Insects on scale
There is a 100 cm long scale with 100 insects on it. At time t = 0, each starts moving either towards 0 or towards 100 at a speed of 1 cm/s. They fall off the scale when they reach any one of the ends.
You have to tell me this: what is the minimum time at which I can be sure that the scale has no insects left.
You have to tell me this: what is the minimum time at which I can be sure that the scale has no insects left.
Labels: mathemagic, mathschallenge, thinktank
posted by Rajesh Lal at
9:18 AM
51 Comments:
assuming that they r put in a line, from 0 to 100, the time should be 50 sec, as at tht time, the 50th ant, the middle one, will reach any one of the side.
assuming that they r put in a line, from 0 to 100, the time should be 50 sec, as at tht time, the 50th ant, the middle one, will reach any one of the side.
The most time would be spent by an insect at position "0" or position "100", and heading toward the other end. ... it would take such an insect 100 seconds to make the trip.
T+101 seconds - if the insects can balance on the two ends ie on 0 and on 100 there would be 101 insects, so to clear all insects would take 101 seconds if they went to their furthest drop off point.
Think of 101 insects on the scale being pushed along, all in the same direction, every second 1 drops off. Withe 101 insects to drop off, it would take 101 seconds.
If the insects have no sense of balance, and there are 99 insects involved, then 100 seconds.
Are these insects killed as a result of this experiment? Or do they go on to lead fulfuilling lives afterwards?
The problem states that the insects fall off when they reach an end....
This is why you should read the question properly..... there are 100 insects on the ruler
But I'm still going with 101 seconds as it would take 100 seconds to get from 0 to 100, and another second to fall off...
If the insect starts at "0" and falls off when he reaches "100" ... wouldn't that take 100 secs?
If there are 100 insects on the ruler, each spaced 1 cm apart, with the last one on the 100cm point, the first one is on the 1cm point, leaving the 0cm point empty... therefore 1 extra second to kill the last insect but inhumanely marching him over the edge.
If there are 100 insects on the ruler, each spaced 1 cm apart, with the last one on the 100cm point, the first one is on the 1cm point, leaving the 0cm point empty... therefore 1 extra second to kill the last insect but inhumanely marching him over the edge.
The problem statement makes no mention of insect location ... it simply states they are on the scale. The insect which would utilize the most time in falling off the scale, would be the insect who happened to be at the "0" or
"100" position. This insect would have to travel 100cm to reach the end. The problem states that the insects fall off upon reaching the end. Therefore at a rate of 1cm/sec., the insect would fall from either end when 100 seconds has expired.
I agree that there is no mention of where the insects are placed, but assuming that the 2 at either end are the most important as they will cling to life the longest by taking the lengthiest route to extinction.
One is at 0cm and the other is at 100cm. They have started there, and have not fallen from the ruler. At 100 seconds they will have reached the opposite ends of the ruler but will still need to go over the edge.... Hence 100 seconds, + 1second = 101 seconds.
Karl ... I like your argument. however, the problem doesn't state the insects can't sit at either end. It does state that they fall off upon reaching the other end. Therefore, if they fall upon reaching the end, there is no additional second. Right?
Zaux, I too see where you are coming from. If they fall off when they reach the end, ie the 0 and 100cm points, then they cannot have started at these points. We are being directed by the phrasing of the question to think in 1 cm increments. If the insects are terefore spread between the 1 cm and 99cm point, then it would only take 99seconds to clear the ruler as they would not have to travel the full 100cm.
I think the "trick" to this question is that 100 seconds is too easy an answer....
I agree ... but it never states they can't sit at an end ... but only that they fall upon reaching an end. Fun argument!
Fun? I've been struggling with 100 insects the last few hours trying to get them to follow some simple instructions! Then I'll prove my theory with video footage on YouTube!
Beats working!
I like my 100 second solution (and the resoning)... but I bet the anticipated correct answer is 99secs.
Looks like we'll have to wait a week....
However in the meantime, consider the question if it we're a 100cm square, or even a 100 cm cube?
With the square,the distance 100 times the sq. rt. of 2.
With the cube, the distance would be 10 times the sq. rt. of 3.
Did that quickly ... does that sound right?
