Chain, Chain, Chain, ...Chain of squares...

or All roads lead to two or one... guess Rick Regan over at Exploring Binary will like this.. and I think the actual next line in that song was "chain of fools", (rock me, Aretha) and yet I continue.... If you take a number, square each of its digits and find the sum you will get a new number(usually, is there more than one number that produces itself as its own iterate under this process). Do the same to that number, and the sequence continues. But eventually you have to come back where you started. Numbers with more than three digits will always produce a smaller number and three digit numbers will always be less than 9²+9²+9²= 243, so eventually, wherever you start, you end up with a number less than 243 (and 243 --> 4+16+9 = 29 so it gets even smaller).. what would be the last number that produces a number larger than itself? Ok, simple stuff, if you start with one, you get one and that's dull so let's go on. If you begin with two you produce the sequence 2--> 4--> 16--> 37--> 58--> 89--> 145-->42-->20-->4 and then repeats the cycle of eight numbers forever. Now the big conjecture... start with ANY integer (oooohhhh, he said you could pick ANY integer, how bold) and eventually, either it gets to one, or it jumps into this chain. Now I wonder; is there a way to prove (short of hacking them all out, which I have done) that there is no "other" chain that some numbers might drop into? 

John Cook did a computer search of all the numbers less than 1000 and they all went to one of these two absorbing states.

 I also have a feeling that as the numbers go off to infinity, the proportion of numbers that go off to one has some non-zero limit; in fact, I suspect it might be around 1/7 or just a tiny bit more, but don't have a clue how to prove that. Any takers? All conjectural rationale will be considered. I think the same kind of questions could be asked about forming the sum of the cube of the digits... would all roads lead back to one or two then? A quick answer trying only a few numbers... NO, Follow the orbits of 2 or 3 or 7 and they all go to separate self-replicating numbers or one-cycles.. 4 goes into a triplet of 55 250,133,55; so I guess my new question about cubics is... How many of these finite cycles are there?

Sunday Morning Math and Coffee

Sunday morning in East Anglia, and I am sipping my coffee as I read through some of my favorite blog sites... Do get over to Concurrencies where Steve Phelps has put up his favorite math movie, a really great explanation of Mobius transformations that should be seen by more HS students.

And since I have several blogs about coincidences, I should point out that Jon Ingram, also here in England somewhere (sorry Jon), has a nice simple problem about coincidences as well at his "Lessons taught, Lessons learnt" . Here is a teaser to get you started, right from Jon's page...

The average of a set of 64 numbers is 64.
The average of the first 36 numbers is 36.
What is the average of the last 28 numbers?

Then he very nicely answers the question:

Solution

The total of all 64 numbers is 64 X 64 = 4096.
The total of the first 36 numbers is 36 X 36 = 1296.
The total of the last 28 is therefore 4096 - 1296 = 2800,
which makes their average ²⁸⁰⁰/₂₈ = 100.

and adds, "and is it a coincidence that 64 + 36 = 100?"

OH! and I couldn't remember this, or where I saw it, but I found it again on Jon's blog... did you know that "1% of a day plus 1% of an hour is exactly 15 minutes"??.
Time for a refill, and then to touch off the notes on my next post...

A Serendipitous Coincidence? The First-Ever Pursuit Problem.

Just reading through some old copies of the Mathematical Spectrum from my great source of mathematical periodicals, Dave Renfro. Intrigued by a couple of posts about a problem from the ancient Chinese Chiu Chang Suan Shu or Jiuzhang suanshu (Nine Chapters of Mathematical Art...about 150 BC) submitted by David Singmaster.

The problem: "A water weed grows 3 feet on the first day, and its growth on each succeeding day is half that on the preceding day. A reed grows 1 foot on the first day and its growth on each successive day is twice that of the preceding day? When are they of equal size?"

Go ahead, stop reading for a minute and try to solve it because I give an answer (actually two different ones) below and I don't want to spoil the fun.

The interesting thing to me, was the two letters of solution. My pre-calc students came through exponential growth and decay a few chapters ago, so they would approve of the first solution that was submitted. It suggested that we assume that the height of the water weed was growing according to the exponential function hw= 3 (1.5)^d-1. The reed would reach a height of hr=3^d-1. Setting these equal we would find the heights are equal when d= log₂6; or at about 2.585 days. They also pointed out that the mutual heights would be 5.705 feet.

