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Carl1
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Can You Explain the Puzzle Toad Proof?
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In summary, the proof is made clearer by going through the edges of E in order, starting with the shortest edge and finishing with the longest. At each stage, the walkers at the endpoints of the edge change places, with the possibility of having already moved along shorter edges. This results in each walker traveling along an increasingly longer sequence of edges. With a total of 2m edges traveled and n walkers, the mean number of edges traveled by a walker is 2m/n, meaning at least one walker must have traveled at least 2m/n edges. The Puzzle Toad site describes the proof as beautiful.
- #2
Opalg
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It would be easier to understand that proof if the second sentence was made more explicit:
At stage $k$ of the procedure, two walkers travel along the edge $e_k$. So the total number of edges traveled during the whole procedure is $2m$. The number of walkers is $n$. So the mean number of edges traveled by a walker is $2m/n$, and therefore at least one of the travellers must have walked along $2m/n$ or more edges.
As the Puzzle Toad site mentions, it is indeed a beautiful proof.
Then go through the edges of $E$ one by one in order, starting with the shortest edge $\color{red}e_1$ and finishing with the longest edge $\color{red}e_m$ ... .
At the $k$th stage of this procedure, when you come to the edge $e_k = (u,v)$, the walkers currently at its endpoints $u$ and $v$ change place by walking along $e_k$. If they have not previously moved then that will be the first leg of their walk. But if one or both of them have already moved during a previous stage of the procedure, they will have moved along edges shorter than $e_k$. When the procedure ends (with the walkers at the endpoints of $e_m$ changing places), each walker will have traveled along some increasingly long sequence of edges.At stage $k$ of the procedure, two walkers travel along the edge $e_k$. So the total number of edges traveled during the whole procedure is $2m$. The number of walkers is $n$. So the mean number of edges traveled by a walker is $2m/n$, and therefore at least one of the travellers must have walked along $2m/n$ or more edges.
As the Puzzle Toad site mentions, it is indeed a beautiful proof.
- #3
Carl1
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Thank you so much. Now I understand the proof. It is beautiful.
FAQ: Can You Explain the Puzzle Toad Proof?
How does the Puzzle Toad Proof work?
The Puzzle Toad Proof is a mathematical proof that helps explain the behavior of toads when they are placed in a maze with two exits. This proof uses the concept of "looping paths" to show that no matter how many times the toad takes a certain path, it will eventually reach the other exit. This behavior is known as the "toad problem" and has been studied by mathematicians for many years.
What is the purpose of the Puzzle Toad Proof?
The Puzzle Toad Proof has several purposes. One is to help us understand the behavior of toads in mazes, which can have real-world applications in fields such as biology and robotics. Another purpose is to demonstrate the power of mathematical proofs in solving complex problems and providing concrete explanations for phenomena.
Are there any real-life examples that can be explained by the Puzzle Toad Proof?
Yes, the behavior of toads in mazes is just one example. Other real-life examples that can be explained by the Puzzle Toad Proof include the movement of animals in search of food or shelter, the flow of traffic in a city, and even the behavior of crowds in certain situations.
Can the Puzzle Toad Proof be applied to other animals besides toads?
Yes, while the Puzzle Toad Proof was initially developed for toads, its principles can be applied to other animals as well. In fact, the proof has been used to study the behavior of other animals such as ants, bees, and even humans. This demonstrates the versatility and applicability of mathematical proofs.
Is the Puzzle Toad Proof difficult to understand?
The Puzzle Toad Proof can be quite complex and may require some background knowledge in mathematics to fully understand. However, there are many resources available online that break down the proof into simpler terms and provide step-by-step explanations. With some dedication and effort, anyone can grasp the concepts behind the Puzzle Toad Proof.