Simon tosses a fair coin repeatedly. Let Hn denote the number of heads among the first n tosses of the coin and let Rn denote the remainder when Sn is divided by 4. (a) What is the state space for the MC (R1,R2,R3,...)? (b) What is the PTM for this MC?

hello from mrbilldoesmath!

answer:

(7 x + 6 y) (49 x^2 - 42 x y + 36 y^2)

discussion:

the trick is to recognize that 343 = 7^3 and 216 = 6^3 and, in general,

a^3 + b^3 =   (a + b) (a^2 - a b + b^2).

in our case

343x^3+216y^3   =   (7x)^3 + (6y)^3

so take a = 7x and b = 6y in the above equation to get the answer.

you,

mrb