TRURO — Voters narrowly approved one of four zoning amendments late Tuesday night at the annual town meeting. But town officials were still looking at the exact vote count on that article yesterday.

In a vote of 136 to 70, voters passed a new time limit on how quickly a cottage colony, cabin colony, motel or hotel can be converted to condominiums. The new limit requires that those properties be in operation for three years before being converted to condominiums.

The idea behind the zoning amendment is to slow the pace of condominium development in Truro and preserve more affordable accommodations for tourists, according to citizens proposing the warrant article.

Currently Truro does not allow condominiums complexes to be built outright in its zoning bylaws. Instead, property owners must build a cottage colony, cabins, motel or hotel first and then covert it to condominiums through a special permit.

The exact count of the vote — 136 to 70 —had town officials hitting their calculators yesterday. The zoning measure needed a two-thirds vote to pass. A calculation by town accountant Trudy Brazil indicated that 136 votes are two-thirds of 206 total votes, said Town Clerk Cynthia Slade.

But is it?  Is 136 a sufficient number of votes to be considered two-thirds of the total 206 votes?  Let’s check:

If you use the fact that and then proceed to multiply 206 by 0.66 you get 135.96.  There were 136 votes in favor which is  more than 135.96 so that means it passes, right?  If you think so, then you’d be WRONG!!! 

The main problem is the rounding.  In fact, or using repeated decimal notation, .  When you round, you are actually creating an error that, in this case, makes a pretty significant difference.

Think of it another way, lets compare 136 / 206 to 2 / 3.  First, just do it by decimal approximation:

My calculator cannot exactly represent either of these fractions but its accurate to 12 decimal places and I can clearly see that 136/206 < 2/3 so the vote should not pass.

Do you remember another way you can compare fractions?  Find a common denominator and convert each fraction, then compare. 

So, here we see that, again,

This second method of checking is even better than the first because there are no approximations involved.  We’ve confirmed, absolutely, that 136 votes out of a total of 206 does NOT constitute two-thirds.

Fortunately, a good citizen made an anonymous call in Truro, MA, to clear this up.  What perplexes me is that they decided they needed to let the State Attorney General’s office decide on the correct count. The mathematical explanation wasn’t good enough. Can you say quantitative illiteracy?

Read the entire story here.

Now, when I say “quick little code”, I meant an easy algorithm to implement, but not necessarily an elegant or amazingly small code that would accomplish the solution.  Here is definitely the smallest (shortest) code I’ve seen that will do it.

def r(a):i=a.find('0');~i or exit(a);[m
in[(i-j)%9*(i/9^j/9)*(i/27^j/27|i%9/3^j%9/3)or a[j]for
j in range(81)]or r(a[:i]+m+a[i+1:])for m in'%d'%5**18]
from sys import*;r(argv[1])

Here’s one that is slightly longer (185 bytes as opposed to the 178 above)

use integer;sub R{for$i(grep!$A[$_],@x=0..80){
%t=map{$_/27-$i/27|$_%9/3-$i%9/3&&
$_/9-$i/9&&($_-$i)%9?0:$A[$_]=>1}@x;
R($A[$i]=$_)for grep!$t{$_},1..9;return$A[$i]=0}
die@A}@A=split//,<>;R

HT: Scott’s Blog

Part of their project is to implement a numerical simulation of their forest fire.  I couldn’t let them have all the fun, so below is an example of my version of the simulation in MATLAB.  I have to hold off on posting the code until after they have handed in their project.

In the graphical representation of my simulation, green represents an unburnt tree, black is burnt and red is currently on fire.  The fire lasts for exactly one time step.  I also implemented a 3-D version, where a height of 1 is unburnt, 2 is on fire, and 0 is burnt.  I’ll confess to having way too much fun with this.

I have used a 200×200 lattice with p = 0.5.

Algebra, says Devlin, is a language, a very precise language written in symbols, and it’s everywhere: in nearly all electronic devices, every statistic and each Internet search engine – and, indeed, in every train leaving Boston.

"You can store information using it. You can communicate information using it," Devlin said. "Google has made billions capitalizing on algebra."

Yet our schools don’t always do a very good job teaching it, Devlin said. Instead of showing students the possibilities and beauty algebra offers, they ultimately steer frustrated and bored students away from math and the 21st century careers that use it – the opposite of the intended result.