Sorry ,, the cube is 100 times the sq. rt. of 3
Yup, going from furthest corner to furthest corner (assuming our insects can only go in a straight line!)
142 seconds to fall of the plane
174 seconds to escape the cube
I've rounded up the seconds to ensure they plummet to their unworthy deaths, rather than stopping short...
Using the distances I calculated, I get the same times:
100 times Sq. rt. of 2= 141.4 (142)
and
100 times Sq. rt. of 3= 17302 (174)
Ooops ... 173.02
173.20?
Jeez ... it is .2 ...sorry
If I could just interrupt the discussion between Zaux and Karl Sharman for a moment, and refer back to the original question . . .
The 100 insects could be at any point on the ruler and could move in either direction. So some could be lined up one end and the rest lined up at the other end. This assumes that they are small enough to be able to line up at each end. They could then all move to the nearest end, i.e. all fall off immediately.
So, the minimum time for all the insects to depart would be whatever fraction of a second it takes for the line of insects at each end to topple off.
Just to be clear, the question asks what is the MINIMUM time to be sure for the scale to clear. As I've stated above the answer is close to zero.
The MAXIMUM time for the scale to clear , i.e. the longest time that I would have to wait, would be the time for any insect at one end to reach the other end, i.e. 100 seconds plus fall-off time.
I like the antipodean logic there - the minimum amount of time.... However, it is the minimum time to be sure the scale is clear.
The shortest time it COULD be cleared is, as you correctly surmise, a little over 0 seconds. To absolutely ensure that the scale is clear has to be the time for one insect to traverse the length of the scale and drop off.
I view the question as if the insects were deadly poisonous, and at what point knowing the length of the scale and the speed of the insects would I dare to pick up the scale knowing no little critters will be hanging on to inject me with their deadly poison.
I'm going back to work and will pick this up in the morning. The boss has just walked back in from a 4hr dinner, looking slightly drunk.....
My guess is 99.9 seconds. Assuming that at the start an insect can remain on the scale within 1mm of each end, but would fall off when at either 0cm or 100cm, then the furthest distance any insect could travel before falling would be 99.9cm, which would equate to 99.9 seconds.
Wiz, your interpretation of "minimum time" does not fit the question. The question asks what is the minimum time that it could be guaranteed that no insects remain, not what is the minimum time that there might be no insects remaining.
If you looked at the scale after 1 second then there migt be no insects (as you have described), but it is probable that there will be. If you looked at the scale after 99.8 seconds then again it is possible that one or more insects remain.
If you looked after 99.9 seconds this is the first instance when it is imposssible for any insects to remain.
It's obviously exactly 100 seconds. Not 101 or 99.9 - what strange thinking!
The only interesting extension is by adding the rule that the insect can't pass each other, so they have to change direction when they meet. But I've got an idea that that is a difficult problem to solve.
As I see it, you have 100 insects on the edge of a rule 100cm long. They are also spaced 1cm apart, and travel at a top speed of 1cm/sec.
As stated, the insects can move toward either end to perform their lemming-like leap towards eternity.
Each insect-being of sound insect mind-choosing to prolong their inevidible demise, opts to head toward the end furthest away from their starting location.
Unfortunately, these particular insects do not posess a mountain goat sure-footedness, so the two insects at 0cm and 100cm immediately plummet to their prospective deaths.
Like-wise, the insects at positions 1cm through 99cm during their respective attempts to perform an about-face on the edge, in order to head in the opposite direction, subsequently lose their footing and make their terminal leap from the edge without making it to the end of the rule.
The total time involved for each and every insect to meet their insect god is .0029 secs as the width of the rule's edge is .029mm/(1cm/sec) at its widest point.
The problem does NOT state the insects are 1cm apart ...
Surely once the insect reaches halfway towards the end i.e. 50cm, it then has half of the remaining distance to go, ie. 25cm. then it has half of that distance i.e. 12.5 cm. and so on until infinity. since it has to cover an infinite distance it cannot ever reach the end. in fact applying the same logic, it can never even start moving (thanks Zeno).
even if it does reach the end after an infinite amount of time, the ruler could easily be a bendy one, bent round on itself, such when it reached one end it would just cross back to the start of the other end, so it would never fall off.
even if the ruler isn't bent and the insect does reach the end after an infinite amount of time, have you ever seen an insect fall of the end of anything? no, it would just crawl round to the bottom side, cos insects aren't subject to the law of gravity - kinda like superman.
