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Consider a rectangular array of numbers, extending infinitely to the left and right, top and bottom. Start with all the numbers equal to 0 except for a single 1. Then go through a series of steps, where at each step each number gets replaced by the sum of its four neighbors. For example, after one step the array will look like \(\begin{array}{rrr} & 1 & \\ 1 & 0 & 1 \\ 1 & \end{array}\) surrounded by an infinite "sea" of zeros, and after two steps we will have \(\begin{array}{llllll} & & & 1 & & \\ & & 2 & 0 & 2 & \\ & 1 & 0 & 4 & 0 & 1 \\\ & 2 & 0 & 2 & \\ & & & 1 & & \end{array}\) a. After \(n\) steps, what will be the sum of all the numbers in the array, and why? b. After \(n\) steps, what will be the number in the center of the array (at the position of the original 1)? c. Can you describe the various nonzero numbers that will occur in the array after \(n\) steps?

Step by step solution

01

- Understanding the process

Initially set the array where all numbers are 0 except a single 1 in the middle. For this example, the four neighbors of each cell will be used to update the array in each step.

02

- Sum of all numbers after n steps

In this step, we need to determine the sum of all numbers in the array after n steps. Note that at each iteration, each cell value is just redistributed, but the total sum of all numbers in the array remains constant. Since initially, there is only one '1' and all others are '0', the sum of all numbers in the array forever remains 1.

03

- Number in the center of the array after n steps

Determine the number in the center of the array after n steps. For each step update, the center value becomes the total sum of its four neighbors. By symmetry and uniform value distribution, the central value after each step remains 0.

04

- Describing the various nonzero numbers after n steps

Identify the pattern of non-zero numbers after n steps. At each step, the array develops a specific pattern based on previous values, but certain positions will always result in non-zero values due to the spreading effect of the initial 1. Generally, these non-zero values appear symmetrically around the center.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Array of Numbers

In this exercise, we start with a rectangular array that extends infinitely in all directions. To simplify, think of it as having unlimited space with zero values everywhere except for one cell that holds a 1. The array's property of being infinitely extendable helps in the distribution process, ensuring that values can spread outwards without any boundaries stopping them. In each step, every cell updates its value to be the sum of its four neighboring cells (top, bottom, left, right). This setup makes it easier to manage and predict how values spread across the array, even when the array is infinite.

Iterative Steps in Arrays

The array undergoes a transformation at each step, making this an iterative process. Initially, only one cell is nonzero (holding 1). In the first step, this value is redistributed to its four neighboring cells, which sums up to one 1 being spread out. This results in something like:
& 1 &
1 & 0 & 1
& 1 &
surrounded by zeros.
In subsequent steps, the values keep spreading outwards. After each step, every cell's value will be replaced by the sum of its four neighbors. This iterative method ensures the values continuously move away from the center. This means observing the pattern's expansion can give insight into future states of the array without calculating each step manually.

Array Value Distribution

As values spread through the array, it's essential to understand how these numbers distribute after each step. From the given example, we see that the center's value quickly becomes 0 due to being the sum of zeros. However, nonzero values start to emerge around the center.

• After one step, you have values of 1 at the immediate neighbors of the initial center.
• After two steps, the array has values spread in a slightly larger pattern, with 1's and 2's appearing symmetrically around the center.

The overall sum of the numbers in the array always stays the same: the original sum, which is 1. This is because the value of 1 is just being redistributed over and over. In essence, the nonzero values form predictable patterns around the initial starting point, creating a visual and numerical spread that reflects the redistribution of the initial number.