[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 500

 
Mathemat:
The diagonals in a non-diagonal are (9-3)*9/2 = 27. Have you got it all figured out, ilunga?

It is enough to place "dangerous" pairs strictly on adjacent corners, this will prevent them from falling on the diagonals.

The solution is not the only one. Example : 8, 1, 6, 2, 9, 4, 5, 3, 7

 
MetaDriver: The solution is not the only one. Example : 8, 1, 6, 2, 9, 4, 5, 3, 7
4*6=8*3. Draw and check.
 
Mathemat 08.08.2011 21:58

From matforum:

How to paint the smallest number of cells on a 9×9 board so that its edges are not visible from the centre of the board (i.e., any ray coming out of the centre touches some painted cell at least at a corner)?

* It is forbidden to colour in cells adjacent to a side or corner as well as the centre cell.

P.S. First try to get some solution, not necessarily "minimal". The main condition: the painted cells must not touch each other, even only in one point.

What did you get, are we going to poke around next?


 
Mathemat:
4*6=8*3. Draw and check.

Nah, I'm not painting. It just turns out not all the dangerous pairs have been written out.

Correction: 8, 1, 6, 2, 9, 4, 5, 7, 3.

 
ilunga: and number the vertices in a circle like this: 1, 6, 2, 9, 7, 5, 4, 3, 8
That sounds about right.
 
MetaDriver: Did you get one of those, are we going to do any more digging?
Yeah, I've got one just like it, only turned 90 degrees. I don't know how to prove it's minimal...
 
Mathemat:
Sounds about right.
He's the most honest man! How can you doubt him?
 
MetaDriver:
He's the most honest man! How can you doubt him?

Right. And how could I have doubted your first solution for the 9-gon?!

Here's another one:

A square with side 1. Each side is divided into three equal parts. Find the area of the shaded square.

 
Mathemat:

1. Right. And how could I have doubted your first solution for the 9-corner?!

Here's more:

2. A square with side 1. Each side is divided into three equal parts. The lines are drawn through the dividing points (see the figure). Find the area of the shaded square.
.

1. Exactly. Shame on you.

2. 1/10 = 0.1

 

Yeah, don't explain 0.1 yet. Let the others suffer.

More:

Some group of psychologists has developed a test, on passing which each person gets a score - the number Q - an indicator of his mental ability (the more Q, the greater ability) Some group of psychologists has developed a test, on passing which each person gets a score - the number Q - an indicator of his mental ability (the greater Q, the greater ability). Suppose that each resident of two countries, A and B, got his or her Q. The average arithmetic mean of the Q values of all the inhabitants of that country is taken as the rating of the country.
a). A group of citizens of country A emigrated to country B. Is it possible that both countries have increased their ratings?
b). After that, a group of citizens of country B (which may include former emigrants from A) emigrated to country A. Is it possible that the ratings of both countries increased again?