You Don't Know Jack

Since the "is closer to the jack than" relation is transitive, what we are looking for is, essentially, the set of all points that are

1. closer to A than to B,
2. closer to B than to C, and
3. closer to C than to D.

For each of these individual inequalities, we find the perpendicular bisector of the segment joining the two points, and shade the side that contains the point that is closer to the jack.

The intersection of these three half-planes is a triangle, whose vertices happen to lie on lattice points. This makes it relatively straightforward to determine that the area of the triangle is $\boxed{30}$ square feet.

I poured sand all over the lawn , counted the goal grains and the total grains, and multiplied the ratio of goal grains to total grains by the total area that I poured sand on.

Python 3.3:

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from math import sqrt
from matplotlib import pyplot as plot
from time import time
from random import random
from pickle import load, dump
search_width = 50
search_length = 50
try:
    x_values, y_values, goals, trials = load(open('jack.p', 'rb'))
except FileNotFoundError:
    x_values, y_values = [], []
    goals, trials = 0, 0
def distance(x1, y1, position):
    x2, y2 = position
    return sqrt(abs(x1-x2)**2 + abs(y1-y2)**2)
position = {'A':(6, 6),
    'B':(4, 10),
    'C':(0, 6),
    'D':(4, 0)}
def goal(x, y):
    return distance(x, y, position['A']) <\
        distance(x, y, position['B']) and\
        distance(x, y, position['B']) <\
        distance(x, y, position['C']) and\
        distance(x, y, position['C']) <\
        distance(x, y, position['D'])
end = float(input("How long? (seconds) ")) + time()
while time() < end:
    try:
        x = random() * search_width
        y = random() * search_length
        if goal(x, y):
            goals += 1
            x_values.append(x)
            y_values.append(y)
        trials += 1
    except KeyboardInterrupt:
        print ("\nBreaking loop...")
        break
# plot the goal positions
plot.plot(x_values, y_values, 'k.', markersize = 1)
# plot original ball positions
letters = 'ABCD'
colors = 'ygbr'
for index, letter in enumerate(letters):
    x, y = position[letter]
    plot.plot([x], [y], colors[index]+'.', markersize = 20)
plot.title(r'Area $\approx$ {0} (after {1} trials)'.format(search_length*search_width*(goals/trials), trials))
plot.show()
# save results for later
dump((x_values, y_values, goals, trials), open('jack.p', 'wb'))

Dependencies:

• matplotlib (if you're on Linux: sudo apt-get install python3-matplotlib )