Logic
# Arithmetic Puzzles

In the game of M2S3, the goal is to use minimum number of steps to get from one number to the next. In each step, you can either

- multiply by 2, or
- subtract 3.

For example, if we wanted to get from 8 to 11, we could do so in 5 steps:

$\large 8 \overset{-3}{\longrightarrow} 5 \overset{\times2}{\longrightarrow}10 \overset{-3}{\longrightarrow} 7\overset{\times2}{\longrightarrow}14 \overset{-3}{\longrightarrow}11.$

What is the **minimum** possible number of steps required to get from 11 to 25?

$\large \left( \square \ + \ \square \ - \ \square \ \right) \times \ \square \ \div \ \square$

You are given that the numbers $1,2,3,4$ and $5$ are to be filled in the square boxes as shown above (without repetition).

Find the **minimum** possible value of the resultant number. Give your answer to 3 decimal places.

$\large \square + \square\square+\square\square\square+\square\square\square\square$

You are given that the numbers $0,1,2,\ldots,9$ are to be filled in the square boxes as shown above (without repetition) such that it represent a sum of a 1-digit, 2-digit, 3-digit, and 4-digit number.

Find total number of possible arrangements of these ten numbers such that the sum of these four numbers is **maximized**.

**Details and Assumptions**:

- For the purposes of this question, 0 is considered a 1-digit number.
- 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 $1 \times \square$.

Using the digits 1 through 9 without repetition, fill out the number grid above. What is the product of the numbers in the 4 corners?

**Note:** The order in which this grid calculates is left-to-right/top-to-bottom **unlike** the usual order of operations. E.g., $1+2\times 3 = (1+2)\times3=9.$

$\large 1 \, \square \, 2 \, \square \, 3 \, \square \, 4 \, \square \, 5 \, \square \, 6 \, \square \, 7 \, \square \, 8 = 9$

There are $2^7 =128$ ways in which we can fill the squares with $+, -$.

How many ways would make the equation true?

**Note**: You are not allowed to use parenthesis.