Consider a five-digit number such that, if the digit 1 is placed at the beginning of the number, the resulting six-digit number is exactly one third of the number formed by placing the digit 1 at the end. What is this original five-digit number?

Introduction / Context:
This puzzle involves an unknown five-digit number that behaves in an interesting way when the digit 1 is placed either at the front or at the back. When 1 is placed at the beginning, the resulting six-digit number is exactly one third of the number obtained by placing 1 at the end. The question checks your ability to translate a word puzzle into an algebraic equation, manipulate place values and solve for the unknown number.

Given Data / Assumptions:

• Let the original five-digit number be denoted by N.
• Placing 1 at the beginning yields a six-digit number: 1N (one followed by the digits of N).
• Placing 1 at the end yields another six-digit number: N1 (the digits of N followed by one).
• The number 1N is exactly one third of N1.
• The answer must be a five-digit number chosen from the options: 51428, 42857, 57142 and 42168.

Concept / Approach:
To solve this puzzle systematically, we translate the words into equations involving place values. If N is a five-digit number, then putting 1 at the beginning is equivalent to 100000 + N, because we place 1 in the hundred-thousands position. Placing 1 at the end is equivalent to 10 * N + 1, because we shift N one place to the left and append 1. The condition that the first result is one third of the second gives us a linear equation relating N and these expressions.

Step-by-Step Solution:
Step 1: Let N be the original five-digit number.Step 2: Number with 1 at the front: 100000 + N.Step 3: Number with 1 at the end: 10 * N + 1.Step 4: According to the puzzle, 100000 + N is one third of 10 * N + 1, so write the equation: 100000 + N = (10 * N + 1) / 3.Step 5: Multiply both sides by 3 to clear the denominator: 3 * (100000 + N) = 10 * N + 1.Step 6: Expand and simplify: 300000 + 3N = 10N + 1.Step 7: Rearrange terms: 300000 - 1 = 10N - 3N, giving 299999 = 7N.Step 8: Solve for N: N = 299999 / 7 = 42857.

Verification / Alternative check:
To verify, compute both constructed numbers. With N = 42857, the number with 1 in front is 142857. The number with 1 at the end is 428571. Now check the ratio: 428571 / 142857 = 3 exactly. Thus 142857 is one third of 428571, which confirms that N = 42857 satisfies the condition perfectly. None of the other options gives this exact relationship when checked in the same way.

Why Other Options Are Wrong:
If you repeat the process with 51428, 57142 or 42168, the resulting numbers 151428 and 514281 (and similarly for the others) will not be in a simple 1:3 ratio. The divisions will not be exact integers, which means those candidates do not meet the condition. Only 42857 produces a pair of six-digit numbers with a perfect factor-of-three relationship.

Common Pitfalls:
A frequent mistake is to misinterpret the phrase "three times smaller" or to set up the equation incorrectly, for example by making 10 * N + 1 equal to one third of 100000 + N instead of the other way round. Another common error is to attempt trial and error without using algebra, which is tedious and prone to mistakes. Translating the word statement carefully into a clear equation is the most reliable method.

Final Answer:
The original five-digit number that satisfies the condition is 42857.