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[VLS]

"Is it better for the fugitive to change hideout each day to one of his 37 houses randomly -knowing that the guards would only seek a randomly chosen one each day- or would it be better to stay in a single house, waiting for him to be lucky to have more time keeping himself there hoping the guards would step him over without looking..... ??? ??? ."

[Gizmotron]

It is far better to have a spy. He should find out where the guard went last and go there. Except on Tuesday if the guard left after 4:32 pm. In that case the use of a magical run-and-hide technique guarantees not getting caught on Wednesday through Mondays. I know. I've been doing it for the last eight years. I've tried to tell everyone under my secret names. My next name is BSzinThe****er. Son of Sam to the great unwashed.

[MarignyGrilleau]

"Is it better for the fugitive to change hideout each day to one of his 37 houses randomly -knowing that the guards would only seek a randomly chosen one each day- or would it be better to stay in a single house, waiting for him to be lucky to have more time keeping himself there hoping the guards would step him over without looking..... ."

It would be better to stay in the run. =) Choosing a single house only when sleeping. =) However the fugitive does not have the house edge, does he? =)
Nice allegory

[TwoCatSam]

This would be an interesting computer study.  I think after a rouzillion spins, it would not matter.