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May 4, 2011:
Here is a skill (recasting) that is very important to being a agile mathematician...(pre-rec is Algebra II)
The sign "^" means "raised to the power of"
If x^4 + x^2 + 1 = 0
What does (x^16 + 3x^8 + 1)/x^8 equal?
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For you chemists...(pre-rec is a first year chem. course where equilibrium and acids/bases have been covered)
If a .1000 Molar solution of Acitic acid (CH3COOH) is titrated to neutralization, what is the pH of the solution when you finish?
Assume you started with 1.000 Liters of solution and titrated with .1000 M NaOH
You may need to know that the equilibrium constand for Acitic acis is 1.8 * 10^-5
(meaning that Kb for the acetate ion is 5.5 * 10^-11)
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For those who want to think about physics...(pre-rec 1 year of physics where waves have ben taught)
I am building an pipe organ, and I want to rattle the listeners with a 16 Hertz base pipe. How long a pipe would I need (minimum length) if I want the pipe to be open at one end and closed at the other? How would the length change if I used a pipe that was open at both ends, or does it matter?
Assume that the speed of sound is 340 m/s
If you think I left something out (and I may have by accident) just e-mail me and I will correct it. Thanks
Have fun!
April 17, 2011:
Here is a problem that came up in a conversation I had with a young lady who told me that she loved Algebra. It was her favorite subject and she enjoyed tutoring young(er) students. I asked if she had studied logarithms, and she said she had, so I gave her this one (only because she bragged that she understood "everything about them")....
9^x + 6^x = 2^(2x-1) where the "^" character means "raised to the power of"
She was not sure how to solve it. Can you? In case the approach is not obvious, fear not. It took me several minutes (about 20 before it hit me), and I think I know Algebra pretty well! I did not berate the girl in any way for not knowing how to do it, and it made a good "teachable moment".
Here is another one (not from the world of logarithms) but rather from an SSAT math level II practice test. If you have the parabola y=x^2 + kx + 54 where k is a constant (real, I think...but don't discount complex numbers), and you are told that one of the roots is twice as big as the other, what is (are) the values of k?
One last puzzle from Algebra...
A^B = 1
What are the constraints on "A" and "B" so that this equation is true. First solve in the Real domain (Hint: there are three seperate cases to consider), then for extra credit include the complex plane.
That' all for today. Thanks for looking at the problems and for listening to the show.
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July 30, 2010
If someone said that they could prove that a circle and a square have equal area and circumference, you would say "That could be if Pi is ...". What value of Pi makes that a geometric "truth"?
Eight spheres fit into a cubic box. The spheres are mutually tangent and also contact the box at three points. How much water will the box hold once you put the spheres in it. Give your answer in terms of L, the length of one side of the cubic box.
May 2, 2010
Last week combustion was discussed, and I devised this little brain "sit-up" for you. If you burn methane (CH4) in pure O2 and you make CO2, CO, and H20. If you manage to make twice as much carbon monoxide as carbon dioxide, what is the molar ratio of the fuel to the oxygen? If you took chemistry in HS or college, dust off those memories and "get cooking". If you never took or studied chemistry, you may find this a bit tough.
August 15, 2009
Suppose that I provide you with two numbers (say 12 and 22...but really you should think of them as "a" and "b"...any two real numbers). Is it possible that the arithmetic and one of the geometric means can be equal? Recall that the arithmetic mean is the average and the geometric mean of two numbers is the square root of the product. If it is possible, what are the implications? Suppose I said that between two numbers there were two arithmetic means and two geometric means (which is the same as saying that a and b together with the two additional numbers between them form an aritmetic or geometric sequence, respectively). This is "doable" if you have had an algebra II course.
Somewhat more non-math-geek friendly: What is the next letter in each of the sequences...
1) F S A S Y A ?
2) G B A L T I L ?
3) O S S F M O G L ?
4) I P A T T F O T U S O ?
5) T M O T T B T M O ?
6) O O T O T O R ?
June 25, 2009
Suppose that I want to tile a floor with either black or white tiles (each measures 1 unit by 1 unit). The floor is ten units by ten units. Neglecting the width of grout, how many possible ways are there to tile the space? To be clear, each tile may be either a white one or a black one. There are no other constraints.
June 18, 2009...
Find a 10 digit number with the following properties:
1. All the digits from 0 to 9 are used once.
2. The first two digits of the number are divisible by 2 (example: 42 or 56).
3. The first three digits of the number are divisible by 3 (example: 423 or 561).
4. The first four digits of the number are divisible by 4.
5. The first five digits of the number are divisible by 5.
6. And so on...
Is there such a number and if so is it unique?
June 14, 2009...
What is greater than God, lower than the Devil, the wealthy don't have it, the poor do have it, and if you eat it...you will die.
My wife told me this one the other day and it kept me busy for about a minute or so (does that sound arrogant?)...enjoy!!
(March 2, 2009)
Yesterday, I had the pleasure of meeting a gregarious 4th grader named Jake, who shared this puzzle with me. Remember...he's only nine!!
A company (let's say the CIA because I know, for a fact, that they have done this) wants to pack $1,000,000 in a suitcase. They only have $700,000 in Benjies ($100.00 bills). The rest will be $10 and $20 bills. The bills fit in a special sleeve that holds 6 bills. There are 500 sleeves to a "case" and the luggage will hold 10 cases. Can they do it? If so, how many $10 and $20 bills are required?
The solution is not trivial, especially for a 4th grader, yet Jake did it, explained it well, and impressed me with his logic and math skills in addition to his easy going manner. His younger brother, is probably just as smart, but he seemed more interested in decorating (and taste testing) the cupcakes that our hostess was making. The boys mixed up olive drab icing which is rather appealing to a "hawk"...(I took as a tribute to our troops and veterans).
