Archive for paradoxes

Strauss: The coffee pot question

When I became chairman of my department a few years ago, I moved from my office far down at the end of the hall to one much closer to the center of action: the tea room! I make a lot more visits there than I used to, and began to notice a frustrating pattern:

Far more often than seems reasonable, there’s not even a full cup of coffee in the coffee pot! Once again, someone has left a nearly empty pot with no regard to the next person (me, whine)!

This seems to happen so often I began to wonder what kind of boors I’ve been working with all these years. They seem like nice people and all, but…?

And then I realized: there’s a perfectly logical reason, a mathfactor puzzle, if you will, that explains this phenomenon perfectly, no boors required, no special tricks, just sensible activity by all. My faith in my colleagues has been restored.

Why is it that on average I see an emptier rather than fuller coffee pot?

PS let us know what works for when we return…

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GY. Chaitin on the Ubiquity of Undecidability

Greg Chaitin, author most recently of MetaMath!,  discusses the ubiquity of undecidability: incredibly all kinds of mathematical and physical systems exhibit utterly unpredictable, baffling behavior– and it’s possible to prove we can never fully understand why!

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G4G9: Report From the Festivities!

Quick interviews with folks here at the Gathering For Gardner, including Stephen Wolfram, Will Shortz,  Dale Seymour, John Conway and many others. 

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GN. Benford’s Law

Benford’s Law is really quite amazing, at least at first glance: for a wide variety of kinds of data, about 30% of the numbers will begin with a 1, 17% with a 2, on down to just 5% beginning with a 9. Can you spot the fake list of populations of European countries?

  List #1 List #2
Russia 142,008,838 148,368,653
Germany 82,217,800 83,265,593
Turkey 71,517,100 72,032,581
France 60,765,983 61,821,960
United Kingdom 60,587,000 60,118,298
Italy 59,715,625 59,727,785
Ukraine 46,396,470 48,207,555
Spain 45,061,270 45,425,798
Poland 38,625,478 41,209,072
Romania 22,303,552 25,621,748
Netherlands 16,499,085 17,259,211
Greece 10,645,343 11,653,317
Belarus 10,335,382 8,926,908
Belgium 10,274,595 8,316,762
Czech Republic 10,256,760 8,118,486
Portugal 10,084,245 7,738,977
Hungary 10,075,034 7,039,372
Sweden 9,076,744 6,949,578
Austria 8,169,929 6,908,329
Azerbaijan 7,798,497 6,023,385
Serbia 7,780,000 6,000,794
Bulgaria 7,621,337 5,821,480
Switzerland 7,301,994 5,504,737
Slovakia 5,422,366 5,246,778
Denmark 5,368,854 5,242,466
Finland 5,302,545 5,109,544
Georgia 4,960,951 4,932,349
Norway 4,743,193 4,630,651
Croatia 4,490,751 4,523,622
Moldova 4,434,547 4,424,558
Ireland 4,234,925 3,370,947
Bosnia and Herzegovina 3,964,388 3,014,202
Lithuania 3,601,138 2,942,418
Albania 3,544,841 2,051,329
Latvia 2,366,515 1,891,019
Macedonia 2,054,800 1,774,451
Slovenia 2,048,847 1,065,952
Kosovo 1,453,000 984,193
Estonia 1,415,681 841,113
Cyprus 767,314 605,767
Montenegro 626,000 588,802
Luxembourg 448,569 469,288
Malta 397,499 464,183
Iceland 312,384 402,554
Jersey (UK) 89,775 94,679
Isle of Man (UK) 73,873 43,345
Andorra 68,403 41,086
Guernsey (UK) 64,587 34,184
Faroe Islands (Denmark) 46,011 32,668
Liechtenstein 32,842 29,905
Monaco 31,987 22,384
San Marino 27,730 9,743
Gibraltar (UK) 27,714 7,209
Svalbard (Norway) 2,868 3,105
Vatican City 900 656

 Looking at these lists we have a clue as to when and how Benford’s Law works. [spoiler]

In one of the lists, the populations are distributed more or less evenly in a linear scale; that is, there are about as many populations from 1 million to 2 million, as there are from 2 million to 3 million, 3 million to 4 million etc. (Well, actually the distribution isn’t quite linear,  because the fake data was made to look similar to the real data, and so has a few of its characteristics.)

The real list, like many other kinds of data, is distributed in a more exponential manner; that is, the populations grow exponentially (very slowly though) with about as many populations from 100,000 to 1,000,000; then 1,000,000 to 10,000,000; and 10,000,000 to 100,000,000. This is all pretty approximate, so you can’t take this precisely at face value, but you’ll see in the list of real data that, very roughly speaking, in any order of magnitude there are about as many populations as in any other– at least for a while. 

Data like this has a kind of “scale invariance”, especially if this kind of pattern holds over many orders of magnitude. What this means is that if we scale the data up or down, throwing out the outliers, it will look about the same as before. 

The key to Benford’s Law is this scale invariance. Data that has this property will automatically satisfy his rule. Why is this? If we plot such data on a linear scale it won’t be distributed uniformly but will be all stretched out, becoming sparser and sparser. But if we plot it on a logarithmic scale, (which you can think of as approximated by the number of digits in the data), then such data is smoothed out and evenly distributed. 

But presto! Look at how the leading digits are distributed on such a logarithmic scale!


That’s mostly 1’s, a bit fewer 2’s, etc. on down to a much smaller proportion of 9’s.


Comments (7)

Morris: World of Britain 2: Proof and Paradox

paradox-clockIn working out the proof for World of Britain I came across a paradox.  Maybe smarter Math Factorites can help me out?  My sanity could depend on it.

In the puzzle you have five different tasks.  On each day one of these tasks is given at random.  How long do you expect it to take to get all five tasks?

First consider a simple case.  Suppose some event has a probability, p, of happening on any one day.  Let’s say that E(p) is the expected number of days we have to wait for the event to happen.  For example if p=1 then the event is guaranteed to happen every day and so E(p)=1.

How can we calculate E(p)? 

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FV. Singmastery!

David Singmaster, Puzzler Extraordinaire, early master of the Rubik’s Cube, poser of the Singmaster Conjecture, etc, etc, engages in some wordplay.

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Q & A: When Two Spheres Touch…

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!


Follow Up: The Harmonic Series

That the worm falls off the end of the rope depends on the fact that the incredible
harmonic series

1 + 1/2 + 1/3 + 1/4 + . . .
diverges to infinity, growing as large as you please!

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EG. The Colossal Book of Short Puzzles and Problems

Dana Richards, editor of The Colossal Book of Short Puzzles and Problems discusses the amazing Martin Gardner and his legacy!


EC. Skyrocketing Functions!

Faster than an exponential! More powerful than double factorials!! The Busy Beaver Function tops anything that could ever be computed– and we mean ever


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