Our 7th Ski Trip to Taos Ski Valley

Last week, we went skiing at Taos Ski Valley for the 7th time in 7 years! Due to dry weather and relative high temperature there, we nearly changed our plan. Fortunately two days before our scheduled departure time, a winter storm brought a foot of snow to Taos. We were also fortunately to have another family to go skiing with us.

On the Road

The 700 mile drive takes long time, but it is actually always a good twosome time for Lily and I. As Kids were quietly watching a movie, we had time to talk about topics other than kids' education and discipline, such as tax cut, health care reform, "dream" act, ...; we also spent time talking about our work, long term career plannings and expectations, exchange or re-exchange research philosophy, personnel management, team building ...

In this trip, for the first time, Justin actually stopped watching movie, stopped listening to MP4, he listened to our conversation with great interest for extended period of time.

In between movies, we sometimes play traffic related math games, other times we play Whiz Quiz type questions and answers. When we were tired of talking, watching and playing, we dosed off, of course except the driver.

The long trip was not boring at all, and it was not tiring either since Lily and I shared the driving responsibility.

On the Slopes

It was snowing in the mountains as we reached the ski resort for the first day. We all started at the novice slope to warm up. After a few runs, we headed into the mountains. Justin went alone - the rest of the family was too slow for him now; I escorted Lily and Nick. The puffy snow covered trails really helped to build up their confidence in their skiing skills. One run after White Feather ( a green trail), Nick wanted to go on to Porcupine ( a blue trail). I was worried about Lily on White Feather on her own, but she was confident that she could handle it. It was a pleasure to see Nick zigzagging down Porcupine with confidence, it was a even bigger pleasure to see Lily catching up with us at the trail intersection. Due to low visibility, we quit skiing by 3:30pm, but we had already made 5 runs. Justin skied until the last lift, made 8 runs.

The next day was warmer, windy and the cloud broke up from time to time, revealing the crystal blue sky above the mountains and letting in flashes of sunshine. We went to the slope early to catch the freshly groomed trails. We took lift #1, skied down White Feather a bit to lift #2, which took us to the top of the front side mountain. The view from top, ~ 11800ft elevation, was magnificent and beautiful under the sunny blue sky. The easiest way down was Bambi, a blue trail. It was Lily's first run on a blue trail. Nick was in the front leading the way, Lily in the middle, I was in the back so I could easily help them when they fell. We got to the base in 20 minutes or so, not bad at all. We went to pick up Michelle, the daughter of our friends,as planed, for her first run in the mountains. She was nowhere to be found. We went up to the mountains skiing and stayed there using lifts 2 and 8 alternately to avoid the crowds at the base, until lunch time.

It turned out that Michelle and family met her friend Celine and family. Michelle skied with them, and Justin skied with Harvey, Celine's older brother. By the end of the day we enjoyed all opened trails!

The third and last day Michelle skied with us in the morning in the mountains, her parents took ski school. Justin went alone until Harvey arrived. After lunch, Celine and parents, Nick and parents plus Michelle went up together, Justin and Harvey went their separate way.

The two girls, Celine and Michelle, and Nick really got along real well. They were considerate to each during downhill skiing, they had fun playing snow during break, later they rode the lift together not allowing any parents sitting with them. It was a plesant sight: the three of them on a white slope, one after another, in their colorful outfits. They also energized each other: Nick and the girls went up the slope one more time as the 4pm closing time was approaching. He had way more runs than he had ever had before. Justin and Harvey had great time skiing together as well. Even more amazingly, they talked to each other in French since both of them took French class in high school.

After a day at ski school, Michelle's parents skied down White Feather on their own during the last run of the day! I was really pleased to see them enjoy skiing and succeed in keeping up with their daughter on the slopes.

That night, the three families had dinner together at Song's Asian Restaurant at Taos. All 11of us had great time despite relative long wait for the table and food.

Side Trips

Due to the long distance from DFW to Taos, we took side trips from time to time to make the most out of the trip. We have been to Albuquerque to visit friends from our graduate school time, and many other places for sightseeing, including Santa Fe, Banderlier National Monument, Las Alamos National Lab, Palo Dura Canyon State Park, Petroglyph National Monument and Pecos National Historic Park.

This time we went to Rio Grande Gorge Bridge right outside Taos, and we stopped at Trader Joe's at Santa Fe. Michelle and family went to White Sand National Monument in southern New Mexico.

Concluding Notes

Some people were amazed that we could go to the same place skiing so many times. For us, every trip to Taos is different, from dramas on the road, to our improving skiing skills, to different companies we have had, to choices of slopes, to the ever changing magnificent and beautiful mountains at Taos Ski Valley. Even with so-so snow conditions this year, we had a wonderful time there.

We will visit other great mountains for skiing in the future. Even then, I am sure we will go back to Taos Ski Valley from time to time because of its enchantment, its closeness to Texas, our familiarity with it and our fond memories skiing there - We are enchanted.

Turtle and Kids' Education

Here is an interesting observation of our turtle Francis.

Every time I watch the turtle climb the stump in its water tank home, it fails except once. Based on this observation, I would assume that Francis is not able to climb onto that stump on a consistent base. But the matter of fact is that it gets on and off the stump a few times a day, it can scale the the stump consistently!!! It does all this when I am not watching.

The reason I relate this observation to kids' education is that similar situations occur during kids' development. Parents, especially highly educated parents, like myself, tend to rush to help kids at the first sign of failure or potential failure in whatever endeavors they are in. When parents react like this, it robs the kids' chance to work things out on their own; when parents consistently react like this, it may create kids' dependence on parents, make them not able to handle difficult problems, or tough situations on their own.

I know that I have this tendency to intervene when my children seem to struggle at their activities. I have been trying really hard via various approaches to refrain from intervening.

