One thousand people stood in a very large circle. Each starting with the Jennifer Lopez look alike, wore a sign on his or her back with a numeral from 1 to 1000 in a clockwise sequence. They began counting off. The Jennifer Lopez look alike said one in and remained in the circle. The 7'6" basketball player to her left said two out and left circle. The rap singer next in sequence said said three in and remained in the circle. The United States senator who was next in the sequence said four out and left the circle.

So it continued with each person sporting an odd numeral and with every person wearing an even numeral leaving the circle.

It was easy to visualize who remained in the circle when the count off again reached the Jennifer Lopez look alike. Since the person before her said one thousand out and left the circle, she now said one in and stayed in the circle. Continuing this sequence the rap singer said three out and left the circle. This process would keep going on and in until only one person was left in the circle.

What was the numeral of the last person who remained in the circle?

start checking after each round

after round 1, 500 people remain, numbered 2k+1
after round 2, 250 remain, numbered 4k+1
after round 3, 125 remain, numbered 8k+1
at this point, #1 still remains, but starting with the next cycle, 16k+1, overlap starts to set in
1,17,33,...,993
Now #1 has to leave, so the people left standing are of the form
17,49,81,... 32k-15 ..., 977

There are only a few cycles left, and I'm sure you can sort them out.

Is it 497?

To solve this problem, let's break it down step by step:

1. We start with 1000 people standing in a circle, numbered from 1 to 1000. Each person has a sign on their back with their respective numeral.
2. The counting off process begins, and the Jennifer Lopez look-alike starts by saying "one in" and remains in the circle.
3. Next, the person to her left, who is the 7'6" basketball player, says "two out" and leaves the circle.
4. The next person in sequence, the rap singer, says "three in" and remains in the circle.
5. The United States senator, who is next in sequence, says "four out" and leaves the circle.
6. This pattern continues, with odd numerals saying "in" and even numerals saying "out," until it reaches the Jennifer Lopez look-alike again.
7. At this point, the person before the look-alike would have said "one thousand out" and left the circle, so she now says "one in" and stays in the circle.

Now, let's observe what happens when the count off reaches the Jennifer Lopez look-alike again. The person before her said "one thousand out," which means every multiple of 1000 leaves the circle. So, numbers 1000, 2000, 3000, and so on will leave the circle.

Since there are only 1000 people initially, when the count reaches 1000, only one person remains in the circle.

Therefore, the numeral of the last person who remained in the circle is 1000.

In this problem, understanding the pattern and visualizing the counting off process helps in determining the outcome.