We ask an old chestnut on the second Math Factor segment ever aired, from February 1, 2004: The Johnsons have two children; we’re told one is a boy. What is the probability they have two boys?
(Incidentally, the music is R. Crumb’s band, Les Primitifs Du Futur)
We ask: What do Google, flutes and monopoly have in common? In fact, important principles behind this question apply to an astounding array of phenomena!
(Since we’ve been offline for a week or so, due to a tremendous ice storm that has paralyzed the town, we add a special bonus: the very first Math Factor episode ever aired, from January 25, 2004.)
Sometimes rules set up to achieve one result can have exactly the opposite result.
You are a financial trader.
We celebrate five years of the math factor in todays segment!!
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In response to the post “Stacking Cannonballs” Trevor H. writes:
I was very intrigued by the recursive sequence you mentioned in the past two episodes–the sequence that begins with 1 and each successive term is the average of all the previous terms times some constant. I have always been fascinated by Pascal’s triangle and all of its surprise appearances in mathematics. Also, my fist encounter with doing mathematics for fun out of my own curiosity was to find a formula for triangle numbers. Like Kyle, I was inspired by bowling pin arrangements. The experience was very rewarding and I have been in love with mathematics ever since.
Inside every hexagram there is a cube trying to get out but, just like the thin woman hiding inside Eddie, it isn’t that easy to spot.
New Scientist magazine has a regular math puzzle called Enigma. A few years ago they posed this intriguing problem.
‘Assign the numbers 1 to 12 to the twelve vertices of a hexagram so that the sum along each line is the same.’
At first sight this is the worst possible kind of puzzle. You have to try lots of ways of assigning numbers until you stumble across the right answer. Very boring!
The best kind of puzzle is when you spot a neat insight that makes everything easy.
Guess what, that is exactly what we are going to do here!
Chris S. writes:
I was wondering what is the theoretical ‘area’ of contact between two spheres in contact with each other. I was unfortunately not able to locate much (if any) information on this. After some thought into this I’ve realised that the spheres would meet at a single ‘point’ however what would the area of this ‘point’ be? The only source related to this claimed the area of contact, the point, has no area. How can a point have no area? If the spheres touch, musn’t there be an area shared between them? Even if only one atom?
Hi, the issue here is that there is a vast difference between physical, real things and the mathematical ideas that model them.
Real, mathematical spheres don’t exist, plain and simple! Never could, even as a region of space— space itself has a granularity (apparently) at a scale of about 10^-33 meters. There simply cannot exist a perfectly spherical region in physical space, much less a perfectly spherical body.
But as an abstraction, the idea of a sphere is very useful: lots of things, quite evidently, are spherical for all practical purposes.
For that matter, “points” don’t exist either, and are also a mathematical abstraction. (So, too, is “area”. Real things are rough, bumpy and not at all like continuous surfaces, on a fine enough scale) But again, these _ideas_ are very good at getting at something important about lots and lots of physical things, and so have proved useful.
Tangent spheres do indeed meet in a single point, which has no area.
Spherical things meet in some other, messier way.
Hope this helps!
After taping this segment, Kyle and I discovered an incredible way to make LOTS of money. We will extend this amazing opportunity to our listeners next week!
Given a difference table, as we considered back in EV. What’s the Difference , how do we come up with a polynomial that gives the values on the top row?
For example, suppose we have
-1 -1 3 35 143 399 899 . . . . .
0 4 32 108 256 500 . . . . .
4 28 76 148 244 . . . . .
24 48 72 96 . . . . .
24 24 24 . . . . .
What is the polynomial P(n), of degree four, that gives
P(0) = -1 P(1) = -1 P(2) = 3 P(3) = 35 P(4) = 143 , etc.
Can this be expressed simply in terms of the leading values on the left of the table: -1, 0, 4, 24, 24?
For me the holidays end on Monday so I just have time to post this seasonal question.
Why is Christmas the same as Halloween?
Specifically why is Oct. 31 = Dec. 25.
Hint: you need to look at exactly how this is written.
A short problem with a long pedigree.
I found this in Martin Gardner’s book ‘The Colossal Book of Short Puzzles and Problems’, problem 3.9 in my copy. He credits Solomon Wolf Golomb, the inventor of Polyominoes which inspired tetris.
Isaac Asimov based a whole story on this puzzle, ‘A Curious Case of Income Tax Fraud’, part of his Black Widowers series.
I asked some work colleagues and had some amusing answers, none of which were maths related. Maybe you have your own?
Enjoy!
Steve
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