High School Mathematics Physics SMILE
13 October 1998
Notes taken by Alex Junievicz

Alex Junievicz [CPS Substitute]


Tynnetta Stanley [Home Schooling Mother]

She simulated lungs by putting balloons in a bottle. She and her assistant tried to blow up the balloon inside a bottle until the balloon filled the bottle. It was easier to blow up a balloon inside a large bottle than a small bottle.

Bill Shanks [Joliet Central H.S., retired]

Walmart [or, more precisely, Sam's Club, for members only] has a laser for $15.00. Bill suggested that we prepare a "laundry list" of experiments one could do with a laser, such as looking into an ice cube, looking into a light bulb, reflecting off the gradations on a ruler to see fringes, etc.

Betty Roombos [Gordon Technical H.S.]

She is fortunate to have a student who has a digital Camera, and showed pictures of physics activities taken by the digital Camera, and printed on her color printer. It was impressive how well they looked while printed by a Hewlett-Packard 722C ink-jet printer, and on regular paper.

Al Tobecksen [Richards Vocational H.S.]

He showed the solution to the rope puzzle of the last meeting. The trick is to put a loop in the rope, and then feed it through the rope around one the hand of the other person. You either become free in the process, or else become more tightly enmeshed in the process.

Roy Coleman [Morgan Park H.S.]

He announced that there were some changes and new items on the SMILE web page. The URL link is http://www.iit.edu/~smile/ .

Larry Alofs [Kenwood H.S.]

He talked about cow magnets [obtained from the farm store in Kewaunee] and sold them for $2.50 each. They have a variety of uses, including those on the farm.

Porter Johnson [IIT]

He talked about a Geometry problem where twins drove 3 nails at random into a table and formed a triangle. If the nails happened to lie along a straight line, there would be no triangle [or else a very skinny triangle with no interior!]. The probabilities of a new nail lying on the right or left of the line are each 50%, and for a triangle there are three lines, and you have to lie "left" "left" and "left" as you circulate around the boundary [counterclockwise]. By this simple argument, you might expect the probability of being inside as being 1/8 = 0.125. However, in the historic words of the 20th Century Physicist Wolfgang Pauli, "nicht Einfach; aber Falsch"---it isn't simple but it is wrong!

It is convenient to use vectors to decide whether a point is inside a triangle. Let us choose one vertex as special, call the vectors from it to the other vertices r1 and r2, and call the vector from it to the point in question r. Then the criterion for being inside the triangle is

r = a r1 + b r2 .

where a and b are both positive, with a + b < 1.

To gain insight, he employed a pseudo-random number generator and applied the ***Monte-Carlo technique. In a run of 10 million shots there was a computer crash because one of the sets of points was accidentally linear. He modified the program and then safely ran to 100,000,000 shots, without incident [it took several hours on the 80 MHz 486PC], obtaining 7,637,924 hits. The corresponding hit probability is thus .076379 +/= .000100, which is consistent with the number 11/144 = .076388888888... , and lots of other more complicated numbers, as well. However, no clever soul has yet appeared with the solution. A copy of the FORTRAN program was passed out, along with a pseudo-random number generator touted by Liam Coffey [Physics Faculty Member and Computational Physics guru], just in case you also want to waste vast amounts of time on this problem or put your lazy computers to work.

Roy Coleman commented that some pseudo-random number generators are not very successful in producing "random" numbers, as you can see from plotting them in pairs. The "safest" random number sequences involve tabulations of radioactive decay times, but they are difficult to use. The two pseudo-random number generators used here gave similar results.

***Much of twentieth century science involves Monte-Carlo simulations of actual experiments, theoretical models, hypothetical problems, and even "useless and insignificant puzzles". The inventor of the idea was the mathematician Stanislaw Ulam, who was involved in the development of the atomic and hydrogen bombs. Ulam became ill after the war, and spent a lot of time in hospital playing the card game Solitaire. He realized that it is more straightforward to play a few games to "estimate" the probability of winning the game, rather than to try to calculate the odds of winning directly. Ulam realized that many problems too difficult to be solved analytically could be resolved by this technique using fast computers. This story, along with many others, appears in his autobiography, Adventures of a Mathematician [ISBN 0-520-07154-9].