…

Algebra, by the dictionary’s definition, is essentially abstract arithmetic, letters and symbols representing relationships between groups, sets, matrices or fields. It’s a way to find a piece to a puzzle using the pieces you already have in place.

It comes in very handy for engineers, financial analysts and sociologists, not to mention World of Warcraft video game players, some of whom use algebraic formulas to decide which weapon is more effective under certain circumstances – perhaps another hook to lure unsuspecting teens into seeing the useful side of algebra.

…

Laptop computer. The computer is just an implementation in electrical circuits of a special form of algebra (called Boolean algebra) invented in the 19th century. Ordinary algebra is used to design and manufacture computers, and is at the heart of how to program them.

Cell phone. A cell phone is a particular kind of computer. An important feature of cell phones is that your phone receives all the signals sent to every cell phone in the region, but only responds to signals sent to your phone. This is achieved by using signal coding systems built on algebra.

Parking cop. Today’s parking enforcement officers may carry equipment connecting them directly to a central vehicle database that registers your parking fine before you get back to the car and see the ticket on the windshield. Without algebra, such a system could not exist.

Hybrid car. Modern cars often come equipped with GPS, a highly sophisticated system that is designed using enormous amounts of mathematics that builds on algebra.

Delivery truck. Large retail chains use mathematical methods to determine the routing and scheduling of their delivery trucks; algebra is fundamental to those methods.

Stoplight. These days, stoplights are centrally controlled by computers, so there is even algebra involved in turning the light from red to green.

IPod. This is a math device in your hand. The iPod stores music using sophisticated mathematics built on algebra. And the iPod shuffle mechanism uses regular school algebra to order your songs randomly.

…

Even though it is a very pro-algebra article, my favorite quote was by an unknown source:

"Algebra … the intensive study of the last three letters of the alphabet."

From 360 (Unofficial Blog of the Nazareth College Math Department in Rochester, New York):

The Herald reported last week that a Traffic Warden was incorrectly ticketing cars in a Devon, England parking lot because of how he was using a calculator. In this parking lot, drivers would pay for a certain amount of time and then post a slip in the windshield with the time they’d entered and how long they’d paid for. One driver, for example, entered at 2:49pm and paid for 75 minutes.

Now 75 minutes is 1 hour, 15 minutes so the driver was covered until 4:04pm. But the Traffic Warden figured out the expiration time by entering in 14.49 into his calculator (for 1449 military time, which corresponds to 2:49pm) and adding on 0.75 (for the 75 minutes). He got 15.24, which he interpreted as meaning that the driver was only covered until 3:24pm. Since it was already 3:41pm, he issued the car a ticket. The car owner saw all this and tried to explain the error — that hours have 60 minutes, not 100, so standard decimal addition doesn’t apply — but the Traffic Warden didn’t see any problem and continued to ticket cars.

In good news, after appeal the incorrect tickets were repealed and a letter of apology sent.

A follow up email attempted to make the point that we aren’t paying that much more for gasoline considering a significant increase in fuel efficiency over the last 20 – 30 years. The examples cited were anecdotal and encouraged me to do a little research on my own.

I was surprised to see that the increase in fuel economy is a lot less than one might have expected over the last 30 years. According to the National Highway Traffic Safety Administration (NHTSA) the average gas mileage for new vehicles sold in the United States has gone from 23.1 miles per gallon (mpg) in 1980 to 26.7 mpg in 2007. This represents a paltry increase of 15% over the 27 year period. Even if you limit yourself to domestic passenger cars the increase is from 22.6 mpg in 1980 to 31.3 mpg in 2007.

Even more interesting to me is the fact that we have benefited from a relatively low cost of gasoline for an extended period of time. (see here) Adjusting for inflation we see a steady decline in the cost of gasoline dating all the way back to the 1920s. The only exception is the late 70s, early 80s and the last 5 years. Prices are at their upper limit even with inflation considered. When considering only yearly averages, the highest cost occurred during 1981 at $3.17 (adjusted to 2008 dollars). Through March of 2008, this year’s annual average has been $3.08.

