# The SIX-WAY LAMP

Demos with Positive Impact

www.mathdemos.org

Objective: This demo involves a classic floor lamp with four 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 mathematics for elementary teachers, 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).

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 counting 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. (Clicking on Figure 2 will link you to the Excel file. 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.)

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.

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 1st and page 27 of the 2nd and 3rd 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.

DRH 2/17/2005      Last updated 6/14/2005 DRH

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