AMC 12 Mock Exam 4 Problem 20

First ignore the week boundary. Let $a_n$ be the number of length-$n$ train/rest strings with no three consecutive training days. A valid string of length $n\ge3$ ends in rest, rest–train, or rest–train–train, so $a_n=a_{n-1}+a_{n-2}+a_{n-3}$ with $a_0=1$, $a_1=2$, $a_2=4$. This gives $a_3=7$, $a_4=13$, $a_5=24$, $a_6=44$, $a_7=81$.

Now subtract the $7$-day strings that are valid within the week but create a run of at least three training days across the boundary. Such a string starts with exactly $p\in\{1,2\}$ training days followed by a rest, and ends with a rest followed by exactly $q\in\{1,2\}$ training days, where $p+q\ge3$. The days strictly between those two rests can be any valid string of length $7-p-q-2$: $$(p,q)=(1,2):\ a_2=4,\qquad (2,1):\ a_2=4,\qquad (2,2):\ a_1=2.$$ (The all-training week is already excluded from the $81$.) Hence the number of routines is $81-(4+4+2)=\boxed{71}$.