Find the two consecutive natural numbers whose squares add up to 25 in this simple number puzzle.

Introduction / Context:
This question is a straightforward algebraic puzzle involving consecutive natural numbers. You are asked to find two consecutive natural numbers such that the sum of their squares is 25. It tests basic algebra and understanding of how to model consecutive numbers.

Given Data / Assumptions:
- The numbers are consecutive natural numbers, so they differ by 1.
- The sum of the squares of the two numbers is 25.
- Both numbers are positive integers.

Concept / Approach:
Consecutive natural numbers can be represented as n and n + 1. The condition says n^2 + (n + 1)^2 = 25. Expanding the expression and simplifying will create a quadratic equation in n. Solving this equation reveals the pair of numbers that satisfies the condition.

Step-by-Step Solution:
Step 1: Let the smaller natural number be n, so the next consecutive number is n + 1.Step 2: The sum of their squares is n^2 + (n + 1)^2.Step 3: According to the question, n^2 + (n + 1)^2 = 25.Step 4: Expand the expression: n^2 + (n^2 + 2n + 1) = 25 which simplifies to 2n^2 + 2n + 1 = 25.Step 5: Rearrange to standard form: 2n^2 + 2n + 1 - 25 = 0 which becomes 2n^2 + 2n - 24 = 0.Step 6: Divide the whole equation by 2 to simplify: n^2 + n - 12 = 0.Step 7: Factorize: n^2 + n - 12 = (n + 4)(n - 3) = 0, so n = -4 or n = 3.Step 8: Since n must be a natural number, choose n = 3, so the consecutive numbers are 3 and 4.

Verification / Alternative check:
Verify by squaring and adding. The squares are 3^2 = 9 and 4^2 = 16. Their sum is 9 + 16 = 25, which matches the condition. The other algebraic root n = -4 is rejected because negative numbers are not natural numbers in this context.

Why Other Options Are Wrong:
Option 2 and 3 gives squares 4 and 9 whose sum is 13, not 25. Option 4 and 5 gives squares 16 and 25 with sum 41. Option 1 and 4 are not consecutive numbers and their squares sum to 1 + 16 = 17. Option 5 and 6 gives squares 25 and 36 with sum 61. None of these satisfy the required sum of 25.

Common Pitfalls:
Some learners forget that the numbers must be consecutive and instead guess random pairs that square to values near 25. Others make algebraic mistakes while expanding (n + 1)^2 or while factorizing the quadratic. Clearly setting up the pair as n and n + 1 and carefully simplifying avoids these issues.

Final Answer:
The two consecutive natural numbers whose squares sum to 25 are 3 and 4.