"Running the Category" on the Jeopardy! TV Show

On the TV program Jeopardy! under each category, there are five different problems. They pay:

$200
$400
$600
$800
$1000

Sometimes, a contestant knows a lot about a particular subject and answers every question in a category.
Alex Trebec calls this "running the category." Without adding the five numbers, what does the contestant win for "running the category?"

Hint: the average prize is $600; there are two smaller prizes and two larger prizes. Multiply $600 by 5 to get the answer, then add up the numbers to confirm.

In "double Jeopardy" there are also five prizes. They pay:

$400
$800
$1200
$1600
$2000

Without adding the five numbers, what does the contestant win for "running the category" in Double Jeopardy? Use the same method to get the answer; it should be twice the previous answer.

The prizes form an arithmetic sequence. You can use the same idea to add up any arithmetic sequence.

Another "Near Miss" of Fermat's Last Theorem

This is from a Japanese comic book, called "Manga" in Japan:

(951 413)^7 + (853 562)^7 = (1 005 025)^7

You can do this with a scientific calculator. If you don't have a scientific calculator, you can get one by searching Google for scientific calculator. The left side agrees with the right side up to 4 decimal places.

See previous post for two other "near misses" which agree to 8 decimal places.

Higher Math from The Simpsons TV Show

Higher Math from The Simpsons TV Show: Fermat's last theorem

(A long post about higher math which young people can understand. )

The New York Times recently reviewed a book about The Simpsons TV show and higher math. Occasionally, Bart Simpson writes higher math on the blackboard. This is a surprise because Bart Simpson is supposed to be a dunce who hates math. In several episodes, Bart writes equations that seem to refute Fermat's last theorem.

For more on this, Google Fermat Simpsons.

Fermat's last theorem states that x^n + y^n = z^n has no solutions when x, y, and z are integers and n is greater than 2. x^n means "x to the n power" for example x^2 means "x squared."

x squared + y squared = z squared has an infinite number of solutions, for example, x = 6, y = 8, z = 10. These solutions are called "Pythagorean Triples" and there are an infinite number of them.

It is remarkable that (x squared + y squared = z squared) has an infinite number of integer solutions but

x cubed + y cubed  = z cubed has no integer solutions.

One of Bart's "equations" is 1782^12 + 1841^12 = 1922^12.

Try this on a scientific calculator. For example, on the TI-30XS Multiview, the key to raise to a power is a key like this: [^] -- it is the middle key on the left column of keys.

You would key in: 1782[^]12 [+] 1841[^]12 [enter], you will get an 8-digit number in scientific notation, ending in 10 to the 39th power. Then key in 1922[^]12 and you will also get a number in scientific notation. The two numbers appear to be equal, but according to Fermat's last theorem, they cannot be.

If you don't have a scientific calculator, you can get one by doing a search in Google for scientific calculator and one will appear. The key to raise a number to a power is marked [x to the y power]; you would put in for example 1922 [x to the y power] 12 [=].

It is easy to show that the equation above is false. 1782 is even, and 1782 raised to the 12th power is also even; it is 1782 multiplied by itself 12 times. But 1841 is odd, and so is 1841 raised to the 12th power. But even + odd is odd, and 1922 raised to the 12th power is even. So we apparently have odd = even, which is impossible.

To see what is really going on, click on the purple  link to big number scientific calculator . Put in the left side of the equation; you will get a long number, approximately 40 digits long. Put in the right side of the equation; you will again get a long number, approximately 40 digits long. Use the "history" key to see the two numbers together. At least the first 8 digits will be the same, but at some point, the digits of the two numbers will not be the same.

Another Bart Simpson "equation" is:

3987^12 + 4365^12 = 4472^12

This is also false. The numbers 3987 and 4365 are both divisible by 9: the sum of their digits is divisible by 3.

But 4472 is not divisible by 9. If the equation were true, if one side were divisible by 9, both sides would be divisible by 9.

You can go through the same routine with the "equation" above. Evaluate both sides with a scientific calculator. You will get two numbers that end with 10 to the 43rd power and appear to be equal. If you  don't have a scientific calculator, Google scientific calculator to get one. Go to the purple link

link to big number scientific calculator enter the left side and the right side and you will get two 44 digit numbers, use "history" key to see both numbers, you will find that the first 10 digits agree but the 11th digit is 8 in one number and 9 in the other.

