In the following alpha numeric analogy, AKP is related to 1121256. Using the same letter to number pattern, LNO is related to which of the following numbers?

Introduction / Context:
This analogy connects letters and numbers using the squares of letter positions. The mapping from AKP to 1121256 may look complicated at first, but once we see that each part corresponds to a squared position of a letter, the pattern becomes clear. We then apply this rule to LNO and choose the correct numeric sequence from the options.

Given Data / Assumptions:

• First pair: AKP : 1121256
• Second first term: LNO
• Options: 196125144, 144196225, 144225196, 41521196.
• Letters are mapped to positions A equal to 1, B equal to 2, and so on up to Z equal to 26.

Concept / Approach:
The idea is that the numeric string 1121256 is actually a concatenation of squares of the positions of A, K, and P in the alphabet. Once we verify this for the first pair, we can repeat the process: find positions of L, N, and O, square each, and concatenate the results in the same order to form the related number for LNO.

Step-by-Step Solution:
Step 1: Find positions of the letters in AKP. A is 1, K is 11, and P is 16. Step 2: Square each position. 1^2 = 1, 11^2 = 121, 16^2 = 256. Step 3: Concatenate these squared values to get 1, 121, and 256 as 1121256, which matches the given mapping. Step 4: For LNO, find positions. L is 12, N is 14, O is 15. Step 5: Square each position. 12^2 = 144, 14^2 = 196, and 15^2 = 225. Step 6: Concatenate these squared values in order to get 144196225.

Verification / Alternative check:
The derived number for LNO is 144196225. Among the options, this matches option 144196225 exactly. The other numeric sequences involve a different ordering of the squared parts or additional digits. For example, 196125144 would correspond to the order 14^2, 11^2, 12^2 rather than 12^2, 14^2, 15^2. Therefore only 144196225 respects the same pattern as the original pair.

Why Other Options Are Wrong:
Option 196125144 mixes the numbers 196, 125, and 144 in an order that does not match the positions of L, N, and O. Option 144225196 has the squared values rearranged, and 41521196 includes digits that do not correspond cleanly to the squares of the letter positions. Since the rule requires squaring the positions and keeping the order of letters, any numerical sequence that breaks this order is incorrect.

Common Pitfalls:
A common mistake is to misread the concatenated number and assume it is a single arithmetic expression rather than a sequence of smaller meaningful parts. It is also easy to mis compute squares of two digit numbers like 14 and 15. Writing each square clearly and then concatenating them helps avoid such errors. Another pitfall is to ignore the order of letters and match any combination of squares, which would not reflect the analogical structure correctly.

Final Answer:
The number that completes the analogy AKP : 1121256 :: LNO : ? is 144196225.