Problem Analysis:
The task is to place the numbers 1 through 7 into the circles of a hexagonal grid such that no two consecutive numbers are directly connected by a line. The grid has 7 circles, and each circle is connected to its neighbors as shown in the image.
Key Rules:
1. Use the numbers 1 through 7 exactly once.
2. No two consecutive numbers (e.g., 1 and 2, 2 and 3, etc.) can be directly connected by a line.
Solution Approach:
To solve this problem, we need to carefully place the numbers while adhering to the rules. We will use a systematic approach to ensure that no two consecutive numbers are adjacent.
#### Step 1: Understand the Grid Structure
The grid consists of 7 circles arranged in a hexagonal pattern. Each circle is connected to its neighbors as follows:
- The top circle is connected to the two middle circles on the left and right.
- The two middle circles are connected to each other and to the bottom row circles.
- The bottom row circles are connected in a straight line.
#### Step 2: Strategy for Placement
To avoid placing consecutive numbers next to each other, we can start by placing the smallest and largest numbers (1 and 7) in positions that minimize their chances of being adjacent to their consecutive numbers. Then, we can fill in the remaining numbers accordingly.
#### Step 3: Trial and Error with Logical Placement
Let's try to place the numbers step by step:
1.
Place 1 and 7: These are the smallest and largest numbers, so they are less likely to be consecutive with others. Place them in opposite corners to maximize distance.
- Place 1 at the top circle.
- Place 7 at the bottom-right circle.
2.
Place 2 and 6: These are consecutive to 1 and 7, respectively, so they should not be placed next to 1 or 7.
- Place 2 at the bottom-left circle.
- Place 6 at the top-right circle.
3.
Place 3 and 5: These are consecutive to 2 and 6, respectively, so they should not be placed next to 2 or 6.
- Place 3 at the middle-left circle.
- Place 5 at the middle-right circle.
4.
Place 4: The only remaining number is 4, which can be placed in the center circle since it is not consecutive to any of the other numbers.
#### Final Placement:
Based on the above reasoning, the placement of numbers is as follows:
- Top circle: 1
- Middle-left circle: 3
- Bottom-left circle: 2
- Center circle: 4
- Bottom-right circle: 6
- Middle-right circle: 5
- Top-right circle: 7
Verification:
- 1 is not adjacent to 2.
- 2 is not adjacent to 1 or 3.
- 3 is not adjacent to 2 or 4.
- 4 is not adjacent to 3 or 5.
- 5 is not adjacent to 4 or 6.
- 6 is not adjacent to 5 or 7.
- 7 is not adjacent to 6 or 1.
All conditions are satisfied.
Final Answer:
\[
\boxed{
\begin{array}{ccc}
& 1 & \\
3 & & 5 \\
2 & 4 & 6 \\
& 7 &
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of 3rd grade math reasoning worksheet.
