Need of Speed For Average Speed

A man drives 1 mile to the top of a hill at 15 mph. How fast must he drive 1 mile down the other side to average 30 mph for the 2-mile trip?

Here is calculation of that speed needed!

Impossible Average Speed Challenge

What was average speed challenge?

A man drives 1 mile to the top of a hill at 15 mph. That means he took, 1/15 hours i.e.4 minutes to reach at the top of a hill.

To achieve average speed of 30 mph, the man has to complete 2 miles trip in 1/15 hours i.e. 4 minutes. But he has already taken 4 minutes to reach at the top of a hill, hence he can't achieve average speed of 30 mph over entire trip. 

MATHEMATICAL PROOF:

Let 'x' be the speed needed in the journey down the hill.

Average Speed = Total Distance/Total time

Average Speed = (1 + 1)/(1/15 + 1/x)

30 = 2/(1/15 + 1/x)

(1/15 + 1/x) = 2 / 30 = 1/15

1/x = 0

x = Infinity/Not defined.

To conclude, it's impossible to achieve average speed of 30mph in trip.

    
 

Cut The Blue Cube Puzzle

A solid, four-inch cube of wood is coated with blue paint on all six sides.

Then the cube is cut into smaller one-inch cubes. These new one-inch cubes will have either three blue sides, two blue sides, one blue side, or no blue sides. How many of each will there be?

Here is solution of the puzzle! 

Cut The Blue Cube Puzzle : Solution

What is the puzzle?

Apart from the 8 cubes at the center all 4 x 4 x 4 - 8 = 56 will have some paint on ones side at least. See below the 1/4 th cube is taken out.

The cubes at the 8 corners will have blue paint on three sides.

The cubes between corner cubes along 12 edges of big cube will have 2 sides painted. That is 12 x 2 = 24 cubes will painted with blue on 2 sides.

And 4 center cubes on each of 6 faces (left, right, top, bottom, front, back) will have only 1 side painted with blue. That is , there are 6 x 4 = 24 cubes having paint on one side only.

To conclude, out of 56 painted cubes,

24 cubes have paint on 1 side,

24 cubes painted with 2 sides,

8 are painted with three sides.