For example, when my children work on hard math problems, I am really prone to jump in to help at the first sign of their struggle. After a few protests from them, I learned to consciously step away from their desks when they work on their tough problems. Just like the turtle, they can successfully solve many difficult problems after some struggle. This doesn't mean I would not help - I would say "Show me how you solve the problem when you are done", "Try it first. Let me know if you need help". Even with this self imposed constraint, I still cross the line from time to time.

An interesting example occurred at Youth Engineering Fair this summer. Nick was in his grade group for a fun oriented competition - packaging a raw egg with Styrofoam cup and other provided materials and then dropping it from 10 feet height. Though tentatively approaching the competition area, he declined my offer to sit by him. Then the organizer announced that parents should stay away from the tables where kids would work on their projects. As the kids were instructed to start, a few kids apparently knew exactly what do, Nick was slow to start. He did not do anything, he checked the materials, looked around a bit and played the materials and egg a bit. I was inclined to help but could not due to the rule, so I walked away from the area. I watched older kids flying model airplanes, building bridges with candies.

I went back to the "egg" area when it was announced that the organizers would start to drop packaged eggs.

First the eggs were dropped from 3 feet height - everyone passed the first test except one. Nick's package passed the test as well, but a few foam balls fell out of the Styrofoam cup. I was trying to tell him to reinforce his package a bit using left over materials on his table. He declined to do anything except put all stuff back in place, maybe with a little bit rearrangement. Next the eggs were to be dropped from 10 feet height. When Nick's package was dropped from 10 feet height, the cup, which had a balloon attached to it, landed on the floor on one side of its bottom edge, it fell, foam balls rolled out, and then the egg rolled out without breaking!!

There were total of 4 eggs not broken after the 10 foot drop. The tie breaker was the amount of materials used - Nick used least amount of material for his package - he won the competition!

When I allow children to work things out on their own, they may succeed, they may fail. This is what I have been trying to do :

Give children time, let them try, allow them to fail and restrain from intervention.

However, it is easy said than done.

Bass Performance Hall Ballet-The Nutcracker

On December 10^th, I went with my family to watch the Texas Ballet Theater perform Ben Stevenson’s The Nutcracker at Bass Performance Hall in Fort Worth. The well known story of the Nutcracker begins as Herr Stahlbaum throws a Christmas party, and all the guests are arriving, and the children are receiving gifts. Herr Stahlbaum’s children, Clara and Fritz, spend a good deal of time chasing after each other and arguing with each other in this opening scene. Dr. Drosselmeyer, Clara’s mysterious godfather, suddenly appears, and entertains the children with a numerous number of magical tricks, as well as life size wind-up dolls. Dr. Drosselmeyer also presents a wooden nutcracker, which he presents to Clara. Out of jealousy though, Fritz breaks off the head of the nutcracker, but the nutcracker is quickly mended by Dr. Drosselmeyer. The party ends as the guests begin to leave, and the Stahlbaums retire to their respective rooms. During the middle of the night, Clara is woken by the sound of a mouse running across her room. Then, as the clock strikes midnight, giant mice appear and attack Clara. Dr. Drosselmeyer suddenly appears, and uses his “magical powers” to cause the Christmas tree to grow forty feet tall, and sends life size toy soldiers, led by the Nutcracker, to Clara’s rescue. In the final battle between the Nutcracker and the Rat King, the Nutcracker is wounded, and Clara comes to his rescue by throwing her shoe at the Rat King. The Nutcracker then transforms into a prince, and the Nutcracker Prince transforms the house into a Land of Snow, where the Snow Queen and her Snowflakes send off the Nutcracker Prince and Clara to the Kingdom of Sweets in an enchanted sleigh, ending Act I. In Act II, upon reaching the Kingdom of Sweets, Clara is entertained by numerous dances: a Spanish dance, an Arabian dance, a Chinese Dance (two men performing acrobatics with swords), a Dance of Mirlitons (better known as Dance of the Reed Flutes), a dance with Madame Bonbonaire and her children, a Russian dance, and the Waltz of the Flowers. The Sugar Plum Fairy and the Nutcracker Prince end the performance with a grand Pas de Deux (a dance for two divided into five parts: entrée, adage, variation for danseur-male dancer, variation for danseuse-female dancer, and coda). Following a Coda, Clara drifts to sleep and reawakens in her own bed.

I found the Texas Ballet Theater’s performance of The Nutcracker highly entertaining, although I altogether do not understand ballet very well. In Act I, I found Fritz’s devilish antics highly amusing; especially since him and his fellow mischief managers would get caught every time and reprimanded by their parents. Many laughs could be heard during Act I during the Christmas party as multiple hilarious events occurred. After the party, the clock, when chiming midnight, did not actually have its hands set to midnight, but to 11:30 PM to my amusement. The battle between the rats and the toy soldiers was entertaining, if not outright odd. Scenes from the battle include a giant rat waving a white flag when the cannon was fired, and rats and toy soldiers fighting while the soldiers did not really make use of their swords. Most amusingly however, the rats all seem to be afraid of shoes and pillows, and in the final scenes of the battle, when the Rat King wounded the Nutcracker, the Rat King appeared to have passed out or died from a girl throwing a shoe at him. Act II was confusing, for I did not understand the concepts of ballet and such did not understand the multiple dances performed. The music however, was very familiar to me, as it was music from Tchaikovsky’s score for the Nutcracker, among notable pieces included the March, Dance of the Reed Flutes, and Dance of the Sugar Plum Fairy.

Hitch and Bike Rack

Every time we went camping, I wished that we could have brought our bikes to the camp. Just by observing those who brought bikes to camping sites, I knew that I would need to install a hitch under a vehicle, and to attach a bike rack to the vehicle through the hitch.

After years of half hearted trying to install a hitch and bike rack, I finally did some serious research earlier this year on what I should do to buy and install a hitch and bike rack at a reasonable price.

