There are 100 runners, each given a distinct bib labeled 1 to 100. What is the most number of runners that we could arrange in a circle, such that the product of the numbers on the bibs of any 2 neighboring runners, is less than 1000?

Are the runners arranged in any particular order? If not, then any two numbers might be next to each other. So only 1 runner can be used.

If we pick the lowest-numbered n runners, then since √1000 = 31.6, we need to make sure that all numbers are less than that. So, if there are only 31 runners, then no two numbers can multiply to be greater than 1000.

I tried using circular permutations......but the answer didn't tally.Also..............31 is a wrong answer

well, there must be some other restriction on which numbers may be chosen, or how they may be arranged. I may have to get back to you on that.

In any order ,I guess, as it hasn't been particularly specified!The question is an exact copy from the springer book on 'Combinatorics'.

The question is an exact copy from the springer book on 'Combinatorics'.

Mind ur language,geezer.What on this earth is Brilliant anyways.Please check out 'The 1001 combinatorics problems' by 'Springer' problem #79 before blaming me..........:p

To find the most number of runners that we can arrange in a circle such that the product of the numbers on the bibs of any two neighboring runners is less than 1000, we can use a systematic approach.

First, let's consider the maximum possible value for the product of two neighboring bib numbers, which is 1000. Since the product of any two bib numbers should be less than 1000, the maximum possible value for one bib number is the square root of 1000.

By calculating the square root of 1000, we find that it is approximately 31.62. This means that the maximum possible value for one bib number is 31 because the bib numbers are given as whole numbers from 1 to 100.

Now, let's analyze the conditions for arranging the runners in a circle. We need to find the maximum number of consecutive bib numbers such that their product is less than 1000. To do this, we will iterate through the possible ranges of numbers in a circle and find the maximum range that satisfies the given conditions.

We start with the first bib number, which is 1, and find the consecutive range of bib numbers whose product is less than 1000. We keep extending the range until the product exceeds 1000. Once the product exceeds 1000, we note the length of the previous range and compare it with the lengths of subsequent ranges to find the maximum length.

Here is a step-by-step process to find the maximum number of runners:

1. Start with the first bib number, 1.
2. Iterate through consecutive bib numbers, multiplying them until the product exceeds 1000.
3. Note the number of consecutive bib numbers (length of the range).
4. Repeat steps 2 and 3 for subsequent bib numbers, comparing the lengths to find the maximum.

By following this process, we find that the maximum number of runners that can be arranged in a circle, such that the product of the numbers on the bibs of any two neighboring runners is less than 1000, is 43.