Knight's problem

Completado Publicado hace 6 años Pagado a la entrega Termina en 6 días
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A knight in standard international chess is sitting on a board as follows

0 1 2 3

4 5 6 7

8 9 10 11

12 13 14 15

The knight starts on square "0" and makes TT jumps to other squares according to the allowable moves in Chess (so that at each space, it has between two to four valid moves). The knight chooses amongst the allowable moves at each jump uniformly at random and keeps track of the running sum SS of keys on which it lands. See below for specific questions and answers.

After T=16T=16 moves, what is the mean of the quantity SS modulo 13?

What is the standard deviation?

After T=512T=512 moves, what is the mean of the quantity SS modulo 311?

After T=16T=16 moves, what is the probability that the sum is divisible by 5, given that it is divisible by 13?

After T=512T=512 moves, what is the probability that the sum is divisible by 7, given that it is divisible by 43?

Please provide the Python script used to generate this result (max 10000 characters).

Python

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