Where to buy Hitch

Of course the first place to check is the car dealers. It is no surprise that the package price for a hitch at dealers' is extraordinarily high. For my van, the price for a hitch plus other required parts is ~ $1000! It would require ~ $1200 to 1500 to have the hitch and rack installed.

When I finally decided to take actions, I did more research on the hitch and I found two places that sell hitches and trailer package, "The Hitch Store" , "etrailer.com" at good prices. The two places have plenty of information on what other parts are needed, how to install a hitch to a given make/model of vehicle.

There are two common size hitches, 1 1/4" and 2" receiver tube size hitches. It appears that 2" size receiver is a more common size for bike rack, i.e. more bike racks for 2" receiver hitch.

Since the sole function of my hitch was for carrying bikes via a rack, I decided to purchase the hitches listed at the two sites though I believe that dealer's hitch would have better quality especially for trailers. Then we went to amazon.com to search for the exact same hitch I identified at the two sites, and found a even better price. We bought a hitch with necessary accessories for a price under $180!

Install Hitch (DIY)

After reading the installation guide at the thehitchestore.com, and watching a installation video on youtube.com, I realized that it was not hard to install the type of hitch with one possible problem - I need to move the exhaust out of the way.

During installation, Lily and I followed the installation guide step by step without any problems until the step of removing the exhaust from rubber isolator. I went online a couple of times to watch the video and studied the connection at the rubber isolator. We decided that we would damage the rubber isolator if we followed the instruction exactly. Instead of removing the exhaust, we slided the hitch side way into place, and installed it onto the frame of the car.

Where to buy Bike Rack

The two sites I listed above also sell bike racks which match the hitch of one's choice. Upon further research on bike racks, I decided to buy a rack at local sporting goods stores so I could have a look and feel at it.

The Dick's sporting Goods website has an excellent bike rack buying guide. Since I have a hitch, so the mount type of of my bike rack (bike carrier) would be hitch mount. The price for this type of racks ranges from $120 to $450. Two key considerations in selecting a bike rack in addition to price: 1) required receiver size should match the hitch's receiver tube size per bike rack manufactures - they recommend against using receiver adaptor; 2) the style of the rack - it is better to have a swing away or tilt back style which allows rear access to a vehicle without removing the rack. Again for my type of applications, I picked a low end rack priced at $120, which has tilt back mechanism.

There was an interesting episode during the purchase of the rack at Dick's Sporting Goods. We went there on black Friday, Lily and Kids went for sports apparels; I went to the bike section, and located the rack I wanted. There was no sale on the item. I talked to a sales person. I told him that I would like to buy the rack, but the box was kind of damaged. I told him I would buy the item with damaged box at a discount, or he should find another item with good box. He called the bike department manager, who offered me 10% discount!

Assembling the bike rack and mounting it onto the hitch was a snap.

When we tried to mount the bikes onto the rack, we encountered difficulty!! New style bikes don't have a horizontal cross bar, and the tubes on the bikes are not round. When the bikes are mounted, they would tilt. The new style bikes have been around for sometime, there must be ways to mount them on a rack. It turned out that we need false crossbars (British terminology) or bike rack adaptor bars, or bike beams, priced around $20 ~ 30. It would be another week before I could really mount the bikes on the rack for a test run.

Try it Out

I got the adaptor bars last week. Last Saturday evening, I finally got a chance to try the hitch mount bike carrier. With the adaptor bars, mounting of the new style bikes were much easier, and the bikes could sit on the bike holders straight up after mounting. It took me about 30 minutes to identify the best way to mount a bike, the best sequence to mount four bikes, and how to add additional securing mechanism. I believe that it would only take ~ 10 minutes to mount 4 bikes when we go on a camping trip next time

After I was satisfied with the mounting, I drove my van around my block for a trial run. I was really nervous at the beginning. The hitch was solid, the rack was sturdy. Quickly all the uneasiness vaporized. The bikes were steady on the rack during the drive over bumps and potholes, turns and slight acceleration.

We are ready to have more fun in our future family explorations of nature.

Thanksgiving Dinners

It is three years in a row that we had Thanksgiving dinner at a friend's house. Whether the dinner is in our house or other's house, it is usually a multi-family affair because for first generation immigrants like us, friends are our extended family. One advantage of having Thanksgiving dinner at a friend's house is to meet people out of our circle - friends' friends. This time I met a guy who is a manager of a factory, another guy who loves hunting!

It was a very typical Thanksgiving dinner. On top of traditional Thanksgiving food, turkey, cranberry sauce, mashed potatoes and gravy, sweet potato pie (in place of pumpkin pie), we had many many side dishes - salmon, chickens, pork rib, soup ..... all delicious. After dinner, the group naturally divided into three groups - wives were together in dinning room talking about kids, kids' education, religion ....; The husbands formed another group in kitchen area talking about politics, sports, and hunting - a rare topic to me...; Kids were glued together by the video games.

I have always enjoyed Thanksgiving dinner with family and friends. Over the years, two of the dinners are most memorable to me - my first ever thanksgiving dinner, our first Thanksgiving dinner after moving to Texas.

My first thanksgiving dinner was some 20 years ago, arranged by international student office at my school and hosted by a local family. It was a formal traditional Thanksgiving dinner. The host and hostess, along with their children and grand children, plus two international students sat by a huge dinning table. The host started the dinner with a prayer and giving thanks for a good year they had had. We had turkey, cranberry sauce, mashed potatoes and gravy, pumpkin pie and bread, with plates passing around just like what I saw in movies. It was a cold Thanksgiving evening, I was however warmed by the host family's kindness and generosity. How I wished then that I would one day celebrate Thanksgiving with my own family.

