Logic
# Arithmetic Puzzles

\[ \large 1 \, \square \, 2 \, \square \, 3 \, \square \, 4 = 10 \]

There are \( 4^3 = 64 \) ways in which we can fill the squares with \( + , - , \times , \div \).

How many ways would make the equation true?

**Note:**

You are not allowed to use parenthesis.

Obey the order of operations.

\[ \left(\color{green}{\square}5\right)^2 = \color{blue}{\square}25 \]

Each square above represents a positive integer. Let \(m\) and \(n\) denote the values that fill in the green and blue squares, respectively, satisfying the equation. Then what is the relationship between \(m \) and \(n?\)

**Details and Assumptions**:

- This is an arithmetic puzzle, where \( 1 \square \) would represent the 2-digit number 19 if \( \square = 9 \). It does not represent the algebraic expression \( 1 \times \square \).

In the figure above, each letter represents a distinct single digit among \(0, 1, 2, \ldots, 9.\) They are positioned in such a way that \[ \large A\times B\times C=B\times G\times E=D\times E\times F. \]

What digit does the letter \(G\) represent?

\[ \large \square + \square\square+\square\square\square+\square\square\square\square \]

Place each of the digits 0 through 9, without repetition, in the boxes above. Then what is the **maximum** possible sum?

**Details and Assumptions**:

- This is an arithmetic puzzle, where \( 1 \square \) would represent the 2-digit number 19 if \( \square = 9 \). It does not represent the algebraic expression \( 1 \times \square \).

\[ \LARGE \square^{\square^{\square^{\square}}} \]

You are given that the numbers \(1,2,3,4\) are to be filled in the square boxes as shown above (without repetition), forming an exponent towers? . Over all \(4!=24\) possible arrangements, let \(S\) be the minimal value that is achieved. How many arrangements of these numbers would produce the value of S?

**Details and Assumptions**

As an explicit example, a possible value of the resultant number is \(\large 2^{3^{1^4}} = 8 \).

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