High School Mathematics-Physics SMILE Meeting
08 March 2005
Notes Prepared by Porter Johnson

Fred Schaal [Lane Tech HS, mathematics]              RANDINT(1,14,4)+50
Fred used the Pseudo-random Number Generator RANDINT, which is programmed into the TI-83 calculator. He generated a random, equally-distributed set of 200 integers from 1 through 14, and obtained the following number-of-occurrences of the generated numbers:

Generated Number   1 2 3 4 5 6 7 8 9 10 11 12 13 14
Number-of-occurrences   13  12  13  18  17   14  20  15  13  13  17  10  13  12
Does this appear to be a "random" set of numbers? The answer is "Yes", despite the fact that the number-of-occurrences ranges between 10 and 20. On statistical grounds, we would expect the average number of occurrences to be about 200/14 = 14.3, with a spread (standard deviation) of Ö14.3 = 3.8.  Thus, about 2/3 of the number-of-occurrences should lie between 11 and 17. That is consistent with the spread in the data. Curiously, only the number "6" occurs exactly 14 times.

Porter Johnson mentioned that "everybody knows" that it is unlikely for a randomly flipped coin to come up H (Heads) ten times in a row. However, not everybody realizes that the alternating sequence H T H T H T H T H T is equally unlikely. Furthermore, it is quite unlikely that in 1000 coin flips, Heads will occur exactly 500 times.

For a general discussion of Pseudorandom Number Generators see the Wikipedia webpage: http://en.wikipedia.org/wiki/Pseudorandom_number_generator.

Fred also brought in his metal candy box, for which we had taken exterior measurements last time mp022205.html to determine a volume of about 910 cm3.  We took a graduated cylinder filled with water, from which we were able to pour about 800 cm3 of water before the box became full. Our estimated volume was too large by over 10%. Why?

Fred also pointed out that the planet Mercury would be visible next to the New Moon just after sunset in the next few days. Thanks for the ideas, Fred! 

Bill Colson [Morgan Park HS, mathematics]              Poetry
Bill read us these two poems that were used in connection with a No Talent Show at a weekend school event:

Enjoyable, Bill!

Bill Shanks [retired, Joliet New Lenox environs]              Geometry
Bill reminded us of the Pythagorean Theorem for a right triangle of sides (a, b, c = b + d), where c is the hypotenuse:

                           /|
               c = b + d  / |  b
                         /  |
                        /___|
                          a
c2 = a2 + b2
(b + d)2 = a2 + b2
2 b d + d2 = a2 
2 b d = a2 - d2
b =  (a2 - d2 ) / (2d)
Bill pointed out, when a and b are chosen so that b is an integer, we obtain a right triangle with integer sides.  Bill found that Interesting Right Triangles were obtained for the cases d = 1, 2, 8, 9, 18, 25, 49, 50, ... .  These numbers are all of the form  d = 2p nq, where n is odd, and p is either zero or an odd number. For example, with  d = 1, we get b = (a2 -1) / 2, and a must be an odd number for b to be an integer.  We thus obtain these right triangles with integer sides:
a b c=b+1
3 4 5
5 12 13
7 24 25
9 40 41
11 60 61
13 84 85
25 312 313
35 612 613
999   499000   499001
For d =2, we obtain  b = (a/2)2 -1, so that a must be an even number.  We thus obtain the table
a b c=b+2
4 3 5
6 8 10
8 15 17
10 24 26
12 35 37
100 2499 2501
1000   249999   250001
For d = 8, we obtain b = (a/4)2 -4, so that a must be divisible by 4. We obtain these triangles:
a b c=b+8
12 5 13
16 12 20
20 21 29
24 32 40
28 45 53
36 77 85
888   49280  48288
For d = 9, we obtain b = [ (a/3)2 - 9 ] / 2, so that a must be odd and divisible by 3. We obtain these triangles:
a b c=b+9
15 8 17
21 20 41
27 36 45
33 56 65
39 80 89
111 680 6893
999   55440   55449
Bill finds this useful in making up exam problems, when he wants all the sides of the right triangle to be integers. Fred Schaal mentioned Nearly Isosceles Integer Right Triangles, for which the the sides of the triangle are  (I, I+1, J), where I and J are integers.  Here are the first six examples of this infinite set:
I I + 1 J
3 4 5
20 21 29
119 120 169
696 697 985
  4059    4060     5741  
  23660    23661     33461  

Note that the scale of subsequent triangles increases by a factor of about 6 each time.  For discussion see the solution to Problem #152 on this website:  http://www.dansmath.com/probofwk/probar16.html#anchor585356. For a general solution and a relation of these triangles to the Pell Equation, see the following PDF file:  http://hometown.aol.com/jpr2718/pell.pdf.

Thanks for the ideas, Bill!

Ann Brandon [Joliet West, physics]              Magnet Experiments
Ann has a homebound student with MS this semester, so she developed a number of "take-home" experimental setups involving magnetism. Ann began by putting a bar magnet on the desk, and covered it with a large sheet of paper.  She place a small "toy" compass on the paper near the location of the magnet, and drew a short line on the paper to mark the direction of the compass needle (direction of the magnetic field).  Ann moved the compass in the direction of the field, marking the direction at each new location, and repeated the process several times.  She then connected the marks with a solid line, thus graphing a line of force of the magnetic field.  By starting at various positions, Ann traced out several lines of force for the magnetic field.  Next, Ann sprinkled iron filings out of a "salt shaker" and onto the paper.  The iron filings aligned along the lines of force that she had previously drawn. As the amount of fillings increased, the lines of force became three-dimensional.  Very interesting visual patterns!

Ann then placed two Ring Magnets on a wooden rod.  When the North (or South) poles of the magnets were adjacent to one another, the magnets repelled each another.  However, when the magnets had opposite poles adjacent, they attracted each other.  Ann then put several magnets on the wooden rod with the unlike poles adjacent, and held the rod vertically up. The magnets bobbed up and down slightly, coming to equilibrium positions.  Interestingly, the spacing between the magnets was greatest at the top, and least at the bottom. (Why?) Neato!

Ann pointed out that it is often possible to get End Rolls of Newsprint from local newspaper offices: e.g.

South Holland/Dolton Star - 6901 West 159th Street, Tinley Park, IL, 60477-1602
Phone: 708-802-8800 Fax: 708-802-8088
http://www.starnewspapers.com.
End rolls may or may not contain a lot of paper, depending upon what's left over after printing. The printers office will often give them gratis to school teachers, or else at minimal cost. Thanks, Ann!

Leticia Rodriguez [Peck Elementary School]              Ceragem Thermal Acupuncture Massager
Leticia has been using a thermal massage bed, which is described on the Ceragem website:  http://www.ceragem.com/. The following is excerpted from that website:

What is Ceragem?
CERAGEM is a thermal massager that helps soothe body aches and pains associated with daily stress, pressure, and bad posture. It combines the benefits of alternative medicine derived from traditional Eastern medicine with advanced technology to provide the most effective healing and relaxation. Ceragem is easy to use and highly effective, as proven by the positive feedbacks we received from our customers. Ceragem is available for free trial at our distribution centers.

The device consists of a bed with rollers for spinal alignment, with an IR light source to stimulate circulation.  Leticia asked that we each go to test this device (six times, without cost) to assess its effects on our health and happiness, at the following location:

5756 West Belmont Avenue, Chicago IL
(773) 205-1020

Notes prepared by Porter Johnson