Several years ago, we moved to Texas, starting a new chapter in our life. As our first Thanksgiving at Texas was approaching, our old friends from our graduate school years, invited us to have Thanksgiving with them at their Oklahoma home. On the way to their home, I felt like I was on the way to visit a relative, to visit my brother and sister, just like millions other Americans on the road that day. The Thanksgiving dinner was a formal one. Their family including grandparents and ours, total of 10 people sat by the dinning table celebrating Thanksgiving. Since we had not seen each other after we graduated. it was also a reunion. We took many pictures during that visit. One photo I liked the most from the bunch is one that the two couples, each with their younger kid in laps, sat in a living room sofa. We looked young, healthy and content. Sadly, our hostess passed away earlier this year.

To me, Thanksgiving dinner symbolize "Family togetherness", "Enjoy our life" and "Be thankful for what we have and cherish them".

Be Thanksful and Give Thanks!

Autumn, Harvest and Thanksgiving

Autumn comes to North Texas later than usual this year. However, when it comes, it is as good as always - crisp air, blue sky, and splendid foliage. Last Sunday morning, I walked around our blocks to capture the magic of the Autumn.

As I stepped out of the house, I was instantly refreshed by the crisp cool early morning air, and the bright sunshine. My feathered neighbors were busy collecting wigs, nuts, ... and they were chirping all around. The splendid foliage looked so much more brilliant under the clear blue sky.

This red foliage is very common in our neighborhood

The blue sky, and the bright sunshine made this bright yellow leafed bush so much more beautiful. They also made mixed colored brown-yellow leaves look great.

The tall grass with white blossom at a neighbor's backyard is a standout among the common foliage of red, yellow, brown and green.

persimmon

red fruit

pine cone - seed

fruit

yellow fruit

white nut - seed

What really drew my attention this morning was the different kind of fruits or seeds on the trees this autumn. They really symbolize what the season is all about - a season of fruition, a season for harvest.

What a beautiful Autmn we have!

I am thankful to Mother Nature for this brilliant Autumn Sunday morning. I am thankful to my neighbors for their plants/bushes/trees that bring the splendor near to my home.

Let us cherish and protect the planet we rely on, the community we live in.

Fermat's Last Theorem - The Odyessy

It took 360 years for many generations of great mathematicians to finally prove Fermat's last theorem. The theorem itself has no practical significance even in a mathematical sense. During the struggle to prove the theorem, many new techniques were developed, new theorems were proved, in particular the Taniyama-Shimura Conjecture was proved which has tremendous implications in mathematics.

Here is the timeline of major milestones in the establishment and proof of the Fermat's last theorem according to the book "Fermat's Enigma" by Simon Singh.

600BC Pythagorean theorem

1637 Fermat's last theorem proposed

1753 First breakthrough in proving Fermat's last theorem by
Leonhard Euler for n=3

1825 Using a method proposed by Sophie Germain, Gustav Dirichlet
and A. Legndre proved Fermat's theorem for n=5

1839 Gabriel Lame proved the theorem for n=7

1847 Spectacular failures - Proof of the Fermat's last theorem
proposed by G. Lame and A. L. Cauchy was shown to be wrong by Ernst Kummer

1908 Paul Wolfskehl found a mistake in Kummer's paper and corrected it. He created Wolfskehl Prize for the first person who proves the Fermat's last theorem

1955 Taniyama-Shimura Conjecture proposed by Japanese mathematician Yutaka Taniyama and Goro Shimura about the relation between modular form and elliptic equation

1984 Gehard Frey showed that if Taniyama-Shimura Conjecture can be proved, then Fermat's last theorem is automatically proved

1986 Andrew Wiles started working on proving Taniyama-Shimura Conjecture for purpose of proving Fermat's last theorem

1988 The claimed proof by Yoichi Miyaoka was shown to be wrong

1993 Andrew Wiles announced his proof of Fermat's last theorem
1993 An error was found in Wiles' proof by referees

1994 Error fixed by Andrew Wiles and Richard Taylor

1995 The proof published in Annals of Mathematics

1997 Andrew Wiles was awarded Wolfskehl Prize

Fermat's Last Theorem - Numbers

The history of number theory decribed in the book "Fermat's Enigma" provided additional excitment to the reading. It described how common number concepts we learn from elementary school to high school, such as prime number, negative number, irrational number, imaginary number, and the concept of infinity, were developed hundreds and thousands of years ago, along with the evolving story of proving Fermat's last theorem. (Note: Fermat's last theorem: The equation x^n+y^n=z^n (*) has no integer solution if n>2. n is an integer)
After the proof for n=3 by Euler, there was no progress in the proof for nearly 75 years until a woman mathematician Sohpie Germain proposed an approach for certain type of prime numbers, such that p and 2p+1 are both primes. For example 2,3,5 are such primes, 7 is not.

Prime Numbers
Prime numbers are those which can be divided only by itself and 1, such as 2, 3, 5, 7, 11, 13, 17, 19, 23, ...7919 (the 1000th prime number) ...., as one can see from the list, there are much less prime numbers than whole numbers. The Fermat's last theorem seemed to be not that daunting any more once ancient mathematicians realized that if they could prove the Fermat's theorem for prime numbers then they proved the theorem because all numbers can be factorized into prime numbers.
Unfortunately there are still infinite prime numbers!! The proof was initially given by the Greek mathematician Euclid of Alexandria, i.e. Euclid. The proof is remarkably easy to understand. Here is the excerpt of the proof.
Assume there are only finite number of prime numbers, p1, p2, ...pn, then we can construct a new number q
q=p1 x p2 x p3 x ...x pn +1.
If q is not a prime number, then it must be factorized by prime numbers, bu it can not be factorized by existing prime numbers due to its construction. q must be either factorized by new prime numbers not in the known finite list, or q itself is a prime number. So there are infinite prime numbers!
The method of the proof is called proof by contradiction. It is a powerful method.
Perfect Numbers

Almost opposite to prime numbers, the perfect number is a number which equals to the sum of all of its divisors excluding itself, e.g. 6=1+2+3. The first 4 perfect numbers are
6 = 1+2+3
28 = 1+2+4+7+14
496 = 1+2+4+8+16+31+62+124+248
8128 = 1+ 2+4 +8+16+32+64+127+254+508+1016+2032+4064
There are remarkable less perfect numbers than prime numbers. It is very hard to find perfect numbers, up to 2009, only 47 perfect numbers were found, the 47th perfect number has 25956377 digits! (more than 25 million) - here is the current list
Pythagoras and followers studied perfect number extensively and found many interesting properties they posses, such as all perfect numbers can be written as the sum of consecutive counting numbers, e.g. 8128=1+2+3+4+...+127. Another properties they posses is that all perfect numbers can also be written as 2^n * (2^(n+1)-1), 8129= 2^6 (2^7 -1).
Perfect numbers are perfect in themselves. One claim went as far as saying: "6 is a number perfect in itself, and not because God created all things in 6 days; rather the inverse is true; God created all things in 6 days because this number is perfect."

