very interesting puzzle...just try!

India
May 24, 2007 9:34am CST
there are 100 ants on a meter stick, each one can walk with a speed on 1 meter per minute. however, whenever 2 ants collide, they reverse their directions and then go on with the same speed. for an arbitrary distribution of ants, what is the maximum time that any ant can remain on the meter stick? (neglect the length of the ants!) please think twice before giving the answer as 1 minute!
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8 responses
@kamran12 (5555)
• Pakistan
28 May 07
1. You never said that ants MUST collide, so I am assuming in first place that they don't collide and in this case maximum time on stick would be 1 Minute taken by an ant moving from one corner and falling from the other. 2. You said that each one CAN (not MUST)walk with a speed of 1m/sec 2.1. Lets assume that one or many or even all ants are not moving at all and not even colliding, in that case the maximum time will be infinity. 2.2. Let's assume that there are 50 ants on both sides facing each other and no ants but two are moving from each end towards the center. they will meet at center and turn back meeting two ants on both ends making them fall off the meter stick. In this pattern the maximum time will be 50 minutes.
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• India
29 May 07
it is not necessary that the ants must collide. however, i am really sorry about the ambiguous wordings given in the question. it requires all the ants to travel at the speed of 1meter per minute, irrespective of whether they collide or not. no ant remains stationary and infinity is not the purpose of this question. well, the answer otherwise, as you have pointed out 50 minutes is not correct. it is easy to see a flaw in the argument presented. let me give you a hint: the answer is less than 50 minutes.
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@kamran12 (5555)
• Pakistan
29 May 07
Then I would assume that the ants do not collide and the answer is 1 minute:-)
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@singlepixel (2743)
6 Jun 07
no limit. as long as they are on that stick. i guess.
• India
6 Jun 07
you are wrong!
@tonixxx (358)
28 May 07
There are one hundred ants and a meter stick, so that is one ant per centre meter, it takes the ants app 60 sec to cross the stick, i would say that the ants are probably more or less evenly distributes with 50 going in each direction. on average the ants will last around 1.6 seconds. I would like to guess that the ant starting on 1 and the ant starting on 100 could last 30 secs before they follow the same fate as all the others and collide and fall. Be gentle with me lol
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• India
29 May 07
hey tonixxx thanks for replying. however, the average cases will not work here because the problem asks you to find the maximum time that is possible for the ants to stay on the stick for any possible configuration...meaning that whichever way the ants are placed initially, after a certain minimum time there will be none left on the stick.
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@marinarovi (1318)
• Argentina
27 May 07
Ok, I have two theories: one is, if the ants do not fall out of the stick, and just turn back and keep walking, they could stay on the stick forever. The other one is, if all the ants are on the same end of the stick, then any one of them would walk the whole stick without colliding, and just fall over the other end one minute later. So my answer would have to be 1 minute.
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• India
28 May 07
well, i asked the maximum possible time that the ants can stay on the meter stick for any possible initial configuration, which is infinity according to you, and which i a m afraid is the wrong answer.
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@marinarovi (1318)
• Argentina
26 May 07
100 minutes, as long as this ant keeps colliding with the other 99 ants... and then the final meter all by itself...
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• India
27 May 07
hello! thanks for the try...but then how did you arrive at the answer...please provide a brief explaination atleast for the same, so that we can discuss it... however, i m really sorry to say that 100 minutes is the wrong answer!
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• Canada
25 May 07
It is one of the two middle ants that would stay on the longest. With 100 ants on there, on average, they are bound to collide with one within 1 cm With 99 ants, on average, they are bound to collide with another within 100/99 cm With 98 ants, on average, they are bound to collide with another within 100/98 cm I'm inclined to believe that on average, every collision, 2 ants fall off the stick. Therefore, it is 1/100 + 1/98 + 1/96 + ..... + 1/2 metres Approximately 2 1/4 minutes.
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• India
26 May 07
well, thats a good try, though i m afraid that the answer is wrong. too many assumptions there, which are really not valid...they are not bound to collide...infact no ant may ever collide! and see the worst case, not the average, coz u want the max. time that it is possible for the ants to stay on the stick!
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@urbandekay (18311)
8 Jun 07
101 x49/100 49 and a half minutes all the best urban
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• Sri Lanka
26 May 07
This is a wonderful riddle. Very similar to the flying bird riddle that I started, but much tougher. I think we have to approah this logically. The maximum time is spent by the ants in the middle, provided all the ants start towards the center. The center ants will meet and a chain reaction will result and there will be a second meeting. In the 3rd meeting the corner ants will fall off. Irrespective of the number of ants on either side the final meeting of the center ant will be at the exact center and the total number of cetimeters it moves to and fro will be 50 in reaching the middle. Then the final turn where it will fall off will be another 50 centimeters and the ant moves 100 centimeters all together. So 1 minute is the maximum time the ant can stay.
• India
26 May 07
well as u said, the middle ant can move in either direction, then reverses and this continues. what if it is at the end of the rod and in the opposite direction?? then it will have to travel all of 100cm again, and the total distance it travels is much more than 100cm! also it is wrong to assume that the middle ant moves 50cm in either direction...in fact please note that the middle ant need not be at the middle of the stick in the first place!
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• Sri Lanka
26 May 07
Is the answer of 1 minute wrong, or is the answer of 1 minute correct but the explanation to arrive at it wrong?
• Sri Lanka
27 May 07
I did this and arrived at the answer perfectly, but cannot explainin without a web cam. The ants irrspective of the numebers only make the middle ant move towards the center, whereever it is. The ant keeps returning only an amout equal to the distance from the corner to where it originally is. So though it moves to and fro it moves only 50 centimeters to the center. Even if all 100 ants are in one corner this is what happens to the middle ant.