The variant problem I suggested (Dec 15, 6:41 pm) was easier to solve than I first thought. Amazingly, the answer remains at 100 seconds.
I came to the above conclusion, possibly using dodgy thinking. But I just realised a simple way to see that truth. When two insect collide and bounce of each other, that is visually indistinguishable from the case where the insects simply pass through each other - assuming the insects are identical. i.e. it is as if no interaction had occurred. So that more generalised problem is identical to the original problem and in fact makes the original problem's solution even easier to understand.
I disagree with 50secs or 100secs as an answer. Think about this: you cannot put the insects on the 0 mark or the 100 mark because for one they would already be falling off and two because that would mean that you have 101 insects in total. (Draw it out on a piece of paper and you'll see). For there to be 100 insects, no more no less, on the scale, they need to be positioned at points 0.5cm, 1.5cm, 2.5cm etc through to 99.5cm.
Now the question does ask for the shortest possible route so the insect at 0.5cm is NOT going to move to 100, it is going to move to 0. The insects that are further away from the edge are insects 49.5 and 50.5. Therefore my answer is that the minimum time for all the insects to reach the edge of the scale is 49.5 seconds.
Nicole why must they be a whole 0.5 cm from the 0 and 100 mark. Why not 0.0000000....000001 cm?
Nowhere does the question say that that the insects take the shortest route. However, if that was the case the time to clear would be 49.9999999999....999999999 seconds rather than 99.9999.....999999. seconds.
Chris, I'm assuming the insects are evenly distributed across the scale otherwise the answer to the question becomes way too random! I mean, the insects could be 0.1mm big and all be huddled at each edge of the ruler and then it could take just under 5seconds to fall off. In which case then the answer being asked for is not a number but rather an equation of the sort:
t = (insect size x100)/2
As for the direction the insects take, the question says "each starts moving either towards 0 or towards 100". It does not say they HAVE to move towards the end that is the furthest away.
Hi Nicole. Assume that there is an insect whose centre of mass is on one end of the scale and that it takes the long route to the other end. When it's centre of mass gets to the far end of the ruler, exactly 100 seconds will have passed. That is the worst case scenario.
If you make up the rule that each insect takes the shortest route, the answer will be depend on the difference in position of the centre of mass of the insect and the centre of of the insect. Let that distance be L. In the worst case the time, would be 50+L seconds. The worst possible value of L is 50 cm, so the worst case time taken would also be 100 seconds.
So, the 100 seconds is independent of the length of the insect and independent of any preference for one end of the ruler over another. It is the maximum possible time that would be taken.
Thanks for pushing. I wasn't fully aware of the conclusions above when I started responding.
Now to shoot myself in the foot. I was wrong to talk about the centre of mass of the insect. I should have used . There will be two centres of grip, distance L apart. So the worst imaginable case is that the time is 100+2L. It seems reasonable that L could easily be 1 cm. At this point I would say that with these additional refinements, the question can't be satisfactorily answered.
The debate rages on as we mere gods debate the future of 100 pawns in our little game of life and death for these hapless insects. Actually the outcome is always death, but you can't make an omlette without breaking a few eggs...
So far:-
Nicole - 49.5 seconds
Anonymous 50 seconds
Dual Aspect 99.9 seconds
Zaux - 100 seconds
Chris - 100 seconds, but then began to waver...
Karl - 101 seconds
Mr Farenheit - Eternity (give or take)
I don't know why this problem intrigues me so much but I am continuing to watch these posts with bated breath.... Later I will go out and get a life, but that is for another day.
After all the debate, I still think I'm right and you are all wrong;-) I have posed the question with some of my colleagues at work and mostly they came back with 50 or 100 seconds. They are all idiots and know nothing!
Have a look at the question again - it all hinges on 1 insect - ignore the other 99. What is the longest time this insect could remain on the 100cm ruler travelling at 1cm per second before plummeting to it's death? The longest survival time for this insect (I've called him "Rocky") is when you can be sure there is no possibility that he is left on the scale - Rocky may have gone at any time between 1 second and X seconds, but when can you be absolutely sure - when would you bet your life that the ruler is guaranteed to be Rocky-free?
