A unsolvable Maths question

16 Nov 2012

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Kim Soo Ho
Kim Soo Ho's picture
A unsolvable Maths question

Hello Lin, I just read about a Mathematics mystery. I don't know how to understand it.

The Mystery: An archer shot an arrow at a target. Some people recorded the position of the arrow when the arrow reached HALF of the distance remaining. Then, the people measured the half of the half distance remaining. So, let's say the distance is 200 metres. So, the first measurement was taken at 100 metres marking. The second measurement is 50 metres. The third one is 25 metres and so on... If the people keep measuring the middle of the distances between the archer and the the target, the arrow will NEVER reach the target as the distance will keep decreasing but will never reach zero. But the funny thing is that it does when an archer actually shoots an arrow. Why?

Lin Guoyao
Lin Guoyao's picture
Zeno's paradox of the Tortoise and Archilles

Hello Sooho,

I believe the question is derived from the famous Zeno's Paradox of the Tortoise and Archilles. 

I think if you look at it in that angle, its seems that the arrow will never hit the target since the remaining distance will keep becoming infinitely small.

However, if you look at it mathematically, its not true!

so here goes:

200m = 100m + 50m + 25m + 12.5m + 6.25m + 3.125m + 1.5625m + 0.78125m + 0.390625m + 0.1953125m + 0.0976562 + ... + 0.0000238 + ... + infinitely small number.

What happen is, you will still get 200m in total if you add up all the infinite number of sub-divided smaller numbers.

In conclusion, the arrow will hit the target eventually.

You can also think in terms of time taken to reach the target. It will eventually reach the time to hit the target =)

I am The Lin.

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