mn squares of equal size are arranged to from a rectangle of dimension m by n, where m and n are nat

Q: Wait, why can't we have a sequence of numbers like 1, 2, 3 in the grid?

That is a fantastic intuition! If you try to place numbers like 1, 2, and 3 in a grid, you will find that the arithmetic mean condition is violated. For the average of neighbors to equal the value of the cell, the values must be perfectly balanced. If you have a gradient, the average of a point will always be strictly less than the maximum or strictly greater than the minimum, which prevents the condition from holding unless all values are identical.

Q: Does the grid size $m \times n$ change the result?

Not at all! The beauty of this problem is that it is independent of the grid dimensions. Whether it is a $2 \times 2$ square or a $100 \times 100$ rectangle, the logic relies on the connectivity of the grid. As long as the grid is connected, the 'maximum value' property will propagate to every single cell.

Q: Is this just a math puzzle, or is there a deeper physical meaning?

This is actually a discrete version of the 'Maximum Principle' for harmonic functions. In physics, this is exactly how steady-state temperature distributions work in a material. If there are no internal heat sources, the temperature at any point is the average of the temperatures of its surroundings. This problem is a beautiful, simplified window into that profound physical reality!

Question

$mn$ squares of equal size are arranged to from a rectangle of dimension $m$ by $n$, where $m$ and $n$ are natural numbers. Two squares will be called 'neighbours' if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares. Show that this is possible only if all the numbers used are equal.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a vast, empty grid of $m 	imes n$ squares. Each square is waiting to be filled with a natural number.

You are bound by a strict, elegant rule: every single number you write must be the arithmetic mean of its immediate neighbors. This constraint demands absolute uniformity across the entire grid.

### The Maximum Principle

Let us consider the set of all numbers written in our grid. Because the grid is finite, there must be a maximum value, which we shall call $M$.

Locate a cell in the grid that contains this maximum value $M$. Let the neighbors of this cell be $a_1, a_2, \dots, a_k$, where $k$ is the number of neighbors (which could be 2, 3, or 4 depending on the cell's position).

Because $M$ is the maximum value in the entire grid, we know for a fact that every neighbor $a_i$ must satisfy $a_i \le M$. This is the crucial observation.

### The Rigidity of the Average

Now, let us apply the rule. The value of our chosen cell is the arithmetic mean of its neighbors:

$$M = \frac{a_1 + a_2 + \dots + a_k}{k}$$

If we multiply both sides by $k$, we obtain:

$$kM = a_1 + a_2 + \dots + a_k$$

Think about this carefully. We know that each $a_i \le M$. If even one of these neighbors were strictly less than $M$, then the sum of the neighbors would be strictly less than $kM$.

The equation demands that the sum be exactly $kM$. Therefore, the only way to satisfy this equation is if every single neighbor $a_i$ is exactly equal to $M$. There is no wiggle room; the math forces the neighbors to be just as large as the maximum itself.

### The Propagation of Equality

We have just proven that if a cell contains the maximum value $M$, all its neighbors must also contain $M$. Since these neighbors are also maximums, we can apply the exact same logic to them.

Their neighbors must also be $M$. This effect propagates like a wave across the entire connected grid.

Because the grid is connected, this "wave" of equality will eventually reach every single cell. There is nowhere for a different number to hide. Every cell, from the corners to the center, is forced to hold the value $M$.

### The Harmony of the Grid

We have successfully demonstrated that the only way to satisfy this arithmetic mean condition across the entire grid is if all the numbers used are **absolutely equal**.

This is not just a clever puzzle; it is a discrete version of the **Maximum Principle** in harmonic functions. This concept governs everything from steady-state heat distribution to the behavior of electric potentials.

You have just walked through a fundamental principle of analysis. The final conclusion is that the grid must be **uniform**.

Visualized Solution

Visualizing the $m \times n$ Grid

Defining Neighbors in the Grid

The Arithmetic Mean Condition

Introducing the Maximum Value $M$

Setting up the Inequality

Analyzing the Average Equation

The Rigidity of the Average

Propagation Across the Grid

Conclusion and Final Takeaway

The Sigma Insight: Relation Between A.M., G.M., and H.M.

Analyzing the Setup

The Maximum Principle

The Rigidity of the Average

The Propagation of Equality

The Harmony of the Grid

Similar Questions

Let $p$ be the first of the $n$ arithmetic means between two numbers and $q$ the first of $n$ harmonic means between the same numbers. Show that $q$ does not lie between $p$ and $(\frac{n + 1}{n - 1})^{2} p$ .

Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}$ and $G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4} + G_{2}^{4} + G_{3}^{4} + G_{1}^{2} G_{3}^{2}$ is equal to

The product of $n$ positive numbers is unity Then their sum is

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ( $l, n > 1$ ) and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:

If $x_{1}, x_{2}, \dots, x_{n}$ are any real numbers and $n$ is any positive integer, then

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

Let $a, b, c$ be positive integers such that $b / a$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b + 2$ , then the value of $\frac{a ^{2} + a - 14}{a + 1}$ is \dots.

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

Visualized Solution

Visualizing the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } m×n Grid

Defining Neighbors in the Grid

The Arithmetic Mean Condition

Introducing the Maximum Value .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } M

Setting up the Inequality

Analyzing the Average Equation

The Rigidity of the Average

Propagation Across the Grid

Conclusion and Final Takeaway

The Sigma Insight: Relation Between A.M., G.M., and H.M.

Analyzing the Setup

The Maximum Principle

The Rigidity of the Average

The Propagation of Equality

The Harmony of the Grid

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The product of n positive numbers is unity Then their sum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If m is the A.M. of two distinct real numbers l and n (l,n>1) and G1​,G2​ and G3​ are three geometric means between l and n, then G14​+2G24​+G34​ equals:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x1​,x2​,…,xn​ are any real numbers and n is any positive integer, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c are in A.P., a2,b2,c2 are in H.P., then prove that either a=b=c or a,b,−c/2 form a G.P.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a1​,a2​,…,an​ are positive real numbers whose product is a fixed number c, then the minimum value of a1​+a2​+⋯+an−1​+2an​ is

Comprehension Passage

Which one of the following statements is correct ?

Which one of the following statements is correct ?

Which one of the following statements is correct?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a>0,b>0 and c>0, prove that (a+b+c)(a1​+b1​+c1​)≥9.

Visualizing the $m \times n$ Grid

Introducing the Maximum Value $M$

The product of $n$ positive numbers is unity Then their sum is

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ( $l, n > 1$ ) and $G_{1}, G_{2}$ and $G_{3}$ are three geometric means between $l$ and $n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals:

If $x_{1}, x_{2}, \dots, x_{n}$ are any real numbers and $n$ is any positive integer, then

If $a, b, c$ are in A.P., $a^{2}, b^{2}, c^{2}$ are in H.P., then prove that either $a = b = c$ or $a, b, - c /2$ form a G.P.

If $a_{1}, a_{2}, \dots, a_{n}$ are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_{1} + a_{2} + \dots + a_{n - 1} + 2 a_{n}$ is

If $a > 0, b > 0$ and $c > 0$ , prove that $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .