## Thursday, September 4, 2008

### Sudoku solver and some other problems

I got a comment today on my post Algorithm for solving Sudoku puzzles with a link to a program for solving sudoku online. So if you have problems with a particular hard puzzle, give it a try. The program is called Sudoku solver.

I also stumbled on an excellent collection of simple physic problems. You can find the collection here. Unfortunately there are no solution to the problems there, but they seems all to be of the type that requires understanding of physics and not knowledge of math and formulas. Since I am trying to write a math blog, it would be pointless for me to write solutions to these problems. However, I want to solve the first problem:

You are given two identical steel balls of radius 5 cm. One ball is resting on a table, the other ball is hanging from a thin string. Both balls are heated (e.g., with a blow torch) until their radii have increased to the same value of 5.01 cm. Which ball absorbed more heat and why?

The answer - the ball that lays on the table will absorb more heat. The reason to this is that the heat can escape better from the ball when the ball touches another object, in this case the table. When the ball is suspended in the air, the heat also escapes but the rate of the escape is lower. This is due to the fact that air is much less dense that a table.
Since more heat escapes, more hear will be required to heat the ball on the table.

No sure about your explanation. I think you need to know more info about the table and the string to evaluate this aspect. But even if you go with this and start to consider how the heat escapes FROM the supporting object, then the string may be more efficient. Because when the table start to heat, your ball is now on a hot surface radiating heat and sending up hot air directly to the ball.

Anyways, looking the data given by the problem, I think the question is more about the sphere's expanding (if not, why don't just use temperature?). Assuming that the shape stays spheric, the mass of the ball on the table must be lift up (the center move of 0.005 cm), which need some conversion of heat energy to potential energy. On the string, the center of mass is lowered. Since we don't know all other physical details, the 0.01 cm difference of potential energy between the 2 balls is the only usable data, so the probable 'answer' is the one of the table.

Now for the physical and chemical process, I don't know how steel expand, but by analogy with gas, I assume that pressure of the weight works against the expanding and so you need more heat to have the same result.

Anatoly said...