You have a row of 100 light bulbs, all of which are initially turned off. You are asked to toggle the state of the light bulbs (turning them on if they are off, or off if they are on) in a specific sequence:

1. In the first pass, you toggle every bulb (bulbs 1, 2, 3, ..., 100).
2. In the second pass, you toggle every second bulb (bulbs 2, 4, 6, ..., 100).
3. In the third pass, you toggle every third bulb (bulbs 3, 6, 9, ..., 99).
4. This process continues until the 100th pass, where you only toggle the 100th bulb.

The question is: After completing all 100 passes, which bulbs will remain on?

Objective:
Identify which light bulbs will be on after all 100 passes.

Solution

To solve this puzzle, we need to recognize a pattern in how often each bulb is toggled and determine which bulbs end up being toggled an odd number of times.

Step-by-Step Solution The Light Bulbs in a Row Puzzle:

Step 1: Understand the Toggling Process

Each bulb is toggled once for every divisor it has. For example:

• Bulb 1 is toggled only once (divisor: 1).
• Bulb 2 is toggled twice (divisors: 1, 2).
• Bulb 3 is toggled twice (divisors: 1, 3).
• Bulb 4 is toggled three times (divisors: 1, 2, 4).

Step 2: Identify Bulbs with Odd Number of Toggles

A bulb ends up being on if it is toggled an odd number of times. A bulb is toggled an odd number of times only if it has an odd number of divisors.

Step 3: Recognize the Pattern of Perfect Squares

A number has an odd number of divisors only if it is a perfect square. This is because divisors usually come in pairs (e.g., 1 and 36, 2 and 18 for the number 36), but a perfect square has one unpaired divisor (e.g., 6 for the number 36).

Thus, the bulbs that will remain on are those whose positions correspond to perfect squares.

Step 4: List the Perfect Squares

The perfect squares between 1 and 100 are:

• 1 (1x1)
• 4 (2x2)
• 9 (3x3)
• 16 (4x4)
• 25 (5x5)
• 36 (6x6)
• 49 (7x7)
• 64 (8x8)
• 81 (9x9)
• 100 (10x10)

These positions correspond to the bulbs that will remain on.

Final Answer

After completing all 100 passes, the bulbs that will remain on are those in positions 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.