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About Fermat's Last Theorem and "Pure Mathematics"

(For more about this, Google Pierre Fermat)

Pierre de Fermat was a French lawyer and amateur mathematician who lived from 1601 to 1665. That is a long time ago. For example, The Pilgrims landed on Plymouth Rock in 1620. Fermat wrote his "conjecture" in the margin of a book about 1637, claiming to have a "marvelous proof" that would not fit in the margin.

All of Fermat's writings, including scribbles in the margins of books, were collected and printed as a book by his son. Fermat's theorem is "last" because it was the last to be proven. For hundreds of years, mathematicians searched for a proof. At last, it was proven in 1995, about 358 years after Fermat first wrote it.

Meanwhile, the Fermat problem gave rise to a branch of mathematics called Number Theory. Number Theory is so called "pure mathematics" because it had no apparent applications when it started. In fact, many applications have been found. But, if you want a sample of "pure mathematics" Fermat's Last Theorem, Bart Simpson's "equations" and the simple proofs that they are wrong, and even the use of "big number" calculators will give you a taste of Number Theory. 

Number Theory courses at the University of Minnesota have numbers in the 5000 range, meaning that they are intended for graduate students majoring in Mathematics.

A Cartoon about Pi

A Problem from Delta Airlines In-flight Magazine

A cow is tethered to a stake by a 14-foot rope. The cow eats 100 square feet of grass a day. For how many days does the cow get enough to eat?

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HINTS: is 14 feet the radius or the diameter? Draw a picture.
Do you know the formula for area of a circle? Pie are square (no! Pie are round).

Warning! When a multiple choice question says 'you may use 3.14 for Pi," DO SO! Otherwise, your answer will not match any of the multiple choices.

But, for here and now, you can do this without a calculator by using 22/7 for Pi and using appropriate cancellations.

Also from continued fractions, Pi almost equals 355/113. This is neat because it uses odd numbers in pairs: 11 33 55. I use it to test calculators.

Coded Answer: Fvk qnlf.

Click this: ==>  Link to rot13.com to decode. This answer is so short that you can write it down and type it in. Or, as always, copy and paste.

The Spirit of St. Louis Movie - a good movie for students interested in aviation, navigation, and engineering

The Spirit of St. Louis movie has been criticized as "dull" but it is not dull for anyone interested in aviation, navigation or engineering.

How to get the movie: The Hennepin County Library has no copies but the Saint Paul Public library has three copies in DVD format. Go to any Hennipin County Library, go to "inter library loan" and they will get you a copy from St. Paul.

I really enjoyed the navigation part. Lindbergh approximated the great circle route from Roosevelt Field (Long Island, NY) to Le Bourget Field (Paris, France) with a string of 100-mile-long straight lines, drawn in pencil on a map. This map still exists and I saw it in the Map Room of The Art Institute of Chicago about 4 or 5 years ago.

There used to be a full-scale model of The Spirit of St. Louis in the Lindbergh Terminal of MSP airport. I wonder if it is still there. (Lindbergh was born in Minnesota.)

Lindbergh's route is 3602 statute miles from NY to Paris. He flew it in 33 hours. 3602 miles / 33 hours is 109 miles per hour. Lindbergh himself was surprised by this speed; his indicated air speed was only 90 miles per hour. He supposed, correctly, that he had a tail wind.

Current flying time from New York to Paris is about 7 hours.

Here is a review of the movie from the New York Times:

February 22, 1957

'Spirit of St. Louis' Is at the Music Hall

By BOSLEY CROWTHER

fter thirty long years, the story of Charles A. Lindbergh's historic flight across the Atlantic from New York to Paris is re-enacted on the screen in Leland Hayward's and Billy Wilder's production of "The Spirit of St. Louis," which came to the Music Hall yesterday. And at last the magnificent achievement of the solo flier is dramatically portrayed in the medium most apt for its portrayal, with James Stewart playing the leading role.
As a straight-away visual re-creation of the background and facts of Lindbergh's flight, with colorful flashback fill-ins of his prior barnstorming and air-mail flying days, this large-screen color picture would be hard to beat. It has drama, sentiment, humor and a slight dash of destiny. It pictures the dogged perseverance of the youthful flier in simple, standard terms. And it details his trans-Atlantic passage in exciting and suspenseful episodes.
Particularly fine and fascinating are the re-created scenes of Lindbergh's take-off from Roosevelt Field, L. I., on May 20, 1927, in the rain-drenched dawn. The night-long vigil that preceded the take-off, the readying of the plane, the excitement of the crowd, the fateful decision of the flier, the establishment of the peril and then his simple, casual comment as he climbed into the cabin, "Well, I guess I might as well go, " and gunned his engine for that race down the runway--these are beautifully shown.
Except for a bit of lily-gilding in having the wheels snap a telephone wire as the plane barely clears the obstacles beyond the runway, it is thoroughly credible.
Likewise, the manifold adventures that occurred as Lindbergh flew up the coast, took leave of the continent at St. John's, Nfld., went out across the night-shrouded sea and finally picked up the coast of Ireland and flew on to France and Le Bourget--all these, including his struggles with ice and sleepiness, are well depicted, too.
The blunt disappointments of the picture are the familiar and superficial way in which the background of Lindbergh is presented and the abruptness with which the drama is brought to an end. In the flashback scenes, which are injected as the flier awaits the take- off in his hotel and as he wings his way over the Atlantic, the young pilot whom we are brought to know is pretty much the same clumsy, lovable hero we've seen in any number of films.
He is the lad who buys an old, beat-up bi-plane, joins a barnstorming "circus," flies air mail and then patiently goes about the business of getting money and a plane for a Paris flight. We see very little of his basic nature, his home life or what makes him tick. As Mr. Stewart plays him, with his usual diffidence, he is mainly a type.
That's too bad, for after all these years of waiting, it would be interesting if we could see what it was about the fellow that made him uniquely destined for his historic role.
And the suddenness with which the film is ended after the landing of Lindbergh at Le Bourget and a brief news clip of his actual New York reception blanks all visioning of his place in history. This film treats his remarkable achievement as an adventure and little more.
Others in the film are not impressive. Bartlett Robinson as the builder of the plane is the only one besides Mr. Stewart who has any more than a brief role. And he, Murray Hamilton as a "circus" companion, Patricia Smith as a take-off spectator and Marc Connelly, well-known playwright making his screen debut, as a priest who takes flying lessons are conventional.
However, a haunting recollection of one of the thrilling events of our times has been handsomely staged by Mr. Wilder, and for that you should see the film. It runs for two hours and eighteen minutes, which is about how long it will take to fly to Paris some day.

Copyright 1957 New York Times Company.

Speed Distance Time: a movie about Charles Lindbergh

Last night I was watching the movie The Spirit of St. Louis about Charles Lindbergh and his flight from New York to Paris, France. Lindbergh was the first to make this non-stop flight. He did it in 1927.

In the movie, Lindbergh does a test flight from San Diego California to St. Louis Missouri in 14 hours and 25 minutes. The distance from Ryan Field in San Diego to Lambert Field in St. Louis is 1557 miles.

What was his speed? (coded answer below)

(There are two kinds of miles, "Statute Miles" and "Nautical Miles." This problem uses statute miles, these are the common miles of 5280 feet. I will make a separate post about nautical miles.

A way to remember 5280 is "Five tomatoes - five two eight oh")

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Hints: you have to divide the distance (1557 miles) by the time, 14 hours and 25 minutes. 25 minutes is 25/60 hours. If you have an ordinary, 4-function calculator, you can divide 25 by 60, write it down, then add to 14 and write that down. You will get 14.dddd where dddd stands for digits. Then divide the distance by the decimal time you just wrote down.

Another way to do it is to clear the memory (use MC if you have it, if not, press MRC twice). Then do 25 [divided by] 60 [=] and you will get a decimal, larger than 0.4 and less than 0.5. Then press [+] 14  [=]. You will get 14.dddd where dddd stands for digits. Now press [M+] to store in memory. An M will appear to indicate the number is stored. Then enter the miles and press [divided by] MR or MRC once and you will get the answer; you divided by the time stored in memory. This may seem a tedious process, but if you practice it, you will find it pretty quick.