Simple, quick, and "Wrong" according to the next commenter. They pointed out that a careful reading of the problem stated that the water weed "growth on any succeeding day is half that of the preceding day", would meant that it grew 3 feet the first day, and then successively it would grow 3/2, 3/4, 3/8 ... feet on each day.. to find the height we should sum this geometric series. So he suggests the height of the water weed would be hw=. This would result in the weed being somewhat shorter after each day than in the previous solution. In the same way, the reed should, on successive days, change by 1, 2, 4, 8 etc feet, so it would have a height of 2^d-1 after d days. Setting these equal we see that the two plants will both reach a height of five feet (exactly) after d= log₂6; or at about 2.585 days.???? WAIT, that sounds familiar....where have I ... Oh YEAH!!!, that was the answer to the problem done the first way? WOW, what a lucky coincidence........ Well, NO... the good Professor who posed the problem stepped up to assure us that, in fact, if you used those same two approaches to similar problems they would always reach equal heights at the same time... (can you prove that???)

Try for yourself. I assumed similar means that the shorter grows at r times its previous days amount, and the taller at 1/r times the previous day. Try a few. In fact, it is frequently the case that the second method gives an integer solution, and when it is not, it seems pretty easy to adjust the growth on the first day of the two plants to make it come out an integer. If we call the first day growths W and R, then the solution works out to log_r(rb/a); where r is the common ratio of the plant whose growth each day is increasing.

Dr Singmaster points out that this ancient text solves both quadratics and cubics, as well as systems of equations (including indeterminate systems with more unknowns than equations) using the "modern" method of elimination. It also is the earliest known text to have used negative numbers, and includes the rules for all the arithmetic operations. It is also the oldest know source of chase or pursuit problems, such as this one. This is problem 11 of the 7th chapter. I had the good fortune to sit with my beautiful Jeannie as the only two non-Chinese speakers at a discussion about the text at the Needham Research Institute in Cambridge by a group of English and Chinese experts. The amount I was able to take in is a credit to the patience of the gracious hosts.

An Average Dismissal? rant about education and bad statistics

It was on my I google news, I had to read it...
"(CNN) -- A school board in Rhode Island has voted to fire all teachers at a struggling high school, a dramatic and controversial plan aimed at shoring up education in a poverty-ridden school district.
On Tuesday night, the board approved the plan by Frances Gallo, superintendent at Central Falls School District, to discharge 88 teachers at Central Falls High School."

ALL the teachers, NOT A SINGLE ONE that was capable...Did they fire the principal that hired and managed 88 incompetent teachers? (bet they didn't)...

SO HOW BAD WERE THEY? you ask.

"The firings come over the district's concern that teachers refused to spend more time with students to improve test scores." AHA, Was the superintendent ready to PAY them to spend additional time... time outside their contracted 5, 6 or 7 hours a day of classroom instruction??? (oh, they didn't say anything about that.)... Try to imagine another profession where that kind of demand would even be suggested... Hmm, are all the bankers who got us into this economic mess actually putting in extra hours at lower pay to help us get out...(pssst.. the correct answer is No.)..

But a teachers' union spokesman called the firings "drastic."

As my students would say....DUH!!!

And now the part I love, they use statistics.... (ummm poorly)...
"The spokesman also cited a 21 percent rise in reading scores and a 3 percent hike in math scores in two years."
Ok, What does that mean? What scores went up 21%.. Try to guess. Does that mean the median national percentile for the school went up by 21 percentile points ? Were they at the 20th percentile and now they are at the 24th (a 21% increase) or did they perhaps go from the 20th percentile to the 41st percentile (an increase of 21 percentile points).. Does it mean that EVERY kid in the school improved 21% over his previous (raw score? natl percentile score ?) WHAT DID THAT MEAN???

And of course, the article concludes with a stunning judgment of the school and the community...."Central Falls is one of the lowest-performing schools in Rhode Island." (Wow, at some schools they probably just took the teachers out and shot them...)

The statistics students in my class have already learned to ask..."Measured HOW?" Do they have the smallest number of people in the theater program? (see, that is a pun on "lowest performing"... rim shot please.... and it might be the truth.. we really don't know)

I have no idea how good, or bad the teachers in Central Falls are, but I am willing to wage heavy odds that they almost certainly are not the 88 worst teachers in Rhode Island. In fact (no elementary or middle school teachers were dismissed???) they may not be the 88 worst teachers in Central Falls... Wait one more wager... I be the superintendent is about to become much more familiar with the ins and outs of class action law suits...
and amazingly, there will be lots of people lined up to interview to work there next year... Eddie Izzard does a nice routine on the Anglican church in which the punch line is "Death or cake?" and in this economy, there will be people lined up to say.. "Death, please."...And now Mr. Ballew has left the soap box...next speaker...