Finding perfect number is hard, verification of a large perfect number is not easy either. I used excel to verify 8128, it would take a computer program with a smart algorithm to check on the very large perfect numbers. Verification of reasonably large perfect numbers in Excel would be good mental exercises.
Mathematical Revolution due to Power of Logic

Typical thinking is that if a person is logic in nature, he is less inventive because being logic means you follow existing rules. This is not true in the world of math, the the revolutionary concepts in the history of math came from exact logic and deep thinking.

In mathematics, there is need for completeness, which simply says that for a given rule of calculation, however you apply it, you should get a results. The concept of negative number was developed by Hindus because of the need of completeness. Say 5-3=2 was natural for ancient people, what about 3-5? which can not be expressed as a natural number (1,2, 3, 4, ...) - this need of completeness and logic in math led to the development of negative number concept!

Similar story occurred to the discovery of irrational number in ancient Greece. They believed that all numbers were rational. When people pondered what a number would be square root of 2. It was found that no rational number p/q could represent square root of 2, logic demanded the expansion of math concept to irrational number, a revolutionary concept at the time, which caused the life of its first discoverer. The development of the concepts of zero and imaginary number happened in striking similar fashion. Learning how ancient great mathematicians developed the math concepts we take for granted is really exciting. To replicate what they did is fun for amateurs like me. I learned the concept of irrational number in high school, but I never asked why. Now I can easily prove that square root of 2 is irrational using the method proof by contradiction. - it is exhilarating!

Fermat's Last Theorem - A Primer

Fermat's Enigma, written by Simon Singh, is a book about Fermat's Last Theorem, its origin, history and how it was officially proved in 1997. One day Lily picked up this book for kids' non-fiction reading. I scaned a few pages and could not put it down - I was fascinated.

Fermat's Last Theorem

The theorem was in fact a conjectecture, first proposed by french mathematician Pierre de Fermat, around 1637. It was based on the Pythagorean Theorem:

In a right hand triangle, the square of the hypotenuse (z) is equal to the sum of the square of the other tow sides, i..e. x^2+y^2=z^2.

A peculiar feature of the equation is that it has infinite whole number (integer) solutions, e.g. (3,4,5), (5, 12, 13), .....

It is noted that Chinese and Babylonians discovered the relation one thousand years before Pythogoras. However it was mathematically proved by Pyththagoras of Samos, an ancient greek mathmatician (~ 600BC), and thus the name after him.

Fermat's last theorem
The following equation

x^n+y^n=z^n (*)

has no integer solution if n>2.

It is intriguing that equation (*) has infinite number of integer solutions for n=2, and has no integer solutions at all for n>2. For 350 years since the proposal of the theorem no body can prove or disapprove the theorem.

People behind the Math

The math involved in the history of proving the Fermat's last theorem is interesting. The life stories of the involved mathematicians are fascinating - their life, their work, their failures in proof, triumphs in reaching major milestones. Five mathematicians were of great significance to the Fermat's last theorem and its proof: Pythagoras(600BC), whose Pythagorean theorem inspired P. Fermat to make the conjecture; Fermat, who intrigued and frustrated mathematicians centuries to follow with his notes on the margin of a mathematics book(1637); L. Euler, the 18th century genius, who made the first breakthrough in proving Fermat's last theorem (1753), 100 years after Fermat's death; P. Wolfskehl, who helped to sustain interests in the proof of the theorem by setting up a Wolfskehl Prize in 1908 and Andrew Wiles, the Princeton professor who proved Fermat's last theorem in 1997.

Fermat, who was actually a judge, was brilliant and smarter than most of his contemporary mathematicians. He was into math because it was fun for him. He did not publish any of his work because he did not like to provide full proof of the problems he solved. He usually just wrote down an outline of how he solved the problem. Of course the most famous such sketch is the note he left on the margins of the book II of Arithmetica. He stated his conjecture there and then he continued to write " ...I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain". However he did enjoying challenging others by posting problems he solved. He had particular pleasures to see others struggle on those problems! Thanks to Fermat's son Clement-Samuel, who collected his father's notes and published them in the book Arithmetica Containing Observations by P. de Fermat, we know Fermat's remarkable work in number theory, and the ensuing centuries of struggle to prove his last theorem.

Pythagoras of Samos, an ancient Greek mathematician (~ 600BC) traveled the world to learn math, especially from Egyptians and Babylonians. He laid the very foundation of mathematical proof on top of Pythagorean theorem. A very interesting story was that he paid his first student three silver coins for each lesson taken. After sometime, the boy became enthusiastic about mathematics. Pythagoras then stopped paying his student, pretending that he did not have any more money. The student offered to pay to continue to receive education from Pythagoras.

To advance mathematics, he established Pythagoras Brotherhood, a cult type of mathematical school. He believed everything was a rational number - the whole numbers and fractions. He took this belief so sacredly that it prevented him from accepting the existence of irrational numbers, such as square root of 2. A student of Pythagoras, Hippausas attempted to find a fraction for square root of 2, and eventually he proved that no such fraction existed. Pythagoras did not want to admit the existence of irrational number, but he could not deny its existence by mathematical logic. Tragically he resorted to force, sentencing the student to death by drowning!!

