Brain Teasers
Half the Distance
You can imagine an arrow in flight, toward a target. For the arrow to reach the target, the arrow must first travel half of the overall distance from the starting point to the target. Next, the arrow must travel half of the remaining distance.
For example, if the starting distance was 10m, the arrow first travels 5m, then 2.5m.
If you extends this concept further, you can imagine the resulting distances getting smaller and smaller. Will the arrow ever reach the target?
For example, if the starting distance was 10m, the arrow first travels 5m, then 2.5m.
If you extends this concept further, you can imagine the resulting distances getting smaller and smaller. Will the arrow ever reach the target?
Answer
Yes. This is because the sum of an infinite series can be a finite number. Thus, 1/2 + 1/4 + 1/8 + ... = 1 and the arrow hits the target.Hide Answer Show Answer
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Comments
No the sum of an infinite series is approximately equal to 1
There is a "different" deffinition of sum in the case of an infinite series.
Its taken as value (if any) that the series approaches. That can be taken to be in your case the dart board, but that does NOT mean the sum of an infinite series is finite
Its taken as value (if any) that the series approaches. That can be taken to be in your case the dart board, but that does NOT mean the sum of an infinite series is finite
The thing to point out is that not all infinite series are finite, but certain infinite series can be. It's what happens when you integrate the area under a curve. If the series approaches a limit at a high enough rate, you can add up the area to infinity and get a finite number. It's in every calculus book.
In the physical world it will, but in the mathematics world, it will never,because it keeps going smaller and smaller never reaching the designated amount
it's simple algebra. it can be proven that if a series is convergent, the sum of an infinite geometric series (like this one) is finite. how do we know if it is convergent? take the series t + t(r) + t(r^2) + t(r^3)... which converges to S = t/(r-1) IF r
hmm, cut off... take the series t + t(r) + t(r^2) + t(r^3)... which converges to S = t/(r-1) IF r
one more time (apparently you can't use certain symbols)... take the series t + t(r) + t(r^2) + t(r^3)... which converges to S = t/(r-1) IF r GT 1. in our example, t = 1/2 and r = 1/2, so if you plug it in, you'll get S = 1. (simple explanation: if the numbers in the series get smaller, it converges to a finite sum; if they get larger, it's an infinite sum)
For the curious, this was one of Zeno's three paradoxes. Zeno, a greek philosopher, claimed, based upon this situaiton illustrated in the teaser, that movement was not possible: Since before the arrow could get to a certain point it would always have to first get though half the distance to that point, the arrow would never move. A story, I think in jest, is told in which Zeno and his students were confronted by a group of hungry lions. Zeno there upon told his students that they should not fear since movement was not possible, and therefore the lions could not move!
If a tree falls in the forest, and there's no-one there to hear it, does it make a noise?
The arrow will reach the target in the real world,not because of mathematics but simply because of speed distance and time basics. But mathematically it will never reach as the sum of an infinite series is finite only after taking limits which is an approximation.it is never exactly finite.We take approximation as the difference between the exact sum and the finite approximation keeps getting more negligle as the series progresses
The question disguises the truth. Yes, the arrow must travel the first half of the distance before the second half. Yes, it must travel the third fourth before the fourth fourth. This only says you can cut the time frame on your observation and find another instant when the arrow had yet to reach its target.
The big shocker is that physicists have evidence that time may be ticking away in ultra-tiny jumps which can not be divided.
The big shocker is that physicists have evidence that time may be ticking away in ultra-tiny jumps which can not be divided.
I've always found quantized time is always an odd though
Then you'll love quantized space and gravity.
Kiddies, when you study calculus, please don't read these comments - it fill totally confuse you. For a geometric series with a common ratio between -1 and 1, the sum of the series converges to a finite limit. Cathalmccabe and others comments that it 'nearly' reaches the limit or it can't reach the target because it requires an infinite number of terms in the real world, are way off the mark.
Consider the example given. The arrow goes 5 m. Lets say this take 0.5 sec. Then it goes 2.5 m which would take 0.25 sec the 1.25 m in 0.125 sec and so on. Thus the time taken is the sum of 0.5 + 0.25 + 0.125 sec.
Here's the first good bit. If the arrow goes 5 m in 0.5 sec it is traveling at 10 m/sec. If it goes 2.5m in 0.25 sec it is again doing 10 m/sec. etc. In other words a car driving at a constant speed of 10 m/sec would be keeping exact time with the arrow so after 1 second the car would have done 10 metres (and so would the arrow!)
Here's the second good bit how many terms of the times series (0.5 + 0.25 + 0.125...) does it take to make 1 second? An infinite number! So the arrow has completed an infinite number of these time-frame movements all in one second! YES! This is because each time we are looking at the arrow we are looking for a smaller and smaller bit of time, converging to zero. For anyone who says that this can't happen in reality - of course it happens all the time. Fire an arrow at a target 10 metres away. Are you really going to agree with Zeno's tongue in cheek conclusion that the arrow cannot move?
Lastly, consider what happens in black holes where our knowledge of the space/time continuum seems to break down. This is magical stuff but the calculus is based on calculatable limits - not approximations!
{I hope this is helpful}
Consider the example given. The arrow goes 5 m. Lets say this take 0.5 sec. Then it goes 2.5 m which would take 0.25 sec the 1.25 m in 0.125 sec and so on. Thus the time taken is the sum of 0.5 + 0.25 + 0.125 sec.
Here's the first good bit. If the arrow goes 5 m in 0.5 sec it is traveling at 10 m/sec. If it goes 2.5m in 0.25 sec it is again doing 10 m/sec. etc. In other words a car driving at a constant speed of 10 m/sec would be keeping exact time with the arrow so after 1 second the car would have done 10 metres (and so would the arrow!)
Here's the second good bit how many terms of the times series (0.5 + 0.25 + 0.125...) does it take to make 1 second? An infinite number! So the arrow has completed an infinite number of these time-frame movements all in one second! YES! This is because each time we are looking at the arrow we are looking for a smaller and smaller bit of time, converging to zero. For anyone who says that this can't happen in reality - of course it happens all the time. Fire an arrow at a target 10 metres away. Are you really going to agree with Zeno's tongue in cheek conclusion that the arrow cannot move?
Lastly, consider what happens in black holes where our knowledge of the space/time continuum seems to break down. This is magical stuff but the calculus is based on calculatable limits - not approximations!
{I hope this is helpful}
The answer is no - it's another one of those trick questions. It won't reach the target, because per the instructions, I am only imagining the arrow. So...I can imagine the arrow hitting the target, but in reality it will never actually reach the target.
The "time" comment by Dishu above is correct. The answer is not mathematical but physical. The riddle has a false premise presented as fact. Theoretically the arrow never reaches the target because the remaining distance can always be halved. But in reality time simply runs out for the arrow. It must eventually hit the target.
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