1,496.6K Views

The Paradox of Achilles and the Tortoise is one of a number of theoretical discussions of movement put forward by the Greek philosopher Zeno of Elea in the 5th century BC. It begins with the great hero Achilles challenging a tortoise to a footrace. To keep things fair, he agrees to give the tortoise a head start of, say, 500m. When the race begins, Achilles unsurprisingly starts running at a speed much faster than the tortoise, so that by the time he has reached the 500m mark, the tortoise has only walked 50m further than him. But by the time Achilles has reached the 550m mark, the tortoise has walked another 5m. And by the time he has reached the 555m mark, the tortoise has walked another 0.5m, then 0.25m, then 0.125m, and so on. This process continues again and again over an infinite series of smaller and smaller distances, with the tortoise always moving forwards while Achilles always plays catch up.

Logically, this seems to prove that Achilles can never overtake the tortoise—whenever he reaches somewhere the tortoise has been, he will always have some distance still left to go no matter how small it might be. Except, of course, we know intuitively that he can overtake the tortoise. The trick here is not to think of Zeno’s Achilles Paradox in terms of distances and races, but rather as an example of how any finite value can always be divided an infinite number of times, no matter how small its divisions might become.

John123 Expert Asked on 30th July 2015 in

Achilles will actually meet up with the Tortoise at an exact distance of 556 meters. This you may think doesn’t sound logical. If we apply mathematics to the specified distances using the given set of parameters, we can actually find the mathematical distance at which the Achilles and the Tortoise meetup.

First off we must enter in the equation. 500 + 50 + 5 + 1/2 + 1/4 + 1/8 + 1/16 + ….. + 1/(2^n) = ? Now one can easily sum up the first few numbers 500 + 50 + 5 = 555. The next step is to sum up the infinite series being described here which is typically described in mathematics as Sn = [&Sigma;] 1/(2^n) . One then proceeds to multiply both sides by 2 making it because of a special  2Sn = 2/2 + 2/4 + 4/8+…2/((2)^(n-1))+ 2/(2^n). This reveals an interesting relationship so the equation now becomes 2Sn = 1 + [1/2 + 1/4 + 1/8 + 1/16 + ….. + 1/((2)^(n-1)] = 1 + [Sn – 1/(2^n)]. One then subtracts both sides by Sn making the equation now Sn = 1 –  1/(2^n). Now by applying calculus and taking the limit of 1 – 1/(2^n) this then gives us the value of Sn. Sn = 1

So now we have our final term which is simply 1 So adding all the numbers together gives us 500 + 50 + 5 + 1 = 556 meters is the point at which Achilles will finally meet up with the Tortoise. Now this case only solves this particular paradox because of the poorly worded statement about the last few changes to the distance being a geometric sum 1/2 + 1/4 + 1/8 + 1/16 + ….. + 1/(2^n). Also this was not a solvable equation back in ancient Greece since the invention of calculus did not take place until the 1700s by Isaac Newton. But there you have it, the solution to an age old paradox.

[&Sigma;]

conmcgee Curious Answered on 21st November 2016.

• More puzzles to try-

• What is the logic behind these ?

3 + 3 = 3 5 + 4 = 4 1 + 0 = 3 2 + 3 = 4 ...Read More »
• Defective stack of coins puzzle

There are 10 stacks of 10 coins each. Each coin weights 10 gms. However, one stack of coins is defective ...Read More »
• Which clock works best?

Which clock works best? The one that loses a minute a day or the one that doesn’t work at all?Read More »

Paul, Sam and Dean are assigned the task of figuring out two numbers. They get the following information: Both numbers ...Read More »
• Five greedy pirates and gold coin distribution Puzzle

Five  puzzleFry ship’s pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all ...Read More »
• Magical flowers!!

A  devotee goes to three temples,  temple1, temple2  and temple3  one after the other. In front of each temple, there ...Read More »
• Tuesday, Thursday what are other two days staring with T?

Four days are there which start with the letter ‘T‘. I can remember only two of them as “Tuesday , Thursday”. ...Read More »
• How could only 3 apples left

Two fathers took their sons to a fruit stall. Each man and son bought an apple, But when they returned ...Read More »
• How Many Eggs ?

A farmer is taking her eggs to the market in a cart, but she hits a  pothole, which knocks over ...Read More »
• HARD MATHS – How much faster is one train from other Puzzle

Two trains starting at same time, one from Bangalore to Mysore and other in opposite direction arrive at their destination ...Read More »
• Most Analytical GOOGLE INTERVIEW Question Revealed

Let it be simple and as direct as possible. Interviewer : Tell me how much time (in days) and money would ...Read More »
• Lateral thinking sequence Puzzle

Solve this logic sequence puzzle by the correct digit- 8080 = 6 1357 = 0 2022 = 1 1999 = ...Read More »
• How did he know?

A man leaves his house in the morning to go to office and kisses his wife. In the evening on ...Read More »
• Pizza Cost Math Brain Teaser

Jasmine, Thibault, and Noah were having a night out and decided to order a pizza for \$10. It turned out ...Read More »
• Which letter replaces the question mark

Which letter replaces the question markRead More »
• Which room is safest puzzle

A murderer is condemned to death. He has to choose between three rooms. The first is full of raging fires, ...Read More »
• Richie’s Number System

Richie established a very strange number system. According to her claim for different combination of 0 and 2 you will ...Read More »
• Srabon wanted to pass

The result of math class test came out. Fariha’s mark was an even number. Srabon got a prime!! Nabila got ...Read More »
• Become Normal!!

Robi is a very serious student. On the first day of this year his seriousness for study was 1 hour. ...Read More »
• Sakib Knows The Number!

Ragib: I got digits of a 2 digit number Sakib: Is it an odd? Ragib: Yes. Moreover, the sum of ...Read More »