible: if you know that the knight went ...-R1->C-R2->... then you can continue the path backwards as well as forwards

which means that

if he ends up on an endlessly repeating cycle then he must in fact have started on that cycle, because extending his route backwards can't leave it.

  • $\begingroup$ I don't see how the above reasoning proves that if the knight moves in a loop, this loop will contain the starting castle.I can imagine a loop being created later on in his trip that doesn't include the starting castle. $\endgroup$ Jan 15, 2020 at 15:18
  • $\begingroup$ Then I'll edit it to try to make it clearer. $\endgroup$
    – Gareth McCaughan
    Jan 15, 2020 at 15:19

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