In a course I am developing, I want to give out some math problems for people to work on that should be in the grasp of adults without much math background at all. One such problem is what I call "coded arithmetic puzzles". The commonest example I know is "Send More Money": Solve S E N D + M O R E = M O N E Y, where each of the 8 different letters in the equation represents a different digit.
I would like to find a collection of these types of puzzles that would enable me to give different classes different puzzles. The puzzles should not be too tedious and should not require too much cleverness; in other words of the difficulty of Send More Money, or easier. Perhaps someone has written a program that would generate puzzles of this sort.
Does anyone know if there is a formal name for this type of problem? "Coded arithmetic puzzles" is not a very helpful Google search.
Name Change
Cambridge Math Learning, Inc. is now doing business as Math for the Rest of Us. I think this emphasizes the mission of the company, which is to teach mathematics to the "bottom 80%" of adult math learners; those who have been poorly served by mathematics instruction in the past and most of whom now have anxiety when facing mathematics that they must learn.
More on Ordering a Multiset
I posed problem A3, to find a formula for the k-th largest element of an n-element multiset A. I found a very interesting formula that is unknown to several famous combinatorists, including Donald Knuth, and I have submitted a problem to the MAA Monthly Problems section which asks for the solution that I found, a linear combination of certain symmetric functions. However, Knuth told me that there is a simpler known formula of a different type. Knuth's formula is
min(maxk)
where (maxk) is a set of C(n,k) numbers, each of which is the maximum of a different subset of A of size k.
Pretty cute!
min(maxk)
where (maxk) is a set of C(n,k) numbers, each of which is the maximum of a different subset of A of size k.
Pretty cute!
Mandelbrot Set
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.
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.
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