A friend just sent me a link to a fantastic video: Mandelbrot Fractal Set Trip to e214 by teamfresh. The video runs about 9 minutes and zooms in on the Mandelbrot set to a magnification of 10^214. Wow!
It feels like there must have been some pretty clever programming and lots of computer time used to produce this video. The idea of using video to zoom in on the Mandelbrot set is so powerful that it seems to make the beautiful still pictures that I am familiar with, obsolete. To teamfresh, I say Bravo!
I am amazed and humbled by the incredible complexity that can be contained in the simplest mathematical formulas, as shown in this video. Truly our own inventions can take a life of their own.
E17. A 1-2-3 counting problem
The following problem seems at first to be quite difficult, but if you look at it the right way it isn't.
How many n-digit integers are there that contain no digits other than 1, 2, or 3, subject to the condition that any two consecutive digits differ by exactly 1.
This problem (for the n = 10 case) appeared in the ATMIM newsletter, Winter 2002, where it is credited to http://www.mathkangaroo.org, an interesting math enrichment and contest Web site.
I think this problem is too easy for me to post an answer, but if anyone asks for one, I will.
How many n-digit integers are there that contain no digits other than 1, 2, or 3, subject to the condition that any two consecutive digits differ by exactly 1.
This problem (for the n = 10 case) appeared in the ATMIM newsletter, Winter 2002, where it is credited to http://www.mathkangaroo.org, an interesting math enrichment and contest Web site.
I think this problem is too easy for me to post an answer, but if anyone asks for one, I will.
The buckling train track
Here's another gem from the ATMIM conference last month, suitable for any class where the students know the Pythagorean Theorem. Imagine a length of train track, two miles = 2 x 5280 ft long. To accommodate expansion of the track on hot days, the track is hinged at both ends and at the middle. If the track expands slightly, the middle of the track will rise, forming a shallow isosceles triangle. Supposed the track expands 2 feet. How high with the track be in the middle?
The teacher asks the students for guesses, which tend to be around 1 foot. Then he leads the students through the calculation of the answer. The height is the length of a leg of a right triangle where the hypotenuse is 5281 feet and the other leg is 5280 feet. This works out to be about 102.8 feet!
The teacher asks the students for guesses, which tend to be around 1 foot. Then he leads the students through the calculation of the answer. The height is the length of a leg of a right triangle where the hypotenuse is 5281 feet and the other leg is 5280 feet. This works out to be about 102.8 feet!
A3. Ordering a multiset
Given a multiset of real numbers {a1, ...,an} find expressions e1, ...,en such that {a1, ...,an} = {e1, ...,en}, and {ei} is a non-increasing sequence, where each expression is formed from the ai, the max function, and elementary arithmetic operations.
I will not post the answer to this problem, because I plan to publish it if it is not already known. If it is a known result, I would appreciate a reference.
I will not post the answer to this problem, because I plan to publish it if it is not already known. If it is a known result, I would appreciate a reference.
E16. A Property of rectangles
Someone showed me the following problem at the meeting of ATMIM (Association of Teachers of Mathematics in Massachusetts.) It's proof is a great non-routine use of the Pythagorean theorem. I think the proof is too easy to post, but if anyone can't figure it out, post a comment and I will make the solution available.
Let ABCD be a rectangle, and P any point in the interior. Prove that AP2 + PC2 = BP2 + PD2.
Let ABCD be a rectangle, and P any point in the interior. Prove that AP2 + PC2 = BP2 + PD2.
E15. Area in a square
Given a square ABCD of side length a, let M be the midpoint of AB and N be the midpoint of BC. Draw AN and CM, and let their intersection be O. Find the area of AOCD.
I saw this problem on an Internet math forum, along with an solution involving Cartesian geometry. The solution was straightforward and not particularly difficult. However, I'm including the problem with a challenge to do it without introducing coordinates. I think the solution that I found is much prettier than the coordinate-based solution. What do you think?
Note: There is something wrong with scribd which is not allowing me to post my solution, which is a short pdf file. I will add a link later when I can upload my file.
I saw this problem on an Internet math forum, along with an solution involving Cartesian geometry. The solution was straightforward and not particularly difficult. However, I'm including the problem with a challenge to do it without introducing coordinates. I think the solution that I found is much prettier than the coordinate-based solution. What do you think?
Note: There is something wrong with scribd which is not allowing me to post my solution, which is a short pdf file. I will add a link later when I can upload my file.
E14. Two-block calendar
A calendar consists of two cubes of the same size, about 2 inches on a side. Each cube contains a single digit. When placed together, the front faces of the two cubes display the day of the month, from 01 to 31. Note that single digit days must be displayed as two digits, with a leading 0. Describe what digits to place on each face of the cubes for this to work.
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