Josephus Problem 100 People Explained

The Josephus Problem 100 people puzzle is one of the most famous logic and math challenges in the world. In this classic elimination puzzle, 100 people stand in a circle and every second person is removed until only one survivor remains. The question is simple but surprising: which position survives?

In this article, we will fully explain the Josephus Problem using a clear example where 100 people stand in a circle, and every second person is eliminated until only one person remains. By the end, you will not only know the final answer, but also understand why that position survives.

---

Josephus Problem 100 People Explained

Here is the exact problem statement:

100 people stand in a circle. Starting from person #1, every second person is eliminated. The process continues around the circle until only one person remains. Which position survives?

This type of elimination puzzle is a classic example of circular reasoning combined with step-by-step logic.

Understanding the Elimination Process

Let’s break it down in simple terms:

• There are 100 people, numbered from 1 to 100.
• They are standing in a circle, not a straight line.
• Starting with person #1, we eliminate every second person.
• Once the end of the circle is reached, counting continues from the beginning.
• This process repeats until only one person is left standing.

At first, it may seem like the only way to solve this puzzle is by manually eliminating people one by one. While that method works for small numbers, it becomes impractical for large groups like 100 people.

This is where mathematics helps.

What Is the Josephus Problem?

The Josephus Problem is named after Flavius Josephus, a Jewish historian. According to legend, Josephus and his companions were trapped and decided to form a circle and eliminate every second person rather than surrender. Josephus used mathematics to position himself so that he would be the last survivor.

Today, this problem is widely studied in:

• Mathematics
• Computer science
• Competitive exams
• Logical reasoning tests

---

Solution

Josephus Problem 100 People Step-by-Step Solution

To find the survivor in the Josephus Problem with 100 people, where every second person is eliminated, we can use a simple and reliable mathematical method.

🔹 Step 1: Identify the Largest Power of 2

First, find the largest power of 2 that is less than or equal to 100.
2⁶ = 64
2⁷ = 128 (greater than 100)
So, the largest power of 2 is 64.

🔹 Step 2: Subtract the Power of 2

Now subtract this value from the total number of people: 100−64=36

🔹 Step 3: Apply the Josephus Formula

For the Josephus Problem where every second person is eliminated, the survivor’s position is given by:

Survivor = (2 × remainder) + 1

Substitute the value:

2 × 36 + 1 = 73

Final Answer

The person standing at position 73 survives.

This method works because the Josephus Problem follows a pattern based on powers of two when every second person is eliminated.

---

Why This Method Works

The reason this shortcut works is because eliminating every second person repeatedly creates a pattern based on binary numbers and powers of two. Each full round of eliminations effectively halves the group while preserving the relative order of certain positions.

When the total number of people is exactly a power of two, the survivor is always position 1. Any number beyond that power shifts the survivor forward in a predictable way.

Can You Solve It Without Math?

Technically, yes—but it would be extremely time-consuming. You would need to:

• Eliminate every even-numbered position in the first round
• Renumber the remaining people
• Repeat the process again and again

For 100 people, this approach is inefficient. The mathematical method gives the correct answer instantly.

Formula Used in the Josephus Problem 100 People

If:

• n = total number of people
• k = 2 (every second person eliminated)

Then:

1. Find the largest power of 2 ≤ n
2. Subtract it from n
3. Multiply the remainder by 2 and add 1

This rule works every time.

Why the Josephus Problem Is So Popular

The Josephus Problem is loved because it:

• Looks simple but requires deep thinking
• Combines logic with mathematics
• Appears in coding interviews and exams
• Challenges assumptions about counting and order

It is a perfect example of how a small rule can create a complex and fascinating result.

Final Thoughts

The Josephus Problem with 100 people standing in a circle is a brilliant logic puzzle that rewards pattern recognition and mathematical insight. While it may seem confusing at first, the solution becomes elegant once you understand the role of powers of two.

So next time someone asks this question, you’ll confidently know the answer:

Position 73 survives. 🏆

If you enjoy puzzles like this, try solving the Josephus Problem with different numbers of people—you’ll start seeing the pattern everywhere!

---