If he (Rocky) starts at 0 he will take 100 seconds to get to 100, but at this point he will still be on the scale, after all he started at 0 didn't he? Rocky will need that extra second to get off the scale.
But as Chris indicates.... how big is the insect, where can he actually start? The question guides us to think in 1cm increments - is the insect 1cm long? Where is it's centre of gravity? How long does it take for the insect to accelerate to the 1 cm per second speed? Does this need to be factored in?
It is still 101 seconds....???? Isn't it....????
The insects length is unknown - so you can't say it's 1 cm. There is no reason to assume that the insects start on 1 cm marks.
"If he (Rocky) starts at 0 he will take 100 seconds to get to 100, but at this point he will still be on the scale, after all he started at 0 didn't he? Rocky will need that extra second to get off the scale." - Rocky needs less than a microsecond (in the limit he needs 0 more seconds) to to go past the end, not a whole second. It would take a whole second if he had to walk to the non-existent 101 cm mark (or -1 cm mark).
Chris, I did posit the question how long is the insect. I suggested that the question guides us to suppose that it is 1cm long. The length of the insect, and its centre of gravity has a bearing of how long it would take to fall off the edge... 1 cm insect, 0.5 cm down its length is the CoG it would presumably take marginally over 0.5 seconds to over-balance beyond its centre of gravity.... but we do not know how big the insects are.....
.... that's the point, you don't know how long the insects are, so you can't say 1 cm, that may be too long. There is nothing in the question to hint at their length.
You must assume the worst case and the only logical length to choose is 0 cm.
Logically 0cm? Adult males of the parasitic wasp Dicopomorpha echmepterygis can be as small as 139 μm long... hardly 0cm! At the other end of the scale is Phobaeticus chani or Chan's megastick. A species of stick insect. It is the longest insect in the world, with one specimen held in the Natural History Museum in London measuring 567 millimetres. This measurement is however with the front legs fully extended. The body still measures an impressive 357 millimetres
So, assuming a mid-point CoG, we have two extreme scalesto work to.
I would like to take this opportunity to thank Wikipedia, the font of all accurate knowledge in the universe for the above information....
More banging on about the length .... Assuming the insects cannot overlap, the insect can be no longer than 1cm if all are to fit on the scale, so we are looking at 2 lengths - the Dicomorpha echmeptrygis and 1cm
Hi Karl. I think I see where you're going wrong. Just for illustration, let the insect be 1 cm long and let it start at 0 cm and move to 100 cm. I think you are taking the entire insect to be on the scale, in which case it only moves 99 cm to get to the other end. So that takes 99 seconds, not 100 seconds.
Consider that the insect starts with it's centre of gravity at 0 (plus a tiny amount so that it just stays on the scale), then walks a tiny bit less than 100 cm (so it just stays on the scale). The time taken is a tiny bit less than 100 seconds. Now let the tiny amount be an an infinitesimally small amount, you end up with 100 seconds exactly.
At the beginnng and end, approximately half of the insects body will be hanging over the ends of the scale.
Thanks Chris - thats the best explanation yet... and unfortunately due to my obstinate nature I am going to have to stick with 101 seconds, whilst secretly knowing it is 100cm.
The time for a 139 μm insect to walk through it's own length is 0.0139 seconds (at 1 cm/sec). Hardly the 0.5 or 1 second that you have suggested. I'm suggesting that 0.0139 seconds is 0 as near as make no odds. That's a 0.0139% error on my part.
Hi Karl, the last posts crossed. I had actually covered just about everything imaginable in my previous posts. I think you simply weren't able to follow my reasoning.
If you go only with the given info ... the problem statement never says the insects can not sit at the "0" or "100" position ... however, it does state that they fall off upon reaching either end. Therefore, the ability to sit at the "0" or "100" position, and the neceesity of falling off upon reaching the opposite end, seems clear. 100 seconds, to me, is the most logical answer, considering only the data provided and not mis-interpreting the situation.
After logging off last night, I realized that I, too, with my last comment, have interjected a non-given assumption ... that the insects can sit comfortably at the end positions.
The problem, and the reason for all the arguments, is in the non-specific wording of the problem ... yet a the same time,it is also the reason for such interesting debate.
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