World Flags

The following map shows the proportions of height to width in the flags of the countries of the world.

Now, the gallery of similar flags:

(Australia, New Zealand)

(United States, Liberia, Malaysia)

(Netherlands, Luxembourg)

(Slovenia, Slovakia)

(Austria, Latvia)

(Andorra, Moldova)

(Turkey, Tunisia)

(Mexico, Italy)

(Bulgaria, Hungary, Tajikistan)

(Honduras, El Salvador, Nicaragua)

(Ireland, Côte d’Ivoire)

(India, Niger)

(Guinea, Mali)

(Lithuania, Myanmar, Bolivia)

(Cameroon, Senegal)

(Qatar, Bahrain)

(Yemen, Syria, Egypt, Iraq)

(Colombia, Ecuador, Venezuela)

(Romania, Chad) (the difference is the shades)

(Poland, Singapore, Indonesia, Monaco) (the last two different only by proportions)

Nepal is the only country with a non-rectangular flag:

Flag of Nepal.svg

Paraguay is the only country with a flag with different front and back sides:



South Africa and South Sudan earn the distinctions of having six different colors without using any coat of arms/emblems:

Flag of South Africa.svg (South Africa)

Flag of South Sudan.svg (South Sudan)

There are flags consisting of and only of red, white, and blue, the most popular flag color triplet, in every inhabited conventional continent:

North America: Panama, United States

South America: Chile

Europe: Czech Republic, France, Iceland, Luxembourg, Netherlands, Norway, Russia, United Kingdom

Asia: Laos, North Korea, Thailand

Africa: Liberia

Oceania: Australia, New Zealand

Proving Conjectures True by Brute Force

So the thing about conjectures for all positive integers is that proving them true is a horrendous idea by brute force—there’s an infinite number of positive integers. But wait! What if we invented a computer that could check the first n positive integers in 1 second, the next n positive integers in 1/2 second (I guess something is derivable about the next set from the previous set by some mechanical analogue of strong induction or something?), the next n positive integers in 1/4 second, the next n positive integers in 1/8 second, etc., we’d be able to either prove the conjecture true by exhaustion or find a counterexample in 2 seconds. There is of course a huge realistic problem: we’ll probably find an easier way around this general issue sooner than we invent infinite computer speed or conquer the Planck Time. Thus, this question deals with idealistic usage of computers, which is quite an interesting combination.

The BART Map, to Scale

One thing that has always bothered me is the fact that no one ever drew a map of all the BART stations to scale. But really, I still have not seen a to-scale map of the BART system: not BART’s Map:


not Wikipedia’s map,


which, interestingly, has two stops in the wrong order (UPDATING EDIT: On 28 December 2012, the Wikipedia map South San Francisco/San Bruneo flip error was corrected),

not BART’s old map, which hilariously is the most accurate there is so far (in fact, they probably intended for this to be to-scale, but made quite a few mistakes doing it),


not even Google Maps’ illustration of the BART system, which only has the stations in the proper locations but not the connections between them.

And thus, I set out to create a to-scale version of the BART system map. I’m quite technically challenged (I would say profoundly technically challenged, but as I found out today, I’m evidently the only member of my family that knows that Ctrl.+P is print, not paste), so I welcome you to amply laugh at me as I note that I made the following map using tools in MS Paint and looking at Google Standard and Satellite maps. (Note that although that means there’s some degree of uncertainty, it is far smaller than the error in any of the maps I presented above.)

If you use BART even occasionally, you could probably figure out what the difference between blue elements and black elements in the maps is. Also, just in case you wanted it superimposed on Google Standard Maps:


So much for being a senior in Spring Break.

Congressional Redistricting

BREAKING NEWS! Congress has decided to redistribute Congressional Districts according to AIME scores and USA(J)MO qualifications rather than population. The new district boundaries have been drawn and are shown below.

Okay, not really, this is still a hypothetical. But still, the following map shows the distribution of high achievement on the AMC-series of contests. District 0 has the property that the centroid of the region is not within the region—it is clearly gerrymandered! Also, all district dividing lines in New Hampshire run through Exeter. Shocking. There are a handful of notable districts smaller than Vatican City.