High School Mathematics-Physics SMILE Meeting
02 April 2002
Notes Prepared by Porter Johnson

Don Kanner (Lane Tech HS Physics) -- ¿¿Magic Triangles??
suggested a  variation of the Magic Square problem, in which we put numbers inside a triangular lattice, and seek solutions in which all external line entries and all angle bisector entries add to the same number, such as the following:

/ \
/\ /\
/ \ / \
/\ /\ /\
/ \ / \ / \
For this triangular lattice, ten entries are required---one inside each triangle. In the following case, all numbers add to a total of 36:
/\ /\
/ 8\ 3/ 7\
/\ 19 /\ 17 /\
/10\ /15\ /11\
Can one solve this problem with nine consecutive numbers; say, 1-9? You have definitely put down the gauntlet, Don!

Roy Coleman (Morgan Park HS Physics) -- CPS Restroom Policy Memo
handed out copies of the following memo that was recently passed out in his school:

Date: 01 April 2002

In the past, faculty and staff were permitted to make trips to the restroom under informal guidelines. Effective April 1, 2002, a new Restroom Trip Policy (RTP) will be established to provide a consistent method of accounting for each employee's restroom time and ensuring equal treatment of all CPS employees. This action is necessitated by a marked increase in restroom privilege abuse.

Under this policy, a Restroom Trip Bank (RTB) will be established for each employee. On the first day of the month, employees will be given a Restroom Trip Credit (RTC) of 20. Each time an employee uses the restroom, one trip credit will be deducted from his/her balance. Unused restroom trip credits (URTC) can be accumulated from month to month. However, this balance may not exceed 30.

Currently, the entrances to all restrooms are being equipped with Personnel Identification Stations and Computer-linked Voice Print Recognition. During the next two weeks, each employee must provide two voice prints (one normal, one under stress) to the main office. The Voice Print Recognition System (VPRS) will be operational, but not restrictive, for the month of April. Faculty and staff should acquaint themselves with the stations during that period.  Effective May 1, 2002, all VPRS will be fully activated and those potential restroom users without voice prints on file will be restricted from restroom use.

If an employee's restroom trip bank (RTB) balance reaches zero, the doors to all restrooms will not unlock to that employee's voice until the first day of the next month.

In addition, all restroom stalls are being equipped with timed paper roll retractors. If the stall is occupied for more than three minutes, an alarm will sound. Thirty seconds after the alarm sounds, the toilet paper in the stall will retract, the toilet will flush, and the stall door will automatically spring open.  Enjoy your April 1.

Roy also mentioned that the next AAPT meeting would be held at New Trier High School on Saturday, 13 April 2002.  In addition to exciting Physics presentations, there will be some nice door prizes!   Details on future meetings are given in the ISPP website, http://ispp.info.

Fred Schaal (Lane Tech HS Mathematics) -- Even Magic Squares
showed the solution for the 4 ´ 4 magic square, as given on the website http://mathforum.org/alejandre/magic.square/adler/adler4.html.

 .    .    .    .  
.  .  .   . 
 .   .   . 
 .  .  .   . 

The idea is to put one of the numbers 1-16 into each of the 16 small squares, with no duplications, in such a way that the sums of rows, columns, and diagonal elements is the same --- 34, in this case.  The construction proceeds in two steps.  First, we count the squares across rows, starting across the top, and insert the number for all "diagonal" squares, as shown:

  1           4  
.   6  7  . 
.  10  11   . 
13   . .  16 
Second, pass by the unfilled squares in the same order and fill them with "missing numbers", starting with the largest and going down as unfilled squares are encountered:
 1 15   14    4  
12  6   7   9 
  8 10  11   5 
 13   3  2  16 

Congratulations; your magic square is completed!  But, this provides us with no insight for solving Mr 6 ´ 6.This way of constructing a 4 ´ 4 magic square can be found in a book by Jerome S. Meyer, Fun With Mathematics (Cleveland: World Publishing Company, 1952). Comment by PJ:  For information on 6 ´ 6 magic squares, see the website How to Create 6 ´ 6 Magic Squares:   http://www.guru.gr.jp/~issei/msq/msq6.html. For the example given below, the numbers all add to a total of 111.

