In a special number pattern, 15 × 26 is written as 6512 and 29 × 36 is written as 6923. Using the same system, what should 46 × 54 be written as?

Introduction / Context:
This reasoning question involves a digit rearrangement pattern rather than ordinary multiplication. Two examples show how two two digit numbers are combined into a four digit result. Your task is to decode the rule by observing how the digits are being rearranged, then apply it to a new pair of numbers. Such questions assess observational skills and pattern recognition using place value.

Given Data / Assumptions:

- 15 × 26 is represented as 6512.
- 29 × 36 is represented as 6923.
- We are asked to find the pattern for 46 × 54.
- The symbol × here indicates a relationship, not ordinary multiplication.
- All numbers involved are two digit numbers, and the outputs are four digit numbers.

Concept / Approach:
Instead of performing direct multiplication, we look at the digits individually. Every example combines the four digits of the two given numbers into a new four digit number. The key is to understand the ordering of the digits based on their positions and sizes. In particular, we consider the smaller and larger of the two numbers and track their unit and tens digits separately, then see how these appear in the result.

Step-by-Step Solution:
Step 1: Analyse 15 × 26 = 6512. Numbers: 15 and 26. Smaller number: 15 (tens 1, units 5). Larger number: 26 (tens 2, units 6). Result 6512 can be split as digits 6, 5, 1, 2. Observe the order: unit digit of larger number (6), unit digit of smaller number (5), tens digit of smaller number (1), tens digit of larger number (2). Step 2: Check this rule for 29 × 36 = 6923. Numbers: 29 and 36. Smaller number: 29 (tens 2, units 9). Larger number: 36 (tens 3, units 6). Apply the same order: unit of larger (6), unit of smaller (9), tens of smaller (2), tens of larger (3). This yields 6, 9, 2, 3 which matches 6923 given in the example. Step 3: Apply the rule to 46 × 54. Numbers: 46 and 54. Smaller number: 46 (tens 4, units 6). Larger number: 54 (tens 5, units 4). Using the pattern: unit of larger (4), unit of smaller (6), tens of smaller (4), tens of larger (5). Thus the required result is 4, 6, 4, 5 or 4645.

Verification / Alternative check:
We can quickly validate by checking whether any other consistent order of digits can explain both examples. Many alternative arrangements (for example ordering by tens digits first) will fail for at least one of the given mappings. The discovered sequence unit larger, unit smaller, tens smaller, tens larger works perfectly in both cases and matches one of the answer options for 46 and 54, which confirms its correctness.

Why Other Options Are Wrong:

- 5464, 4564, and 4465 all use the correct set of digits {4,4,5,6}, but the order does not match the pattern derived from the examples.
- Attempting to use those orders for 15 and 26 or for 29 and 36 would not reproduce 6512 or 6923.

Hence, they cannot represent the intended rule and must be rejected.

Common Pitfalls:
Learners sometimes try to involve actual multiplication, addition, or subtraction of the numbers instead of looking at how digits are rearranged. Others may spot only part of the pattern, for example matching the first two digits but not the last two. Systematically listing the tens and unit digits of the smaller and larger numbers and checking their order in the results is a reliable strategy.

Final Answer:
According to the discovered pattern, 46 × 54 should be written as 4645.