Leonhard Euler, the great mathematician from Swiss, worked in Russia most of his life. Contrary to Fermat, Euler liked to write papers. He proved the theorem for n=3 using imaginary number to plug holes in his proof. His attempts to extend the proof all ended in failure. Euler showed exceptional talent in math at young age but his father wanted him to study theology, and he dutifully obeyed. Luckily the Bernoulli family - one of the Bernoulli was responsible for the Bernoulli principal in fluid mechanics, was in the same town Euler lived and two of the Bernoullis were Euler's good friends. The Bernoullis appealed to Euler Sr. to let Euler pursue a career in math. Because Euler Sr. had great respect to the Bernoullis, he reluctantly gave in. The world got one of its greatest mathematicians.

Paul Wolfskehl was an average mathematician from a wealthy family. He created Wolfskehl Prize due to an aborted suicide. The story goes like this. A failure in his pursuit of a beautiful woman led to his plan to kill himself. He set a date and would kill himself at exactly midnight that day. He was so efficient in carrying out his plan, he finished the preparation for his suicide attempt ahead of time. So he went to the library to scan through mathematical publications and he happened to read a paper on why two major efforts to prove Fermat's last theorem by Lame and Cauchy respectively failed. He was quickly absorbed into the paper and found an error in it. By dawn he remedied the error in the paper, and his despair evaporated. He tore his will and wrote a new one to create a prize for the proof of Fermat's last theorem.

Andrew Wiles, the Princeton math professor, who worked in secret for 7 years during his pursuit of the proof, has a life long obsession with Fermat's last theorem starting at 10 after he read a history book on math. After he announced that he had proved Fermat's last theorem at a math conference, his proof was subject to peer review. Referees found a potential crushing error in the proof. Wiles sweat for about a year and with substantial help from others, finally fixed the error in the proof. The proof was eventually published in the journal Annals of Mathematics in 1995. The Wolfskehl Prize was presented to him in 1997.

(to be continued ... Numbers)

Our Yellowstone Trip Plan

Trip plan helps us to enjoy a trip. As for any major trips we have had, I prepared a detailed trip plan for our Grand Teton/Yellowstone trip.

The first two days and the last day of the trip we stayed at Jakson Hole. During the Yellowstone portion, we stayed in a hotel at West Yellowstone, a small town just outside the west enstrance.

Day 1: Flight from DFW to Jackson Hole arriving at noon. (Flight was delayed due to mechanical problem, afertnoon activity canceled, rescheduled to the end of the trip)

Afternoon activity – Ride gondola and hiking at Jackson Hole Mountain Resort

Day 2: Grand Teton – whole day

Morning: Scenic Drive - Jenny Lake Scenic Drive, Hike to Hidden Falls/Inspiration point

Afternoon – white water rafting

http://www.davehansenwhitewater.com/whitewater.shtml#8manraft

Day 3: Yellow Stone National Park – day 1 - Geysers

Morning - stop at Ox Bend before entering Yellowstone from south entance

1) Experience Old Faithful, the most popular geyser in the world, and hundreds of other geysers and hot springs

2) Kepler cascade: Kepler Cascades is the most easily reached waterfall in the district. A marked pullout just south of Old Faithful and a short walk from the car offers the visitor easy access to view this 125-foot cascade.

3) Mystic Falls Trail
This trail follows a lovely creek through a lodgepole pine forest before reaching the 70- foot falls. By following a series of switchbacks, an overlook of the Upper Geyser Basin can be reached before looping back to join the main trail.

4) Go to West Yellowstone for lodging

Day 4: Yellow Stone National Park – day 2 - Canyon Village

1) On the way – stop at Madison, Artist Paint Pots

2) Gibbon Falls

This 84-foot (26-meter) waterfall tumbles over remnants of the Yellowstone Caldera rim. The rock wall on the opposite side of the road from the waterfall is the inner rim of the caldera.

3)Norris Geyser Basin is the hottest, oldest, and most dynamic of Yellowstone's thermal areas.

4) Grand Canyon of Yellow Stone

Day 5: Yellowstone National Park – day 3 - Mammoth hot spring

On the way - Roaring Mountain

1) The Gardner River and Gardner River Canyon
The North Entrance Road from Gardiner, Montana, to Mammoth Hot Springs
2) Mammoth Hot Springs
3) Beavers Pond Trail
4) Ranger's program
5) Lemar Valley

Day 6: Yellowstone National Park – day 4- Tower-Roosevelt

To via Mammoth, back via Canyon village

1) Petrified Tree Specimen Ridge
Specimen Ridge, located along the Northeast Entrance Road east of Tower Junction, contains the largest concentration of petrified trees in the world. There are also excellent samples of petrified leaf impressions, conifer needles, and microscopic pollen from numerous species no longer growing in the park. Specimen Ridge provides a superb "window" into the distant past when plant communities and climatic conditions were much different than today.

2) Tower Fall
Tower Fall is the most recognizable natural feature in the district. The 132-foot drop of Tower Creek, framed by eroded volcanic pinnacles has been documented by park visitors from the earliest trips of Europeans into the Yellowstone region. Its idyllic setting has inspired numerous artists, including Thomas Moran. His painting of Tower Fall played a crucial role in the establishment of Yellowstone National Park in 1872.

3) Natural Bridge Trail
The natural bridge is a 51 ft. (18 m) high cliff of rhyolite rock that has been cut through by the erosional forces of Bridge Creek. The trail from the campground meanders through the forest for 1.2 mile (0.8 km). It then joins the road and continues to the right (west) for 1 mile (1.5 km) before reaching the Natural Bridge. The short but steep switchback trail to the top of the bridge starts in front of the interpretive exhibit. To protect this fragile resource, the top of the bridge is closed to hiking. However, good views may be attained next to the bridge. The bicycle trail to the bridge begins just south of the marina off the main road.