   1  35   33     4   32     6 
 30   29  10    9  26    7
 18   20   22  21  17  13 
 19   14  16  15   23  24 
 12   11  27  28    8  25
 31    2    3  34    5  36

Earl Zwicker (IIT Physics) -- Getting a Full Night's Sleep
passed out copies of a newspaper article describing a study published in February 2002 in the Archives of General Psychiatry showing a correlation between the amount of sleep per night versus longevity.  Specifically, sleeping more than 8 hours per night was shown to correlate with high mortality.  Here is the abstract of that article, take from the website http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Abstract&list_uids=11825133:

Mortality Associated With Sleep Duration and Insomnia, Daniel F. Kripke, MD; Lawrence Garfinkel, MA; Deborah L.Wingard, PhD; Melville R. Klauber, PhD; Matthew R. Marler, PhD, Arch Gen Psychiatry. 2002;59:131-136

Background Patients often complain about insufficient sleep or chronic insomnia in the belief that they need 8 hours of sleep. Treatment strategies may be guided by what sleep durations predict optimal survival and whether insomnia might signal mortality risks.

Methods In 1982, the Cancer Prevention Study II of the American Cancer Society asked participants about their sleep duration and frequency of insomnia. Cox proportional hazards survival models were computed to determine whether sleep duration or frequency of insomnia was associated with excess mortality up to 1988, controlling simultaneously for demographics, habits, health factors, and use of various medications.

Results Participants were more than 1.1 million men and women from 30 to 102 years of age. The best survival was found among those who slept 7 hours per night. Participants who reported sleeping 8 hours or more experienced significantly increased mortality hazard, as did those who slept 6 hours or less. The increased risk exceeded 15% for those reporting more than 8.5 hours sleep or less than 3.5 or 4.5 hours. In contrast, reports of "insomnia" were not associated with excess mortality hazard. As previously described, prescription sleeping pill use was associated with significantly increased mortality after control for reported sleep durations and insomnia.

Conclusions Patients can be reassured that short sleep and insomnia seem associated with little risk distinct from comorbidities. Slight risks associated with 8 or more hours of sleep and sleeping pill use need further study. Causality is unproven.

We generally discussed whether the "Science Lite" presentation of this and other studies in the media are simplistic.  This apparent correlation may be an effect, rather than a cause, of poor health, and it would be hard to separate the effects.

Earl also mentioned  an opening for a High School physics teacher at West Chicago High School District #94.  You may write to JoAnn Tunt, Science Department Chair, at 326 S Joliet Street, West Chicago IL 60185, call at (630) 876-6430, or email at jtunt@d94.org.  There were reported to be several openings in the Barrington High School District, as well.

Bill Blunk (Joliet Central HS Physics) -- Comets, Stamps, and April Foolishness Revista
first mentioned that the Comet Ikeya-Zhang (magnitude 3.5) is visible in the Northwest sky after sunset.  For additional information check the Sky and Telescope website, and in particular http://www.skyandtelescope.com/news/3305696.html?page=1&c=y

Bill then made an admittedly anti-phenomenological presentation, passing around a set of stamps for marking papers, with the following inscriptions:

 Method OK   Units?  Algebra Error  Show Basic Equation 
 List Quantities  Sig. Fig.  Arithmetic  Show Substitution
 Show Work  Decimal   Math Error  Solve Basic Equation
.. What?? ..
Finally, Bill did a replay of the magic trick, in which he apparently placed wax paper over n water-filled olive jar with a very large opening, turned it upside down, and -- to the surprise of some -- the water remained in the jar. But then he held the jar -- still upside down -- over a container and carefully removed the wax paper-- and still -- the water remained in the jar!  After giving us time to see that this was really happening, he shook the jar vigorously, and the water dumped out.  With some reluctance, he decided to show us how this was done.  He reached into the container of water and retrieved the wax paper, along with a thin sheet of plastic acetate film that he had cut to fit just over the opening of the jar.  When he earlier had placed the wax paper over the mouth of the upright, water-filled jar, the plastic cover (not visible to us) had been sticking to the wax paper, so that it actually covered the jar's mouth, with the wax paper sticking to it on top.  With a big smile, he remarked that "He who acetates is lost". Very sly, Bill!

