Need help starting a proof about an infinite square grid.

In summary, the conversation addresses a problem of proving a claim regarding the burning of a city represented by an infinite square grid. The claim states that if on Day 0 there are only n okay blocks remaining in the city, the entire city will be burned by Day n + 1. Some questions are posed to help the person start thinking about the problem and potentially come up with a proof, such as considering the best and worst case scenarios for the layout of the n blocks, the possibility of having all okay neighbors, and the number of cases to consider.
  • #1
Felixander
1
0
Hi, I need help with simply starting to prove this. I'm terrible with proofs and even the easy ones are hard for me. Any tips to get started in the right direction would be appreciated.

A city represented by an infinite square grid of blocks is on fire. Every day, the status of city block is either burned or okay. Due to winds from the northeast, the status of a city block on Day i is the majority of the statuses of (1) that block, (2) its immediate northern neighbor, and (3) its immediate eastern neighbor from Day i - 1. Prove the following claim: If on Day 0 there are only n okay blocks remaining in the city, the entire city will be burned by Day n + 1.
 
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  • #2
I notice this was your first post, so hello and welcome to PF!

Now, I know you are anxious to prove this statement, but first I propose you think about a few questions.

First, since there are [itex]n < \infty[/itex], what would the best possible layout for those n blocks be--grouped, spread out, partially grouped, etc? Moreover, is their a worst-case for the best case scenario (this one specifically deals with [itex]n[/itex] specifically.

Next, is it possible for all of the [itex]n[/itex] blocks to have an "okay" neighbor to the north and to the east?

Finally, how many cases do you think you need to consider--is it possible to prove this by considering only 1 case?

All of these questions have answers so hopefully after you think about them and we talk you will not only have an idea where to start, but also how to prove it.

I look forward to what you think about these.
 

FAQ: Need help starting a proof about an infinite square grid.

1. What is an infinite square grid?

An infinite square grid is a theoretical concept in mathematics, representing an unbounded set of points arranged in a grid-like pattern. Each point has a unique coordinate, typically denoted by integers, and the grid has an infinite number of rows and columns.

2. How do you start a proof about an infinite square grid?

The first step in starting a proof about an infinite square grid is to clearly define the problem or statement to be proved. This may involve identifying any given information or assumptions, as well as what is to be proven. From there, one can proceed to develop a logical argument using mathematical principles and techniques.

3. Can you provide an example of a proof about an infinite square grid?

Sure, one example of a proof about an infinite square grid is the proof of the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be proven using an infinite square grid by showing that the area of the square formed by the hypotenuse is equal to the sum of the areas of the squares formed by the other two sides.

4. What are some common techniques used in proofs about infinite square grids?

Some common techniques used in proofs about infinite square grids include mathematical induction, contradiction, and direct proof. These techniques involve using logical arguments and mathematical principles to prove a given statement or problem.

5. Are there any real-world applications of proofs about infinite square grids?

While an infinite square grid may seem like a purely theoretical concept, it has many real-world applications. For example, it can be used in computer algorithms to solve complex problems, in physics to model the behavior of particles, and in statistics to analyze data. Proofs about infinite square grids are also important in the field of graph theory, which has numerous applications in computer science, engineering, and other fields.

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