Day 7: Yellowstone National Park – day 5: Yellow Stone Lake – West thumb area

Go there via old faithful

1) Craig Pass
Craig Pass, at 8,262 feet on the Continental Divide, is about eight miles east of Old Faithful on the Grand Loop Road. In 1891, road engineer Captain Hiram Chittenden discovered Craig Pass while he was surveying for the first road between Old Faithful and West Thumb. It was probably Chittenden who named the pass for Ida M. Craig (Wilcox), "the first tourist to cross the pass" on Chittenden's new road, on about September 10, 1891. At the time that her name was given to the pass, Ida Wilcox (1847-1930) had been married 24 years. So why did Chittenden use her maiden name? Perhaps it was to honor her singularly for being the first tourist to cross the pass. It is also possible that through his connection with the military, Chittenden knew her father (Gen. James Craig) or her brother (Malin Craig, Sr.) and was really honoring the Craig family.

2) Isa Lake
Hiram Chittenden of the U.S. Army Corps of Engineers claimed to have discovered this lake on the Continental Divide at Craig Pass in 1891. Isa Lake is noteworthy as probably the only lake on earth that drains naturally to two oceans backwards, the east side draining to the Pacific and the west side to the Atlantic.

3) Boating/Kayaking on Yellowstone lake – time permitting

4) Cascade Lake Trail This hike takes 3 hours and is an enjoyable walk through open meadows and over small creeks for those with limited time. Look for wildlife and wildflowers in season. Most years, this trail remains very wet and muddy through July. Trailhead: Cascade Lake Picnic Area, 1.5 miles north of Canyon Jct. on the Tower-Canyon Road. Distance: 4.5 miles (7.2 km) roundtrip Level of Difficulty: Easy

Day 8: Return to Jackson hole

1) Kayaking @ Jenny Lake, Grand Teton NP in the moring
2) Visiting Jakson hole ski mountains in the afternoon

Day 9: Return to DFW

Note:
1) We chose the hotel outside the park for two reasons - first hard to resever lodging inside the park, and inside lodging condition was not good; second we did not want to move daily from one place to another inside the park - which would take a lot of time. Hindsight - staying outside made us drive more inside the park and thus more chances to see wildlife
2) best wildlife sighting time is early morning and early evening
3) got to have good binoculars and high zoom camera to view and capture wildlife far awy from viewing locations
4) plan a trip to Yellowstone early of the year, book hotel and plane ticket early

Wildlife We Saw at Yellowstone

(Note: Double click small size pictures, the full size photos will appear for better viewing of the far away animals)

During our trip at Yellowstone, we stayed outside the park at a hotel in the town of West Yellowstone - very close to the park's west entrance. It worked out pretty well as far as wildlife viewing and traffic was concerned. Getting in early in the morning (7am or so), leaving a bit late (~ 7pm) helped us to avoid traffic jams. Even though we drove about 30 miles of the same sections of roads every day, we actually had more chances for wildlife sightings. This might be one case that doing the same thing again and again we could expect different outcome.

The first day we entered the park from the west entrance in early morning, ~ 7am. About 10 miles into the park, we saw cars lined up by the side of road ahead of us. "Animals!" As we drew closer, we saw a large group of deers on the hill slope by the road. Despite the frequent sighting of deers, we were still very excited since this was our first animal sighting at Yellowstone. As we were about to leave the area, one deer came down the hill and crossed the road in front of our car!

One day on the way out of the park, we saw another line up of cars by roadside, we slowed down, and there it was - a red fox by the road side. We parked on the other side of the road, and observed and video taped the fox observing passers by, stretching its front legs, yawning and simply enjoying the late afternoon Sunshine.

Another day as we were on the road from one major area to another at noon time, we saw a lot of people standing by road side looking at an area with a few huge fallen trees. We were told by other onlookers that there was a male adult moose under a big tree trunk. It took us quite a bit of time to locate the moose which has a big antler. Someone must have really sharp eyes and excellent sense of moose to spot the good camouflaged animal under the fallen trees.

One time, there was an eagle fishing in a river, frequently resting on a log. A ranger set up a telescope aiming at the log, so people who did not have good binoculars could see it. I took a picture of the eagle through the lens of the telescope!

It is great to see wildlife when someone else found it. It was much better and way more exciting and thrilling if we were the first one to spot a hardly found animal, such as the grizzly bear and the trumpeter swan. The key to wildlife viewing in such an large area is to be vigilant, to be alert and to be on the look out.

The mammoth spring was a big disappointment of our Yellowstone trip because most springs were dormant that year. We went to a ranger educational program about the Yellowstone ecosystem there- which was interesting. At the end of the program, Lily asked the ranger about the best area to view gray wolf - a unique and controversial animal to the area. It was Lemar Valley - a 30 ~ 40 mile stretch to the east entrance of the park. We drove to the valley later that day. When we got there, there was not much to see - brown grass at the bottom of the valley, low grayish green bush buffered between the grass and pine trees further away, all the same along the stretch. We did see a couple of pick-ups parking along the stretch, old people with binoculars in hand waiting for gray wolfs to show up - as the ranger told us. It was getting late, we turned around, and drove through the valley again, around 6:10pm (per time stamp on pictures), Lily spotted something moving along the seam between grass and bush, "Wolfs?" "A pack of wolfs!" Justin confirmed via his binoculars. The three wolfs were walking along the seam in a line. How exciting! We were exuberant. We saw a couple of bighorn antelopes in the same area, wishing to see wolfs to prey on them. Unfortunately, the wolfs were going in the opposite direction. We watched the wolfs until they disappeared into the bushes.

Be observant, and be patient would go a long way as well.

