some 5 minute brain teasers

[phil1882]

1. if all borks are grogs, and all grogs are snorts, are all snorts borks?

2. prove that GCD(A,B) = GCD(A+B,A) = GCD(A+B,B)

3. how much water, approximately, would you need to cover all land on earth in 3 feet of snow?

4. whats the shortest distance between any two points on a globe? A) straight line B) curved line

5. and just to be silly: what's the speed of dark?

 

 

[bonanova]

1. No
    
2. Let  f = GCD (A, B) so that A = f a and B = f b where GCD (a, b) = 1.
   Then (A+B) = f (a+b).
   GCD (A+B, A) = f GCD (a+b, a) = f  = f GCD (a+b, b) = GCD (A+B, B) = GCD (A, B).
    
3. Approximately,
   R ~ 4000 mi;
   land area ~ (3/10)(4) pi R² sq miles;
   3' of snow ~ 3" of water = 3/(12x5280) miles of water.
   Do the math: (land area) x (water depth) cubic miles of water would be needed..
    
4. It depends.
   A) if "on a globe" refers just to the points then the distance between them is the length of a straight line;
   B) if "on a globe" refers to the distance then the distance on the globe's surface is the length of a curved line.
    
5. The speed of light.