DANGER Coding Pattern Analysis

This problem requires us to decipher a coding pattern used for the word 'DANGER' and then apply that same pattern to find the code for the word 'MACHINE'. The provided code for 'DANGER' is 10-7-20-13-11-24.

DANGER Code Breakdown

First, let's identify the standard alphabetical position for each letter in the word 'DANGER':

• D corresponds to the 4th position.
• A corresponds to the 1st position.
• N corresponds to the 14th position.
• G corresponds to the 7th position.
• E corresponds to the 5th position.
• R corresponds to the 18th position.

So, the sequence of alphabetical positions for DANGER is 4, 1, 14, 7, 5, 18.

The given code for 'DANGER' is 10-7-20-13-11-24.

Coding Rule Found

Let's compare the alphabetical positions with the numbers in the code to discover the underlying rule. We can check if a simple arithmetic operation links them. Let's test adding a constant value:

• For D: Alphabetical position is 4. The code is 10. The difference is $10 - 4 = 6$.
• For A: Alphabetical position is 1. The code is 7. The difference is $7 - 1 = 6$.
• For N: Alphabetical position is 14. The code is 20. The difference is $20 - 14 = 6$.
• For G: Alphabetical position is 7. The code is 13. The difference is $13 - 7 = 6$.
• For E: Alphabetical position is 5. The code is 11. The difference is $11 - 5 = 6$.
• For R: Alphabetical position is 18. The code is 24. The difference is $24 - 18 = 6$.

We've found a consistent pattern! For each letter, its code number is generated by adding 6 to its alphabetical position. The coding rule is: Code = (Alphabetical Position) + 6.

MACHINE Code Calculation

Now, we apply this discovered rule to the word 'MACHINE'. First, list the alphabetical positions for each letter in 'MACHINE':

• M is the 13th letter.
• A is the 1st letter.
• C is the 3rd letter.
• H is the 8th letter.
• I is the 9th letter.
• N is the 14th letter.
• E is the 5th letter.

Next, apply the rule (Position + 6) to each letter's position:

• M: $13 + 6 = 19$
• A: $1 + 6 = 7$
• C: $3 + 6 = 9$
• H: $8 + 6 = 14$
• I: $9 + 6 = 15$
• N: $14 + 6 = 20$
• E: $5 + 6 = 11$

MACHINE Final Coded Word

By combining these resulting numbers in order, separated by hyphens, we get the code for 'MACHINE':

19-7-9-14-15-20-11

This sequence represents the code for 'MACHINE' based on the pattern derived from 'DANGER'.