One time we saw a group of pelicans swimming in a pond, a minute later a bison walked slowly into the picture. Another time we were witnessing a bald eagle circulating over a hill, a few minutes later, a group of white birds started following the bald eagle! Yet another time, We observed a few ospreys over a pond. Then one osprey starting to flap its wings, stayed stationary in the air, looking down, a few seconds later it dived into the pond - apparently to catch fishes. It was so fast, and so far away, it was really difficult to capture the actions on camera or video though we observed the dive several times.


Timing is hugely important as well.

In our second day at Yellowstone, we hiked along the beautiful and scene ever changing beavers pond trail. We saw a pheasant on the trail, a few seconds later, it disappeared into the woods. When reached the beavers pond, we saw a beaver dam and ducks in the pond but beavers were nowhere to be found. As we were about to continue our hike, I looked into the pond one more time and there it was, the elusive beaver was swimming in the pond, actually from our side of the pond toward its house - the beaver's dam! This was our first ever sighting of beavers in their natural habitat.

Wildlife is not limited by the administrative boundary of the park. On our way to Jackson Hole Mountains, we crossed road again with hard to see wildlife - a mother moose and her 1 day old baby (per a ranger who was monitoring the moose at the scene) were in a pond by the road. The innocent baby was only several yards away from us, it looked at our direction curiously until its mom called. We witnessed the feeding process as well as moose peeing!

For more common animals, such as deers, elks, bison/buffalo on the ground, and pelicans, gees, ravens, gulls in the sky or by the water, all you need to do is to be at the right places. On the beavers pond trail, we met a group of three couples of senior citizens, who told us they had not seen buffaloes - the common big mammals in the park! If you don't know where to see specific animals, ask a park ranger.

There are many buffaloes in the park in the center east area - we saw hundreds of buffaloes at one valley, we saw tens of buffaloes crossing a road at fisherman's pier by Yellowstone lake.

Elks are very common as well, they might rest in a meadow by a road in the early morning, they might walk in a river at noon time, or they might eat grass in late afternoon in a group.

Looking for wildlife is thrilling. Finding some on your own is exciting.

Yellowstone: Water and Mountains

Water of various forms represents a very important facet of Yellowstone area - numerous waterfalls, rivers and streams, ponds and lakes, and most famous of all - geysers, the steaming water.

Shortly after we entered the park from the south, we arrived at the Moose falls. Not many people stop there, we had the whole area for ourselves for 30 minutes or so, listening to the sound of rushing water, feeling the breeze, enjoying the nature. Of all the water falls at Yellowstone, the most impressive one is the lower falls on Yellowstone river, which has a height of 300feet. The water was rushing so fast and forceful, it sent mists into the air forming a light fog. As we went down to the bottom of the fall via Uncle Tom's trail, we saw rainbows!

In addition to Yellowstone river, the larger rivers include the Firehole river which originated from geyser basin. The snake river runs through Yellowstone, and the most beautiful scenery along the river is at its Grand Teton section. We had white water rafting the first day at Grand Teton National Park.

The biggest lake is Yellowstone lake. It is huge and the water is crystal clear. It is so huge we decided to drive a motor boat instead of our usual favorite kayaking. At one corner of the Yellowstone lake, there is a large marsh land - which is said to be bears favorite place, but we did not see any bears there. There are so many ponds we did not even try to remember their names, except the beavers pond - because a trail we hiked is named beavers pond trail and we actually saw beaver's dam and a beaver swimming in the pond just for minute or two when we hiked there.

The most famous form of water there is geyser. After touring the geyser basin, we could appreciate how amazing it is that Old Faithful erupts regularly. Among the less famous geysers, this long horn like dormant geyser caught my eyes. Geysers are amazing, but they smell big time.

The pictures listed below are 1) old faithful- one of its smaller eruption I caught on camera, 2) a dormant geyser, 3) steamboat geyser - what shown in the picture is not a eruption, it was a continuous steam from the boiling geyser, 4) water seeping out of rocks from one side of Calderon geyser

The mountains at Yellowstone are not high. They, along with blue sky, white clouds, remnant of glaciers, lakes, trees and green vegetation, form a beautiful tapestry of Yellowstone.

Bear Encounter at Yellowstone

Watching grizzly bears in the wild was one thing we looked forward to during our 7 day stay at the Yellowstone National Park two years ago. We hiked on the trails which had good chances of bear sighting, we went to marshes that bears usuaully hang around ...despite many attempts, we did not catch even a glimpse of the bears in the first 6 days there.

The last day, we had a fantastic morning - driving a motor boat in the Yellowstone lake, first ever for all of us. In the afternoon we went to Cascade Lake Trail for an afternoon hike, and also our last chance of bear sighting.

It was a breezy, beautiful sunny afternoon. The blue sky was spotted with snow white clouds. After about a quarter mile in the woods, we walked into open Meadows full of wild flowers - white, purple, red and yellow, crystal clear water running in shallow creeks. There were few people there. When we reached cascade lake, we were all by ourselves as far as eye could see, with the company of two hawks flying above the valley.

After snack by the lake, we were on the return hike. As we entered the wooded area, I sensed something moving near a stump by the side of the trail ahead of us. We stopped, the rest of the group saw nothing. "False alarm! You want to see bears too much." Lily teased me.

A few seconds later, a dark brown animal was in sight. "Grizzly bear!" The boys exclaimed, starting to run to the bear. "Stop!" I shouted out in a suppressed voice. "Be careful!" Lily's voice was a bit shaky. We stopped there, watching the bear and took a lot of pictures. The bear apparently did not notice us, snorting around, occasionally raising its head to smell the air, walking slowly toward the lake direction. We stood there watching the bear until it disappeared into the meadow. What a sight! Wow!

We were lucky.

On the way back to htotel, Lily spotted something white in a pond we drove by. We decided to turn around to check it out - We were thrilled to see a trumpeter swan. The lucky day just got luckier for us.

Exploration made it so much fun.