In a certain code, Z is written as 2197 and R is written as 729. Using the same rule, how would the letter J be written in that code?

Introduction / Context:
This coding and decoding question links individual letters with three digit and four digit numbers. You are told that Z is written as 2197 and R as 729 in a certain code, and you need to find the code for J. The pattern involves alphabet positions and cubes of smaller integers, which is a standard trick in reasoning exams.

Given Data / Assumptions:
Letter Z is coded as 2197.
Letter R is coded as 729.
You must deduce the general rule that maps a letter to a number.
Then you must apply that rule to letter J.
Alphabet positions are taken as A = 1, B = 2, C = 3 up to Z = 26.

Concept / Approach:
The numbers 2197 and 729 are well known cubes: 2197 is 13 cubed and 729 is 9 cubed. The link to letters likely involves taking half of the alphabet position (for even positions) and then cubing that value. This is because Z is the 26th letter and half of 26 is 13, while R is the 18th letter and half of 18 is 9. Once we confirm this pattern, we can apply it to J, which has position 10 in the alphabet.

Step-by-Step Solution:
Step 1: Determine the alphabet position of Z. Z is the 26th letter. Step 2: Note that 2197 equals 13 multiplied by 13 multiplied by 13, which is 13 cubed. Step 3: Observe that 13 is exactly 26 divided by 2. So for Z, the code is (position of Z divided by 2) cubed. Step 4: Now check R. R is the 18th letter of the alphabet. Step 5: Half of 18 is 9, and 9 cubed is 9 multiplied by 9 multiplied by 9, which equals 729. This matches the given code for R. Step 6: Thus, the coding rule is: for a given letter, take its alphabet position, divide by 2 and then raise the result to the power 3 (cube it). Step 7: Now apply this rule to J. J is the 10th letter of the alphabet. Step 8: Divide 10 by 2 to get 5. Step 9: Compute 5 cubed: 5 multiplied by 5 equals 25, and 25 multiplied by 5 equals 125. Step 10: Therefore, the code for J is 125.

Verification / Alternative check:
You can verify the pattern by checking whether any alternative simple relationship exists that maps 26 to 2197 and 18 to 729 consistently. The cube interpretation fits both pairs perfectly. Also, 2197 and 729 are standard cube numbers widely used in aptitude problems, so it is very unlikely that a more complicated rule is intended. Applying the same rule to J yields 125, which is 5 cubed, another standard cube. The pattern is therefore consistent and complete.

Why Other Options Are Wrong:
216: This is 6 cubed and would correspond to a letter whose half position is 6, that is, letter with position 12. J has position 10, not 12, so 216 cannot be its code.
124: This is not a perfect cube and does not fit the pattern of cubing half the letter position.
512: This is 8 cubed and would correspond to a letter whose half position is 8, that is, position 16, which is P, not J.

Common Pitfalls:
Sometimes candidates attempt to use squares instead of cubes or try to link the numbers directly to alphabet positions without noticing the halving step. Recognising 2197 and 729 as cubes is the key insight. Always check for common powers like squares and cubes when large three or four digit numbers appear in such coding problems.

Final Answer:
According to the rule used for Z and R, the letter J is written as 125 in the given code.