Larry Alofs (Kenwood HS Physics) -- Vibrating Pipes
showed us a cylindrical whistle with a sliding plunger.  He blew on the whistle, and as he moved the plunger back and forth, we heard the pitch of its sound vary smoothly higher and lower, not unlike a siren. He then pulled the plunger all the way out, leaving its bottom end open, and we heard the sound reach its lowest pitch.  Next, he covered the open end with his finger, and we heard the pitch go up by about one octave.  After repeating this a few times so that we could be certain of what we observed, he excited the higher harmonics by "over-blowing" (blowing very hard) on the open-ended whistle, showing that the harmonics were about one and two octaves higher than the fundamental.  By contrast, the first harmonic of the closed end pipe had three times the frequency of the fundamental, corresponding to and octave and a fifth [i.e. » 2**(1 + 7/12)]; for details see the 23 November 1999 SMILE lesson, ph112399.htm.

Larry "played" the open pipe, first tapping and bouncing one finger off the end, and then "closing" the end with each finger tap.  We could certainly see and hear the difference in "open end" and "closed end" modes; the transient sounds were about an octave apart.  Pretty!  Larry next filibustered a bit on the conceptual errors caused by most textbooks, that display longitudinal sound waves in air as transverse waves.  He showed the following alternative display (of his own invention) of the first few modes of a pipe with one end open:

                 Nodes: ©       Antinodes:  <--->      
|© <-----> Fundamental
|© <---> © <---> First harmonic
|© <--> © <--> © <--> Second Harmonic

Using such pictures, you can clearly indicate the resonant sound as a longitudinal displacement wave.  Larry indicated that he was showing nodes and antinodes of the displacement , and that for pressure or density (as measured with a transducer and shown on an oscilloscope), the nodes and antinodes would be reversed.

Ann Brandon (Joliet West HS Physics) -- Sinking of Straws
passed out an instruction sheet for an experiment used by Physical Science teachers at her school, which containing the following information:

  1. Problem: How can we predict the quantity of BBs that it will take to sink a straw to a chosen depth in water.
  2. Procedure:
      Conducting the Experiment:
    1. Put a rubber band around the straw at a distance of 4 cm from the plugged end.
    2. Predict the quantity of BBs it takes to sink the straw to the 4 cm mark.  Record your prediction on the table.
    3. Put the straw into a container of water and add BBs until the straw sinks to the 4 cm mark (at the rubber band).  Record the results of the table and your observations.
    4. Repeat steps 1-3 with the rubber band of distances 5 cm, 6 cm, and 7 cm.  Each time you do the experiment, predict how many BBs will be needed to sink the straw to marked depth, record your predictions, conduct the experiment, and record your results and observations.
  3. Data: Sinking of the Straw Data (Individual Team Results)
    Length of straw
    below the surface (cm)
    Predicted Number of BBs Actual Number of BBs Observations
     4 cm      
     5 cm      
     6 cm      
     7 cm      

    Collect the data from each team and create a Class Data Table

    Number of BB's Needed to Sink Straw to Indicated Depths
     Team  4 cm 5 cm 6 cm 7 cm

  4. Analysis:
    1. Record your team's data for 4, 5, 6, 7 cm lengths.
    2. Graph your team's data (length on horizontal, BBs on vertical) ... use pencil.
    3. Record each team's data.
    4. Calculate the average class data for each length.
    5. Graph the average class data on the same graph paper ... Use ink.
    6. Using the class average graph, interpolate to determine how many BBs are required to sink the straw to 4.5 cm, 4.8 cm, 5.3 cm, 6.7 cm.
    7. Using the class average graph, extrapolate to determine how many BBs are required to sink the straw to 7.5 cm, 8.2 cm, 8.8 cm, 9.2 cm.
  5. Conclusion Summary:
    Review what you did, what you discovered, and discuss any variations or similarities in the results obtained by each different team.

Ann obtained BBs and straws from a wide selection available at WALMART.  We found that 4 BBs were necessary to sink the straw to a depth of 4 cm, and that with 5 BBs the straw went down to a depth of 6 cm.  Very interesting, Ann.

Monica Seelman (St James) Digital Numbers, Multiplication Facts, and Geometrical Figures
was ill and unable to attend today, but will present the lesson next time, 23 April 2002.

Notes taken by Porter Johnson