Suppose a trek begins and on this trek the road is paved by squares with labels on them. The warning sign next to the beginning of the first square, labeled $1$, states:
Beware that due to natural circumstance this path is forced to be muddy amongst every block labeled a number not a multiple of three!
So the brave hikerman trails, but as he traverses the trail, he loses a strength point $S$ every time he crosses a blob of mud.
A) If the hikerman starts with $56$ strength points, how many blocks may he be able to traverse before $S = 0$?
B) When spring comes around, the blobs are dispersed, so now the blobs are on steps with multiples of two but not multiples of eight. If the hikerman, stronger than ever, begins with $S = 500$, how many steps will it take for the collapse at $S = 0$?
C) Can this formula be generalized such that the blobs appear every multiple of $x$ and disappear along every multiple of $y$? What is the closed form of this formula?
D) What are other ways of solving this exercise besides the uses of algebraic techniques?
Here are my BIG hints.
Using modular arithmetic:
Notice that in part A) the hiker does not lose strength every third step. If we treat the hiker's step just before he starts a labeled block, and call it block $0$, then we are wrapping around the modulus $3$. Recall at block $0$, no strength has been used up. What does this imply for blocks $3a : a \in \mathbb{N}$? Can the blobbed blocks be further grouped from here?
Re-framing the exercise:
Every time that the hikerman crosses a non-blobbed square, no strength is lost. Therefore, to find the length of the trail, we can just count the number of strength points he has and treat those as blobs. This is incomplete, however, because of the non-blobbed blocks he has crossed, preserving his strength. Is there a way to treat these blocks as a "bonus"?