import turtle
import math

turtle.color("White")
turtle.sety(100)
turtle.color("Black")
turtle.circle(-100)
turtle.seth(90)
turtle.color("Red")

Vx = 0.3
Vy = 0.3

SecondStage = False

for sec in range(3500):
    CurrentX = turtle.xcor()
    CurrentY = turtle.ycor()
    d = math.sqrt(CurrentX ** 2 + CurrentY ** 2)
    f = 17 / (d ** 2)
    theta = math.atan2(CurrentY, CurrentX)
    fx = math.cos(theta) * f
    fy = math.sin(theta) * f

    Vx += -fx
    Vy += -fy
   
    if turtle.pencolor() == "Red":
        turtle.goto(turtle.xcor() + Vx, turtle.ycor() + Vy)

    if d < 100:
        turtle.color("white")

    if sec == 250:
        SecondStage = True
        New = turtle.Turtle()
        New.color("white")
        NewCurrentX = CurrentX
        NewCurrentY = CurrentY
        New.goto(NewCurrentX,NewCurrentY)
        New.color("blue")
        VNx = Vx + 0.05
        VNy = Vy - 0.2

    if SecondStage == True:
        NewCurrentX = New.xcor()
        NewCurrentY = New.ycor()
        d = math.sqrt(NewCurrentX ** 2 + NewCurrentY ** 2)
        f = 17 / (d ** 2)
        theta = math.atan2(NewCurrentY, NewCurrentX)
        fx = math.cos(theta) * f
        fy = math.sin(theta) * f
        VNx += -fx
        VNy += -fy
        
        New.goto(New.xcor() + VNx, New.ycor() + VNy)

        if d < 100:
            New.color("white")

turtle.exitonclick()

from random import randint
import time

def f(x):
    return {
        1: 1,
        2: 2,
        11: 0,
        12: 2,
        21: 0,
        22: 1,
    }[x]

def drandomu():
    drandomint = randint(1, 2)
    if drandomint == 1:
        drandomint = -1
    else:
        drandomint = 1
    return drandomint

def drandom(iDirection):
    drandomint1 = f(iDirection * 10 + randint(1, 2))
    return drandomint1

CurrentTime = time.clock()
vArray = [[[0 for x1 in range(3)] for y1 in range(3)] for z1 in range(3)]
vMaxArray = [[[0 for x2 in range(3)] for y2 in range(3)] for z2 in range(3)]


iMax = 0
iTotalBlock = 1
vChain = [3, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2]
m = 1000000
limit = int(m * 1)
for lSim in range(limit):
    try:
        GiveUp = "False"
        iDirection = 0
        iTotalBlock = 1
        vCurrent = [0, 0, 0]
        for x in range(3):
            for y in range(3):
                for z in range(3):
                    vArray[x][y][z] = 0
        for iLink in range(17):
            if GiveUp == "False":
                if iLink == 0:
                    for iBlock in range(vChain[iLink]):
                        if iBlock > 0:
                            vCurrent[0] += 1
                        vArray[vCurrent[0]][vCurrent[1]][vCurrent[2]] = iTotalBlock
                        iTotalBlock += 1
                else:
                    iDirection = drandom(iDirection)
                    iUpOrDown = drandomu()
                    for iBlock in range(vChain[iLink]):
                        if vCurrent[iDirection] == 0:
                            iUpOrDown = 1
                        vCurrent[iDirection] += iUpOrDown
                        if vArray[vCurrent[0]][vCurrent[1]][vCurrent[2]] != 0:
                            if iMax < iTotalBlock:
                                iMax = iTotalBlock
                                for x in range(3):
                                    for y in range(3):
                                        for z in range(3):
                                            vMaxArray[x][y][z] = vArray[x][y][z]
                            GiveUp = "True1"
                            break
                        else:
                            vArray[vCurrent[0]][vCurrent[1]][vCurrent[2]] = iTotalBlock
                            iTotalBlock += 1

        iTotalBlock = 0
        for x in range(3):
            for y in range(3):
                for z in range(3):
                    iTotalBlock += vArray[x][y][z]

        if iTotalBlock == 378:
            print(vArray)
    except:
        pass

print(int(time.clock() - CurrentTime), "Seconds")
print(int((time.clock() - CurrentTime)/60), "Minutes")

for x in range(3):
    print(vMaxArray[x])

print(iMax)
 

Since $X_i$ is a geometric random variable, we know that $\mathbb{E} [ X_i ] = \frac{100}{101-i}$

Giving us:

$\mathbb{E} [ S ] = \mathbb{E} [ \sum\limits_{k=1}^{100} X_i ] = \sum\limits_{k=1}^{100} \mathbb{E} [X_i ] = \sum\limits_{k=1}^{100} \frac{100}{101-i} = 100 \sum\limits_{k=1}^{100} \frac{1}{101-i} = 100 \sum\limits_{k=1}^{100} \frac{1}{i}$
How to compute $100 \sum\limits_{k=1}^{100} \frac{1}{i}$ ?
This is a famous sum in mathematics. The value of this sum for a given upper limit is known as a harmonic number. The nth Harmonic number $H_n$ is defined to be $\sum\limits_{k=1}^{n} \frac{1}{i}$.
In this case, using wolfram alpha to look up the value of $H_100$ tells us that $H_{100} = 5.18737751$
Therefore on average it will take us $100 H_{100}$ visits to 5 guys to collect all 100 tickets. This is approximately equal to $519$ visits, which is a lot of burgers!