The uploaded worksheet focuses on logical reasoning with Triangular Numbers. Here's a breakdown of the instructions and how to approach solving the problems:

Key Concepts

1. Triangular Numbers:

   • These are numbers that can form an equilateral triangle when arranged as dots.
   • Example sequence: $1, 3, 6, 10, 15, \ldots$
   • Formula: $T_n = \frac{n(n+1)}{2}$, where $n$ is a positive integer.

2. Directions:

   • Use numbers only once per puzzle.
   • Subtract neighboring numbers to get the number below them.
   • Example: $4 - 3 = 1, 1 - 5 = -4,$ and so on.

Steps for Solving Each Puzzle

1. Understand the Pattern:

   • In the example, numbers $1$ through $6$ are used.
   • Start at the top row and work downwards, ensuring that subtraction of any two adjacent numbers gives the value below.

2. Strategy:

   • Start from the largest triangular number that fits at the top row.
   • Work step by step to ensure the subtractions fit the triangular number pattern.

3. Check:

   • At each stage, confirm that all numbers are used exactly once, and subtraction logic is valid.

Questions to Ask Yourself While Solving

1. Are the subtractions between adjacent numbers producing the correct result?
2. Did I use each number only once in the sequence?
3. Is the triangular pattern maintained throughout?
4. Can smaller triangular numbers be used for intermediate rows?
5. Does the final answer align with the example provided?

Questions for You

1. Would you like me to solve the "Numbers 1-10" puzzle for you step-by-step?
2. Do you want a detailed explanation of triangular number logic or their applications?
3. Should I calculate more triangular numbers beyond 15 for additional examples?
4. Do you need help visualizing how subtraction forms the triangular pattern?
5. Would you like tips on organizing your solution efficiently?

Tip

Always verify each subtraction as you proceed. Write intermediate steps to ensure you don't lose track of the logic or accidentally reuse a number!