In 1927, planes flew at about 100 miles per hour. Modern planes are much faster. It took Lindbergh more than 14 hours to do a flight that takes about 3 hours today.

If you have a scientific calculator, the calculation is much simpler. If you need help, write to me using margolis.stephen at Yahoo.com, tell me what kind of scientific calculator you have, and I will tell you how to do the calculation.

Use the @ sign in your email, the "at" is put in above to discourage spam.
washington.george at yahoo.com means washington.george@yahoo.com

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Coded answer: Bar uhaqerq rvtug zvyrf cre ubhe.

This answer is in Rot13 code. To decode, do a copy operation on the coded answer, go to Rot13.com, paste the coded answer in, and press cypher to decode .Link to rot13

If you prefer manual decode, you can use:

abcdefghijklmnopqrstuvwxyz
nopqrstuvwxyzabcdefghijklm

Dr. Steve's Wife Drinks Latte

Dr. Steve's wife decided to change to whole milk for her Latte (a mixture of strong coffee and warm milk). She likes to have 300 milliliters of milk and to shop once a week.

A half-gallon of milk contains 1890 millileters. Can she drink 300 milliliters per day and shop once a week?

Suppose she decides to shop once a week and use less milk. How much milk can she use per day, in milliliters?

You may be surprised to find that the answer comes out to be a whole number. Look at the divisibility rules and concentrate on divisibility by 7, printed in red below.

The following is borrowed from "Ask Dr. Math" published by Drexel University in Philadelphia PA.

Divisibility by:

2	If the last digit is even, the number is divisible by 2.
3	If the sum of the digits is divisible by 3, the number is also.
4	If the last two digits form a number divisible by 4, the number is also.
5	If the last digit is a 5 or a 0, the number is divisible by 5.
6	If the number is divisible by both 3 and 2, it is also divisible by 6.
7	Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also.
8	If the last three digits form a number divisible by 8, then so is the whole number.
9	If the sum of the digits is divisible by 9, the number is also.
10	If the number ends in 0, it is divisible by 10.
11	Alternately add and subtract the digits from left to right. (You can think of the first digit as being 'added' to zero.) If the result (including 0) is divisible by 11, the number is also. Example: to see whether 365167484 is divisible by 11, start by subtracting: [0+]3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11.
12	If the number is divisible by both 3 and 4, it is also divisible by 12.
13	Delete the last digit from the number, then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number. 1890 / 7 Consider the number 1890, what is it divisible by? Double zero and remove from 1890, leaving 189. Double 9 and subtract from 18, leaving zero, which is clearly divisible by 7. So, 1890 is divisible by 7. Dr. Steve's wife gets exactly 270 milliliters of milk per day, which is more than a cup, and works fine. This is a real-life story. Try the number 7560, for more practice. II The last digit is zero, which is even, so it is divisible by 2. III The sum of the digits, 7 + 5 + 6 + 0 = 18, which is divisible by 3, so the number is also divisible by 3. IV The last two digits, 60, are divisible by 4, so the number is divisible by 4. V The last digit is zero, so the number is divisible by 5 VI The number is divisible by 3 and 2; consequently, it is divisible by 6. VII For 7560 to be divisible by 7, 756 must be divisible by 7. Knock off the last digit and consider two numbers, 75 and 6. Double 6 and subtract from 75; 75 -12 = 63.We recognize that 63 is divisible by 7, 63/7 = 9. So 7560 is divisible by 7. VIII The number is divisible by 4 and 2, hence is divisible by 8. Or, looking at the last three digits, 560 divided by 8 is 70, so 7560 is divisible by 8. IX The sum of the digits is 18, which is divisible by 9, so 7560 is divisible by 9. X The last digit is zero, so the number is divisible by 10. The number is not divisible by 11 or 13. XII The number is divisible by 3 and 4, hence is divisible by 12. 7560/12 = 630. In fact, I "cooked up" 7560 as 3 x 5 x 7 x 8 x 9, since 8 x 3 = 24, the number was sure to be divisible by 2, 4, 6 and 12. Methods for divisibility by 11 and 13 will be covered in a subsequent post.