High School SMILE Meeting
04 April 2006

Our ranks were a bit thin this afternoon, since many of our group attended the ISPP meeting today in Lake Forest College.

Fred Schaal (Lane Tech)                      Looking for 6174
Years ago, when calculators were new, Fred came across a very interesting phenomenon. Write down any 4 digit number (such as 1234) and arrange the digits so that there is the largest number possible (4321) and the smallest possible number (1234) and subtract the smaller from the larger. In this case you get 3087. Repeat the process with the new number 3087 (i.e., 8730 - 0378 = 8342). Keep repeating the process.  We obtained

1234 ® 3087 ® 8352 ® 6174 ® 6174 ® ...
Eventually (in the case in question by 4 iterations) the process produces a result  which no longer changes (in the case in question (7641 - 1467 = 6174). Porter pointed out that starting with any 4 digit number, the result will be the same repeating  number 6174.  We asked ourselves why this works, but could not come up with an answer. We then tried it with a two digit number to see what happened.  Here is what we obtained:
12 ® 09 ® 81® 63 ® 27 ® 45 ® 09 ® ...
For this case, the process has reached a sort of  loop in which the same five steps (09 81 63 27 45) occur over and over.  Fred also wondered if it would work in a base other than  base 10? We have no idea, just as we don't know for base 10 numbers of various lengths.  For four digit numbers of base 8 we obtain the following sequence:
1234 ®3065 ® 6143 ® 5063 ® 6143 ® ...
Thanks for simulating our thinking, Fred.

Walter O. MacDonald (VA Hospital and CPS)                The EnergyCel
brought a video describing the Energy Cel. It claims to increase the efficiency of internal combustion engines when placed around the fuel line. The theory is that the magnetic field produced by the magnets in the device breaks up clusters of fuel molecules, resulting in more complete combustion. One dynanometer test overseen by a news team showed a 10 % increase in gas mileage and a road test (overseen by the same news team) showed a 27 % increase in gas mileage. The cost of  the device plus installation is about  $300. Despite the reported results our group was skeptical! Walter, however, put one on his car and his anecdotal report is that his car seemed to run better (although he did not calculate fuel efficiency before and after). Porter also pointed out that if the EnergyCel results in a leaner mix of gas to air, it might cause too hot a temperature in the cylinder, which might result in burned valves. A spirited discussion of modern internal combustion engines, modern gasoline, etc. ensued!  Let us know how your gas mileage goes!  Thanks, Walter.

Karlene Joseph (Lane Tech)              Go Figure
Karlene brought a children’s book called Go Figure by Johnny Ball: http://www.amazon.com/Go-Figure-Totally-Numbers-Nonfiction/dp/0756613744.  It is a book about numbers. One section asks the question, “What if we had no numbers?” We couldn’t report sports scores, TV listings, and a million other things! There is also a section on the pyramids (Karlene loves ancient cultures!). The dimensions of the pyramids are arranged so that several interesting relationships occur. Another interesting item is that about 100 years ago the State of Indiana tried to pass a law decreeing that p would be exactly 3.2!  For details see The Indiana Pi Billhttp://www.agecon.purdue.edu/crd/Localgov/Second Level pages/indiana_pi_bill.htm.

Here’s another one about p (we thank Archimedes for this one). A circle inscribed inside a square of side 1 has a diameter of 1 and a circumference of p. The perimeter of the square is 4, which means p must be less than 4. Repeat with a hexagon inscribed inside the circle (hexagon perimeter 3.0); now p must be greater than 3. Archimedes iterated this up to a 96-sided polygon and found that 223/71 < p < 220/70 (or 3.14084507 < p < 3.142857143).  For details see Archimedes Traps Pihttp://physics.weber.edu/carroll/archimedes/pi.htm.

Then Karlene gave us each a triangle cut from a piece of paper (random shapes and sizes). We then talked about the things we know about triangles. One is that the total of the three angles is 180 degrees (the same as a straight line). Karlene then had us tear off the three corners and use the little angles to tuck into one another and see if we got a straight line and we surely all did! One can do the same with a planar quadrilateral to obtain 360 degrees. Neat stuff!  Thanks, Karlene.

Terri Donatello (retired)                          Newspaper Articles
brought in two recent newspaper articles. In one a team of MIT scientists tried to recreate Archimedes death rayhttp://web.mit.edu/2.009/www//experiments/deathray/10_Mythbusters.html. While they did not disprove that it actually happened, they showed that it was unlikely to have worked. A second article was about the current neutrino project at Fermilab (in which IIT is significantly involved). It described a continuing experiment in which neutrinos made at Fermilab sre beamed 450 miles to a detector 0.5 mile below the surface in the Soudan Underground Iron Mine in northern Minnesota:  http://www.dnr.state.mn.us/state_parks/soudan_underground_mine/index.htmlThanks, Terri.

Our next SMILE meeting will be on Tuesday April 18, 2006. See you there!

Notes prepared by Ben Stark and Porter Johnson.