Now back to the original point, on average the cost (in 2008 valuation) per mile was 12.8 cents in 1981 (when gas averaged $3.17 per gallon in 2008 dollars and the average fuel economy was 24.6 miles per gallon) . The average cost per mile, currently, is 13.6 cents (with a current national average of $3.63 per gallon and average fuel economy of 26.7 mpg). In the end, while it seems that we are paying a ghastly amount at the pump we aren’t that far above the historical high, nevertheless we are, in fact, paying more than ever.

Watson postulates that for intelligent observers to evolve, a small number (n) of very difficult evolutionary steps must be passed. Once passed, evolution occurs quickly until the next stage is reached. Complex and intelligent life evolved quite late on Earth and Watson suggests that this may be because of the difficulty in passing these stages. He suggests that n is less than 10 and most likely equal to 4. These stages include the emergence of single-celled bacteria, bacteria with complex cells, cells allowing complex life forms, and intelligent life.

Professor Watson uses the Earth’s fossil records to establish upper bounds on the probability for each state.

The work supports the Rare Earth hypothesis which postulates that the emergence of complex multicellular life (metazoa) on Earth required an improbable combination of astrophysical and geological events and circumstances.

Read more about his paper at Plus Magazine.  At the time I am writing this entry, the article is freely available at the Astrobiology Journal site.

Statistical methods have been developed to help provide an accurate recount procedure that takes into account the election results to verify that a sampling recount is 99% likely to be the same as a full hand recount.

The story below appeared on MathTrek.

By Julie J. Rehmeyer

Counting is hard. Neither people nor machines seem to be able to do it reliably. And that’s a nightmare for election officials who need an accurate ballot count to decide elections.

Eighteen states require officials to double-check the machine counts by hand for a portion of the ballots. But election officials have had little guidance on what to do with the recount results. If the election is close and the recount finds a few errors, should a registrar call for a larger recount or go ahead and certify the result? Most laws left it to their discretion.

Now Philip Stark, a statistician at the University of California, Berkeley, has developed a recount method that guarantees a 99 percent chance that the result is the same as it would be with a full hand count. Several counties in California plan to try out the method on ballot measures during the presidential primaries this year. If this trial and others go smoothly, California could adopt the method statewide.

…

MathTrek: Checking It Twice

The story below appeared on KHOU.com:

The Math Behind Houston’s Traffic.

By Lee McGuire / 11 News

It doesn’t take a genius to know that accidents cause traffic jams. But it did take a genius to figure out why sometimes rush-hour traffic slows to a bewildering stop for no apparent reason at all.

In the last few months, two mathematicians at the University of Exeter in England came up with this equation, which has become something of a validation for engineers urging new approaches to traffic management. It’s changing things in Houston already.

Rice University’s Rolf Ryham explains the man.

He said for the first time, this equation takes driver “reaction time” into account. What’s revealed here could hold the key to solving Houston’s traffic problems.

Basically it’s mathematical proof that on a crowded freeway, when one driver just taps on his brakes, the driver behind him reacts and brakes a little more. The driver behind him brakes even more, and so on until drivers actually stop moving. It’s the kind of jam that happens on 290 every day.

“Slowing traffic down a little bit during peak periods is OK,” Highway 290 Expansion spokesman Stephen Hrncir said. “It’s the near-stop conditions we need to avoid.”

It’s called “cascading.” You can see it by speeding up video of 290 at rush hour. There are waves of congestion moving backward. The ripples start when someone up front does something that causes drivers behind him to brake, starting a cascade of slowing traffic.

…

The math behind Houston’s traffic | TOP STORIES | Breaking Houston News, Weather, Sports, Traffic, Video from KHOU.com | 11 News

Try out the Bulbous Blob Puzzle.

Professor Albert von Braun, noted food researcher and author of the Chinese Laundromat Cookbook, has just made a major breakthrough on his long sought after Grand Unified Meal Theory. Working in his laboratory late into the night he has, at long last, proved the mathematical link between the five foundation sauces of French cuisine and the thirty-one basic flavors of ice cream.

Your job is to free the reb blob from the laboratory. Can you accomplish this task before the tasty blobs of death make a meal out of you?

Plus, if you really enjoy yourself you can order the low-tech version that you can play without a computer!

Tags: math-puzzles, mathematics, brain-teasers