Ants III

iwanttobelieve · November 28th, 2006, 10:38 am

In a square of side 1, you have been able to capture 51 ants, who are possibly panicking.You have a glass, whose radius is 1/7.Show that you can at any moment position the glass so as to encompass at least 3 of them.

vixen · November 28th, 2006, 11:01 am

Ok, without giving away too much and spoiling it for others,1. Pigeon Hole Principle2. sqrt(2)/5 < 2/7

cdmurray80 · November 29th, 2006, 1:28 am

I was getting stuck for like 15 minutes before I realized...don't try to cover a square with 17 circles...that can't be done (or at least I don't know how to). Try with 25...

fn075425 · December 4th, 2006, 10:00 am

i believe dividing into 25 sub-square is the best choice:first we divide to 25 parts, which means at least one of them contains 3 ants.then the size of this area is 0.04, and the size of the glass is pi/49so we can get the glass can cover at least one area with 3 ants.Hope this doesn't make any mess.

iwanttobelieve · December 5th, 2006, 12:00 pm

What do you mean by 'size' ? The area is not relevant. Lower area is a sine qua non condition for a shape to be coverable by another but it's obviously not sufficient. The answer by vixen is far more clearer. This is the fact that the diagonal of the square is less than the diameter of the circle that matters.

rimaephosie · December 5th, 2006, 1:34 pm

The ants are panicking, so they are uniformly repartited in the square.The number of ants in an area of Pi/49 is 51*(Pi/49) > 3 thus it's ok ?!

iwanttobelieve · December 5th, 2006, 1:52 pm

I am afraid that's not ok rimae[. You are making a typical mistake.The result you give is in 'expectation', whereas I was asking for an 'almost sure' result.Generally speaking, in finance people are more interested in almost sure results.

fn075425 · December 5th, 2006, 4:37 pm

QuoteOriginally posted by: iwanttobelieveI am afraid that's not ok rimae[. You are making a typical mistake.The result you give is in 'expectation', whereas I was asking for an 'almost sure' result.Generally speaking, in finance people are more interested in almost sure results.to make it simple, we just say pigeon hole theory. If you look at the any books concerned about combinatorics, you can find many similar question

imereli · December 6th, 2006, 1:52 am

QuoteOriginally posted by: rimaephosieThe ants are panicking, so they are uniformly repartited in the square.The number of ants in an area of Pi/49 is 51*(Pi/49) > 3 thus it's ok ?!Small addition. You do not need uniform distribution of ants in the square. If they are not uniformly repartited in the square, there is always Pi/49 area where you can find more than 3 ants.