Mathematics- Physics High School SMILE Meeting
28 March 2000
Notes Prepared by Earl Zwicker

Announcement by Porter Johnson

Recent Academic year SMILE write-ups have been placed on the SMILE home page. Check them out at the URL The write-ups are being put first on my home page, and then transferred over to the SMILE site.

                     OUR NEXT MEETING...
...will be April 11, 2000
4:15 p.m.
111 LS

Fred Schaal (Lane Tech HS)
first mentioned Weird Science (Lee Marek) on one of Letterman's shows, in which two cans of soda pop are shaken in an apparatus like a paint shaker. The cans are then simultaneously pierced, and streams of pop shoot out to the ceiling in opposite directions. Spectacular - but not done live because of the messiness. Also one may use an ultrasonic cleaner with high frequency vibrations in place of a paint shaker.

Next, Fred brought us back to something he had done with us earlier: The truncated cone. Start with a circle of radius R cut from paper. Draw two radii on the circle from its center, and so define an angle of 360o - ko between them. Cut out the sector between them, as shown:

s = 2 p r = p [k/180] R

Tape the radii together, to form a truncated cone surface with the paper. Let r be the radius of base of the cone, and h its altitude. Then

r = R * [ko/360o]


h = (R2 - r2)1/2 = R * [ 1 - (ko/360o)2]1/2.

The conical volume is given by

V = pr2h/3

= p R3/3 * [ko/360o]2 * [ 1 - (ko/360o)2 ]1/2

"What value should the angle ko have in order for volume V to be a maximum?" was the question Fred asked us. But lacking for any volunteers, Fred left us with the unanswered question. Maybe next time somebody will have an a solution?

Ed Robinson (Collins HS)
posed the following challenge to us. We are given twelve coins, one of which is either heavier or lighter than the others. Using a balance, how - with no more than three weighings - can we determine which coin is the odd one? He noted that 1990s pennies have a zinc cladding, and therefore weigh different from a 1972 penny, which is made of solid copper. This would make it possible for us to do this experimentally on the balance he had placed on the table.

Ed proceeded to show us a theoretical strategy to do this, starting with just 6 coins, and 9 coins---namely, split the coins into 3 sets, and go from there. A guiding principle is that you can tell which coin of 3 is bad by making one weighing. One can expand the same strategy to 12 coins. Ed actually carried out the experiment with a set of coins, and it worked! He also handed out this paper:

A Strategy For Solving The Forged Coin Problem
Edward J Robinson [Collins HS]

There are twelve coins, one of which is forged. The forged coin is either lighter or heavier than the others. By using at most three weighings on a balance scale, find the forged coin and determine whether it is heavier or lighter than the other coins.

Divide the twelve coins into three groups of four, groups A,B,C.

First weighing

Place group A on the left and place group B on the right. Set group C aside. If the balance is equal then the forged coin is in group C. If the balance tends to the left, then the forged coin is either in group A or B. If the forged coin is in group A, then it is heavier; if the forged coin is in group B, then it is lighter. If the balance tends to the right, then the forged coin is either in group A or B. If the forged coin is in group A, then it is lighter; if the forged coin is in group B then it is heavier.

For a JAVA applet to try your hand at solving this problem at a virtual level, see the website Find the Forged Coins;

PJ Comment: Also, check these websites:

You got us thinking, Ed! Thanks!

Sally Hill (Clemente HS)
put us through a paper-and-scissors exercise to construct a geodesic dome. A template, which consists of 21 equilateral triangles, is given on the website We crowded around the table and carefully cut out section after section from the pattern copied onto many pieces of paper. Fitting and gluing (use stick glue to make it easy) 5 sections together will do it!

Sally, thanks for introducing us to this useful idea and website resource!

Porter Johnson (IIT Physics)
passed out copies of "Weighing Problem,"a write-up from five years ago, which is reproduced below: How does it compare with Ed's Approach?

21 MARCH 1995


One is given a set of 12 coins, one of which is counterfeit and either lighter or heavier than the other 11. We are allowed to use a [crude] balance, which can only tell us whether the coins on the left pan weigh less than, the same as, or more than the coins on the right pan. Devise a scheme which involves only three different weighings to determine which coin is counterfeit, and whether it is lighter or heavier than the others.

Remark: To solve this problem one must make full use of the three possible results of weighing, varying which and how many coins are on each side, to develop the scheme. I have found a scheme which works, which I will describe here. It may not be unique.

Let us begin by numbering the coins 1 through 12. [It is important that the coins be distinguishable, in spite of having the same weight--perhaps the dates on them are all different.

Weighing #1: Put coins 1, 2, 3, 4 on the left pan and coins 5, 6, 7, 8 on the right pan.