10 people are standing in a circle. Each person is assigned a number as shown in the figure below. The first person (numbered 1) is removed. Thereafter, in the clockwise direction, every third person is removed and this is repeated till only two people are left. What is the sum of the numbers assigned to the two people who are left at the end?
Show Hint
For elimination puzzles in a circle, it's crucial to be careful with the starting point for counting after each removal and to correctly wrap around the list when you reach the end. Drawing the circle and crossing out numbers can be a helpful visual aid.
Step 1: Understanding the Question:
This is an elimination puzzle, often a variant of the Josephus problem. We need to simulate the removal process step-by-step to find the last two survivors. Step 2: Detailed Explanation:
Let's list the people in the circle: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] Round 1: Initial Removals
- The first person (1) is removed.
Remaining: [2, 3, 4, 5, 6, 7, 8, 9, 10]
- Counting starts from the next person (2). Every third person is removed.
- Count 1: 2, Count 2: 3, Count 3: 4. Remove 4.
Remaining: [2, 3, 5, 6, 7, 8, 9, 10]
- Count 1: 5, Count 2: 6, Count 3: 7. Remove 7.
Remaining: [2, 3, 5, 6, 8, 9, 10]
- Count 1: 8, Count 2: 9, Count 3: 10. Remove 10.
Remaining: [2, 3, 5, 6, 8, 9] Round 2: Continuing the process (wrap around the circle)
- Counting continues from the person after 10, which is 2.
- Count 1: 2, Count 2: 3, Count 3: 5. Remove 5.
Remaining: [2, 3, 6, 8, 9]
- Counting continues from 6.
- Count 1: 6, Count 2: 8, Count 3: 9. Remove 9.
Remaining: [2, 3, 6, 8] Round 3: Continuing further
- Counting continues from the person after 9, which is 2.
- Count 1: 2, Count 2: 3, Count 3: 6. Remove 6.
Remaining: [2, 3, 8] Round 4: Final Removal
- Counting continues from the person after 6, which is 8.
- Count 1: 8, Count 2: 2, Count 3: 3. Remove 3.
Remaining: [2, 8]
The two people left at the end are numbered 2 and 8. Step 3: Final Answer:
The numbers of the two people left are 2 and 8. Their sum is:
\[ 2 + 8 = 10 \]
The sum of the numbers is 10.