(December 28th 2008)
There are three weights connected by massless and stretchless rope. The masses are a: M, b: 2M, c: 3M. These rest on a frictionless surface as shown and are pulled to the right with a force, F. What are the tensions in the rope segments?
Let F=12N and let M=1 kg for simplicity.
M----Tension1----2M----Tension2----3M----Tension3---->Force=F
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(December 17th 2008)
Spaceship Fred is cruising along at .9c and behing him, happily following along is spaceship Kimber also travelling at .9c.
Spaceship Kimber decides to pass Fred at .5c. How fast must she go to accomplish this (PLEASE...don't even think of saying 1.4c since Einstein said that c is the universal speed limit).
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If I drop a rock into a well and it takes me 6 seconds to hear the splash from the time I drop the rock, how deep is the well (from where I dropped the rock to the water level).
You may need: Air speed is 343 m/sec and x=1/2 g t^2 where g=10 m/sec
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NO ONE HAS TRIED THIS WEEK'S PUZZLE. TOO HARD? TOO STRANGE? WHAT KIND OF PUZZLES WOULD YOU LIKE TO SEE HERE??
If you or your company would like to sponsor this weekly puzzle for any length of time, please e-mail the professor.
This week’s puzzle is sponsored by Downtowner Magazine – Not a Location…A Frame of Mind. Be one of the first five (5) people to solve this correctly and inform me by e-mail, and you will win a free subscription to the Downtowner. The first correct solution also wins a "Get Freducated" hat.
The subject should contain the word PUZZLE and your answers:
1E, 2B, 3B, etc.
This will be the format of your solution in general, i.e., PUZZLE brief answer (as subject of e-mail) then contact info within).
The body of the e-mail should contain your contact information:
Mr./Mrs./Ms./Dr./Duke Did
I. Win
YOU DON'T HAVE TO WIN A GET FREDUCATED HAT TO WEAR ONR WITH PRIDE. JUST E-MAIL ME AND I'LL LET YOU KNOW HOW TO ORDER ONE.
The Professor's weekly brain teaser for Sunday, February 24 --
Sorry to post this on Wednesday, but I lost a day and a half to a migraine. (I hope this little problem doesn't give you one!)
I have a steel wire 5.2 meters long whose mass is 4.8 grams. It hangs from an overhead beam. A 12 kilogram brass monkey is attached to the wire (actually the monkey is 50% brass, 30% gold, and 20% silicon). How much energy is stored in the wire once I suspend the monkey? (Answer should be expressed in Joules, please). If you make any unusual assumptions, please list those in the body of your e-mail answer. You probably want to know that steel has a density of 7.8 grams/cubic centimeter and its Young's modulus is 20 x 1010Pascals. Please note that only two sig. figs. are required in your answer. So, you would say: 34 Joules, not 34.254 Joules, OK?
I got this problem from the fifth edition of PHYSICS by Arthur Beiser (Addison-Wesley publishing). I believe that this book is one of the best presentations of physics one can get without the use of calculus. For the "best" treatment of physics with little or no math, try "Conceptual Physics" by Paul Hewitt. It is a nice introduction to why mechanics is the way it is without bogging the reader down in long derivations. It includes a pretty good chapter on Einsteins relativity too. All in all, a really nifty job!
EXTRA CREDIT (for those who 'hate' the 'pure math/physics' puzzles): If 21 chickens can lay 21 eggs in 21 days, how many days are required for 14 chickens to lay 14 eggs. In general, how many days are required for n chickens to lay n eggs? How many days would be required for 2n chickens to lay n/2 eggs? OK, I'll stop. I've talked about this chicken/egg puzzle on the air, so no prize for getting it right.
The Professor’s weekly brain teaser for Sunday, February 17 --
Six radio hosts – Big (Steve), Caldara (John), Douglas (Steve), Earhart (Amelia), Fred (Ebert), and Gunny (Bob) are auditioning for a time slot for a new KOA show. These auditions will take place over four consecutive days, starting on Thursday. Each host will have one audition. The days on which the different hosts will audition must conform to the following rules and conditions:
At least one audition will take place each day.
Big’s audition must take place on Saturday.
No more than two auditions will take place on any one day.
On any two consecutive days there can be no more than three auditions.
Caldara’s audition must take place on a day during which there is another audition.
Fred’s audition must take place on a day before Gunny’s audition.
Douglas’s audition must take place on a day after Earhart’s audition.
Five questions for your consideration:
#1) If there is only one audition on Thursday, which host could have that audition?
A)Big B)Caldara C)
Douglas D)Fred E)Gunny
#2) If Big’s and Fred’s auditions are on the same day, which of the following must be true?
A)
Douglas’s audition will take place on Thursday.
B)
Douglas’s audition will take place on Friday.
C) Gunny’s audition will take place on Thursday.
D) Caldara’s audition will take place on Sunday.
E) Earhart’s audition will take place on Sunday.
#3) If the program director decides to hold two auditions on Thursday and two on Sunday, how many hosts would be eligible to audition on Friday?
A) 1 B) 2 C) 3 D) 4 E) 5
#4) If Earhart and Gunny have their auditions on the same day, which of the following must be true?
A) Earhart’s audition will take place on Thursday.
B) Fred’s audition will take place on Friday.
C) Caldara’s audition will take place on Saturday.
D) Gunny’s audition will take place on Saturday.
E) Caldara’s audition will take place on Sunday.
5) If Earhart auditions on Saturday, which of the following hosts cannot audition on the same day as any other talk host?
A) Big B) Caldara C) Earhart D) Fred E) Gunny

The Answer