In an arithmetic reasoning puzzle, an intelligent boy is assembling a jigsaw with 275 pieces. Each day that he fits pieces together there are fewer pieces left, and it is reasonable to assume that he fits more pieces each day because the number left to sort out diminishes progressively. Hence he is able to fit one extra piece as each new day goes by. On the first day he fits 20 pieces. How many whole days does it take for him to complete the puzzle entirely?

Introduction / Context:
This arithmetic reasoning question describes a boy who is completing a jigsaw puzzle by fitting more pieces each day than on the previous day. The situation naturally leads to an increasing sequence of numbers and tests understanding of arithmetic progressions and summations. The puzzle asks how many complete days are required if the boy starts by fitting 20 pieces and then increases his daily output by exactly one extra piece every new day until all 275 pieces are used.

Given Data / Assumptions:

• Total number of pieces in the jigsaw puzzle is 275.
• On the first day, the boy fits 20 pieces.
• Each subsequent day he fits exactly one more piece than on the previous day.
• He continues this pattern without any break until the puzzle is fully completed.
• We assume that he finishes exactly on a day when the total fitted pieces reaches 275 without exceeding it.

Concept / Approach:
The number of pieces fitted per day forms an arithmetic progression. The first term of this sequence is 20, and the common difference is 1 because he adds one extra piece each day. If he works for n days, then the total number of pieces fitted is the sum of the first n terms of this arithmetic progression. The sum of n terms of an arithmetic progression with first term a and common difference d is S = n/2 * [2a + (n - 1) * d]. We equate this total to 275 and solve for n.

Step-by-Step Solution:
Step 1: Let n be the total number of days he works on the puzzle.Step 2: The first term a is 20, and the common difference d is 1, so the daily pieces are 20, 21, 22, and so on.Step 3: Use the arithmetic progression sum formula: S = n/2 * [2a + (n - 1) * d].Step 4: Substitute S = 275, a = 20 and d = 1: 275 = n/2 * [40 + (n - 1) * 1].Step 5: Simplify inside the brackets: 275 = n/2 * (n + 39).Step 6: Multiply both sides by 2 to clear the denominator: 550 = n * (n + 39).Step 7: Rearrange to form a quadratic equation: n^2 + 39n - 550 = 0.Step 8: Solve the quadratic equation. The discriminant is 39^2 + 4 * 550 = 1521 + 2200 = 3721, and the square root of 3721 is 61.Step 9: Use the quadratic formula n = [-39 ± 61] / 2. The positive solution is n = (61 - 39) / 2 = 22 / 2 = 11.Step 10: Therefore the boy completes the puzzle in exactly 11 days.

Verification / Alternative check:
We can verify by directly summing the terms. The daily fitted pieces are 20, 21, 22, and so on up to the 11th term. The 11th term is 20 + (11 - 1) * 1 = 30. The average of the first and last terms is (20 + 30) / 2 = 25. There are 11 terms, so the total is 25 * 11 = 275, which matches the required number of pieces. This confirms that 11 days is correct.

Why Other Options Are Wrong:
The option 15 days would give far more than 275 pieces because the sequence would continue increasing for too many days. The option 13 days also overshoots the required sum. The option 16 days is even larger and clearly does not match the total pieces. None of these alternatives produce a sum of exactly 275 using the given pattern, so they are incorrect.

Common Pitfalls:
A very common mistake is to divide 275 by 20 or by some average without recognizing that the number of pieces fitted is increasing every day. Another error is to treat the situation as if the number of pieces fitted per day is constant. Some learners might also misapply the arithmetic progression formula by forgetting the factor of one half or misplacing the values of a and n. Careful algebra is required to obtain the correct value of n.

Final Answer:
The boy completes the entire 275 piece jigsaw puzzle after exactly 11 days.