Objective:
This demo involves a classic floor lamp with 4 bulbs and two switches and
addresses the natural question, 'How many levels of illumination are
possible?'Level:
This demo can be used at various levels depending on the background of the
students. It can be used when teaching probability, combinatorics, discrete
mathematics, or any general mathematics class at the high school or college
level.
Prerequisites:
The demo requires only counting capability, but can be adapted for students in
probability and combinatorics using appropriate concepts and notation. (Here
we illustrate only the counting approach.)
Platform:
None required. However, illustrating the illumination levels with a six-way
lamp is recommended. If such a lamp is not available we have included an Excel
file that can be used in class or used by students as part of an out of class
assignment. In addition an animation is available.
Instructor's
Notes:
The six-way lamp is configured as follows:
- There are two switches.
- One switch controls a set of three 60 watt bulbs. As
this switch is rotated, first one bulb is lit; on the second rotation the
other two bulbs are lit; on the third rotation all three bulbs are lit.
- The second switch controls a three-way bulb. The
first rotation turns on the lowest wattage (50 watts), the second rotation
turns on the intermediate wattage (100 watts), and the third rotation
turns on the top wattage (150 watts).
(Click here to see a picture of
a six-way lamp.)
The problem is to count the number of levels of
illumination available in the six-way lamp. I have used this problem for a
long time in a variety of settings. "Recently I saw such a lamp in a Macy's ad
and couldn't resist acquiring it as a classroom prop. This galvanized student
interest and brought forth numerous guesses as to the correct count.
Surprisingly, the responses have been many and varied, with only a very few
reaching the right answer. This is an effective demo for showing students the
power of simple counting methods. Indeed, the lamp became somewhat famous
among computer science majors in our department." [1]
Figure 1 shows a diagram of the six-way lamp
configuration that can be used if one is not available to bring to
class.
Figure 1. |
"It has been my experience that when students are asked
for the number of possible light levels, they respond with a dazzling variety
of proposed counts. Nine, as the product of three outer levels with three
central bulb levels, is a common response. Classes over the years have been
creative in thinking up other counts as well, and they generally ignore the
fact that the "off" position for each switch needs to be considered. Including
"off", there are four levels for each of the two switches, and the
multiplication principle leads immediately to 16 levels of illumination.
Occasionally a bright or experienced student comes up with the correct
enumeration. Even more unusual a response is 15, representing the number of
nontrivial levels of illumination." [2]
"Fortunately, no student in my recollection has proposed
the number commonly given in the ads for the lamp, 'six.' That count not only
ignores the cases when either switch is in the off position, it adds rather
than multiplies the presumed number of possibilities arising from the two
sources of varying wattage. I've pointed out the error of their ways
both to a manufacturer, Stiffel, and to a vendor, Restoration Hardware, but
without effect." [1]
This type of lamp problem is by no means new; it appears
in Feller's classic text [3]. In this setting students are expected to use
basic probability theory (the multiplication principle mentioned above) to
readily compute the correct number of possible illuminations. This principle
is inherent in the display of the Excel file discussed below. See Figure 2.
Figure 2.
The animation below illustrates the
illuminations of the six-way lamp.
Auxiliary resources:
<> The animated gif
above and a corresponding QuickTime movie can be downloaded by clicking
here. These files are zipped and must be
extracted to the same folder.)
<> An Excel file for
experimenting with the illumination levels can be executed or downloaded by
clicking here. WARNING: The Excel file
uses macros so you may need to "enable macros" and possibly reduce the
security level so that the macros can function. After making these choices
from within Excel you may need to exit Excel and reenter it and then start the
routine.
<> For information on some six-way lamps available
click here.
References:
[1] Starr, Norton, Three hands-on classroom demos: Counting, induction,
and data analysis, MAA Session on My Favorite Demo: Innovative Strategies
for Mathematics Instructors, at Joint Mathematics Meeting, Atlanta, Ga., Jan.
5-8, 2005. (Click here to see Norton and the
lamp he takes to class for illustrating the levels of illumination.)
[2] Starr, Norton,
"Nothing Counts for Something", The College
Mathematics Journal, Vol. 29, No. 4 (Sept.
1998), pp. 308-309.
[3] Feller, William, An Introduction to Probability
Theory and Its Applications, Wiley, New York. (Page 24 of the 1^{st}
and page 27 of the 2^{nd} and 3^{rd} editions, 1950, 1957 and
1968.)
Credits:
This demo was submitted by
nstarr at amherst dot edu
Mathematics and Computer Science, Amherst College
and is included in Demos
with Positive Impact with his permission.
The Excel file was created by David R. Hill, with a generous assist by Deane
Arganbright.