In this arithmetic reasoning puzzle, if you write down all the numbers from 1 to 100, how many times will the digit 3 be written?

Introduction / Context:
This is a classic digit-counting problem from arithmetic reasoning, often asked in aptitude and placement exams. You are asked to write all the numbers from 1 to 100 and then count how many times the digit 3 appears. Such questions test your ability to systematically count occurrences of a specific digit in different place values without missing or double counting any cases.

Given Data / Assumptions:

• We list all integers from 1 to 100 (both inclusive).
• We must count every occurrence of the digit 3 in the units and tens places.
• Numbers like 33 contain two occurrences of the digit 3.
• We assume standard decimal representation with no leading zeros for two-digit numbers.

Concept / Approach:
The key idea is to treat the tens place and the units place separately. First, we count how many numbers between 1 and 100 have 3 in the tens place. Then, we count how many numbers between 1 and 100 have 3 in the units place. After that, we add these two counts. For the number 33, which has a 3 in both places, we naturally count each occurrence separately, so it contributes two to the total count.

Step-by-Step Solution:
Step 1: Count numbers with 3 in the tens place. These are 30, 31, 32, 33, 34, 35, 36, 37, 38, and 39.Step 2: There are 10 such numbers (from 30 to 39), and each contributes one occurrence of 3 in the tens position, giving 10 occurrences so far.Step 3: Count numbers with 3 in the units place: 3, 13, 23, 33, 43, 53, 63, 73, 83, and 93.Step 4: Again, there are 10 such numbers, each contributing one occurrence of 3 in the units position, adding 10 more occurrences.Step 5: Note that 33 has already been handled correctly because it contains two 3s, one in the tens place and one in the units place, and we counted each occurrence separately.Step 6: Total occurrences of the digit 3 in all numbers from 1 to 100 = 10 (tens place) + 10 (units place) = 20.

Verification / Alternative check:
To verify, you can quickly list the key sets again. For tens place: 30 to 39 clearly gives 10 occurrences. For units place, the pattern 3, 13, 23, and so on up to 93 also clearly includes 10 terms, generated by adding 10 each time. The systematic separation of places ensures no number is missed. Since there is no additional place value for these two-digit numbers apart from tens and units, and 100 does not contain the digit 3, the total 20 is fully reliable.

Why Other Options Are Wrong:
Option A (11): This undercounts significantly and suggests an incomplete listing of cases where 3 appears.
Option B (18): This misses two occurrences, likely due to incorrect handling of the overlap at 33.
Option D (21): This overcounts, probably by double counting some numbers or misinterpreting the range up to 100.

Common Pitfalls:
Many students either forget that 33 contains two occurrences of the digit 3 or mishandle it by counting it incorrectly. Others miss numbers in the tens or units lists due to unsystematic enumeration. A structured method that separately counts tens and units positions, followed by careful addition, is the best way to avoid these errors in digit-counting problems.

Final Answer:
The digit 3 is written a total of 20 times when listing all numbers from 1 to 100.