To determine the number of different paths that spell "ARCH" starting from the letter 'A' and moving one letter at a time vertically, horizontally, or diagonally, we can approach this problem systematically. Here is the grid structure:
```
A
R R R
C C C C C
H H H H H H H H
```
We'll identify all possible paths that form the word "ARCH."
### Step-by-step Analysis
#### Step 1: Finding 'R'
There are 3 'R' letters, all directly connected to 'A'. Let's denote their positions as R1, R2, and R3 from left to right.
#### Step 2: Finding 'C' after each 'R'
Each 'R' has up to 3 'C' letters it can connect to. We need to count all valid connections:
- **From R1**: Connects to C1, C2, and C3.
- **From R2**: Connects to C1, C2, C3, and C4.
- **From R3**: Connects to C2, C3, and C4.
#### Step 3: Finding 'H' after each 'C'
Each 'C' can connect to 'H' letters below it. Let's count:
- **From C1**: Connects to H1, H2, and H3.
- **From C2**: Connects to H1, H2, H3, and H4.
- **From C3**: Connects to H2, H3, H4, and H5.
- **From C4**: Connects to H3, H4, H5, and H6.
- **From C5**: Connects to H4, H5, and H6.
### Counting All Paths:
By following the steps outlined above, we can count the various combinations for paths:
#### Paths through R1:
- A -> R1 -> (C1 or C2 or C3)
- From C1: Connects to H1, H2, H3 (3 paths)
- From C2: Connects to H1, H2, H3, H4 (4 paths)
- From C3: Connects to H2, H3, H4, H5 (4 paths)
Paths through R1 = 3 + 4 + 4 = 11 paths
#### Paths through R2:
- A -> R2 -> (C1 or C2 or C3 or C4)
- From C1: Connects to H1, H2, H3 (3 paths)
- From C2: Connects to H1, H2, H3, H4 (4 paths)
- From C3: Connects to H2, H3, H4, H5 (4 paths)
- From C4: Connects to H3, H4, H5, H6 (4 paths)
Paths through R2 = 3 + 4 + 4 + 4 = 15 paths
#### Paths through R3:
- A -> R3 -> (C2 or C3 or C4)
- From C2: Connects to H1, H2, H3, H4 (4 paths)
- From C3: Connects to H2, H3, H4, H5 (4 paths)
- From C4: Connects to H3, H4, H5, H6 (4 paths)
Paths through R3 = 4 + 4 + 4 = 12 paths
### Total Paths:
By summing all possible paths through R1, R2, and R3:
- Paths through R1: 11
- Paths through R2: 15
- Paths through R3: 12
Total Paths = 11 + 15 + 12 = 38
Thus, there are a total of **38 different paths** that spell "ARCH".
Starting with the A and moving one letter at a time vertically, horizontally, or diagonally, how many different paths spell ARCH?
A
RRR
CCCCC
HHHHHHHH
1 answer
1 answer