4 added 418 characters in body

n walkers ${A}_{i}$ start out from X to Y simultaneously with speeds ${a}_{i}$, $i=1,2,...,n$. ${a}_{i}\neq {a}_{j}$ if $i\neq j$. At the same time, a motorcyclist M with speed $m=1$ starts out from Y to carry them (as shown in the illustration attached). The motorcyclist can pick up any person he meets. He can carry at most one person. But he's free to drop off the person he carries at any time. Walkers just walk straight forwards (from X to Y), while motorcyclist is free to drive either forwards or backwards.

The motorcylist's goal is to find a plan under which all the walkers arrive at Y simultaneously, if such a plan exists. We call such a plan a feasible plan.

Define the configuration of the problem to be the n-tuple $({a}_{1},{a}_{2},...,{a}_{n})\in {\mathbb{R}}_{++}^{n}$.

Define ${B}^{n}\subseteq{\mathbb{R}}_{++}^{n}$ to be the set of all feasible configurations (i.e., configurations for which a feasible plan exists).

Trivially, we know that ${B}^{1}={\mathbb{R}}_{++}$ and ${B}^{2}={\mathbb{R}}_{++}^{2}$. Also it's easy to see that ${B}^{n}\subseteq {(0,1)}^{n}$ for $n>2$. However, it is nontrivial to find ${B}^{n}$ for $n>2$. Without loss of generality, assuming ${a}_{1}<{a}_{2}<...<{a}_{n}$, I find that ${B}^{3}=\{({a}_{1},{a}_{2},{a}_{3})| 2{a}_{2}\leq{a}_{1}+1,\ {a}_{3}<1 \}$, which is neat!

When I tried to figure out ${B}^{4}$, however, I found it to be significantly harder than ${B}^{3}$. So I have two related questions here:

1.How should we tackle ${B}^{n}$? Is there some systematic method we can use to characterize ${B}^{n}$ for successive n's?

2.If not, can we speculate what ${B}^{n}$ will look like, for example by giving some upper and lower bound on it which we are able to characterize?

EDIT: $n=4$ is EXHAUSTING, and honestly I have no idea where to start. Maybe it's just impractical to find ${B}^{n}$ for $n>3$. An interesting way to simplify the model is to assume that M can teleport himself anywhere he wants (as long as the motorcycle is not loaded). This assumption may allow a cleaner analysis and nicer answer.

3 illustration display error fixed

n walkers ${A}_{i}$ start out from X to Y simultaneously with speeds ${a}_{i}$, $i=1,2,...,n$. ${a}_{i}\neq {a}_{j}$ if $i\neq j$. At the same time, a motorcyclist M with speed $m=1$ starts out from Y to carry them (as shown in the illustration attached). The motorcyclist can pick up any person he meets. He can carry at most one person. But he's free to drop off the person he carries at any time. Walkers just walk straight forwards (from X to Y), while motorcyclist is free to drive either forwards or backwards.

The motorcylist's goal is to find a plan under which all the walkers arrive at Y simultaneously, if such a plan exists. We call such a plan a feasible plan.

Define the configuration of the problem to be the n-tuple $({a}_{1},{a}_{2},...,{a}_{n})\in {\mathbb{R}}_{++}^{n}$.

Define ${B}^{n}\subseteq{\mathbb{R}}_{++}^{n}$ to be the set of all feasible configurations (i.e., configurations for which a feasible plan exists).

Trivially, we know that ${B}^{1}={\mathbb{R}}_{++}$ and ${B}^{2}={\mathbb{R}}_{++}^{2}$. Also it's easy to see that ${B}^{n}\subseteq {(0,1)}^{n}$ for $n>2$. However, it is nontrivial to find ${B}^{n}$ for $n>2$. Without loss of generality, assuming ${a}_{1}<{a}_{2}<...<{a}_{n}$, I find that ${B}^{3}=\{({a}_{1},{a}_{2},{a}_{3})| 2{a}_{2}\leq{a}_{1}+1,\ {a}_{3}<1 \}$, which is neat!

When I tried to figure out ${B}^{4}$, however, I found it to be significantly harder than ${B}^{3}$. So I have two related questions here:

1.How should we tackle ${B}^{n}$? Is there some systematic method we can use to characterize ${B}^{n}$ for successive n's?

2.If not, can we speculate what ${B}^{n}$ will look like, for example by giving some upper and lower bound on it which we are able to characterize?

n walkers ${A}_{i}$ start out from X to Y simultaneously with speeds ${a}_{i}$, $i=1,2,...,n$. ${a}_{i}\neq {a}_{j}$ if $i\neq j$. At the same time, a motorcyclist M with speed $m=1$ starts out from Y to carry them (as shown in the illustration attached). The motorcyclist can pick up any person he meets. He can carry at most one person. But he's free to drop off the person he carries at any time. Walkers just walk straight forwards (from X to Y), while motorcyclist is free to drive either forwards or backwards.

The motorcylist's goal is to find a plan under which all the walkers arrive at Y simultaneously, if such a plan exists. We call such a plan a feasible plan.

Define the configuration of the problem to be the n-tuple $({a}_{1},{a}_{2},...,{a}_{n})\in {\mathbb{R}}_{++}^{n}$.

Define ${B}^{n}\subseteq{\mathbb{R}}_{++}^{n}$ to be the set of all feasible configurations (i.e., configurations for which a feasible plan exists).

Trivially, we know that ${B}^{1}={\mathbb{R}}_{++}$ and ${B}^{2}={\mathbb{R}}_{++}^{2}$. Also it's easy to see that ${B}^{n}\subseteq {(0,1)}^{n}$ for $n>2$. However, it is nontrivial to find ${B}^{n}$ for $n>2$. Without loss of generality, assuming ${a}_{1}<{a}_{2}<...<{a}_{n}$, I find that ${B}^{3}=\{({a}_{1},{a}_{2},{a}_{3})| 2{a}_{2}\leq{a}_{1}+1,\ {a}_{3}<1 \}$, which is neat!

When I tried to figure out ${B}^{4}$, however, I found it to be significantly harder than ${B}^{3}$. So I have two related questions here:

1.How should we tackle ${B}^{n}$? Is there some systematic method we can use to characterize ${B}^{n}$ for successive n's?

2.If not, can we speculate what ${B}^{n}$ will look like, for example by giving some upper and lower bound on it which we are able to characterize?

1