PUZZLE
# 025
NOVEMBER 21, 2016
PUZZLE
# 025
You have five ping pong ball sized spheres, all with
a diameter of 4 cm. You also have an airtight
rectangular box to store them in. The base of the box
has an inner dimension of 8 cm x 8 cm.
FIVE BALLS
IN A BOX
You put four of the balls in a two by two grid, so that
they all touch each other and the base of the box.
Then, you put the fifth ball in the middle, on top of
that grid, so that is rests in the little cradle touching all
four other balls (Fig 1 & 2).
When you put the lid on the box, it fits perfectly, so it
just barely makes contact with the top ball (Fig. 3).
The Question: What is the height of the inside of the
box?
Fig 1: Five balls in an 8 x 8 cm box
AA
We can first consider how the top ball
interacts with the four bottom balls. We
need to find the amount of vertical
overlap there is between the bottom
and top layers.
8 cm
4 cm
2 cm
To start, we can consider the points
where the top ball touches the lower
ones. To do so, we need to look at a
diagonal section through the box
(Fig. 5).
2 cm
4 cm
4 cm
PLAN
How We Can Look At The Problem:
Space Between Balls:
AA
In order to consider the points where
the balls come in contact with each
other, we need to know the diagonal
distance at the center of the first layer
of balls.
8 cm
Fig 2: Plan of fifth ball resting on 2 x 2 grid of 4 balls
x = Height of box
4 cm
y
Distance
Overlap
Height
between fifth
above balls
ball and base
at base
ELEVATION
r = [(√42 + 42) - 4 cm] / 2
r = 0.828 cm
The diagonal distance between the two
balls is 2r = 1.66 cm.
(Fig. 4)
Triangle Between Midpoints:
If we make a triangle between the
midpoints of the 3 balls that we see
through the diagonal section, we realize
that the distance between the midpoint
of the top ball and the bottom ones is 2
times the radius, or 4 cm.
8 cm
Fig 3: Elevation of fifth ball resting on the four others
Height of Triangle:
4
cm
DISTANCE
BETWEEN
BALLS
(Fig. 5)
0.
4 cm
86
8
cm
The base of the triangle is 2 times the
radius, plus the space between the two
bottom balls, 1.66 cm. This gives a total
of 5.66 cm.
To find the height of the triangle, we
can use the Pythagorean theorem:
0.
r
86 r =
8
cm
h = √42 - 2.8282
h = 2.828 cm
(Fig. 5)
Fig 4: Plan of diagonal distance between lower balls
2
cm
2
6.828 cm
cm
2 cm
h = 2.828 cm
HEIGHT
BETWEEN
MIDPOINTS
2 cm
2 cm
SOLUTION
Interior Height Of The Box:
2 cm
The inside of the box is the radius of the
bottom balls (2 cm), plus the height
between the midpoint of the bottom
and midpoint of the top ball (h = 2.828
cm), plus the radius of the top ball (2
cm):
2 cm + 2.828 cm + 2cm
6.828 cm
Therefore, the inside of the box needs to
be 6.828 cm
4 cm
1.66 cm
8 cm
4 cm
Fig 5: Section AA, with triangle formed by midpoints
(Fig. 5)