Question

suppose that 6 numbers are randomly positioned around a circle. prove that there always exists 3 consecutive numbers around the circle whose sum is at least 11. use the pigeon hole principle for this question

Answer 1

Using the pigeonhole principle, we can prove that there always exists 3 consecutive numbers around a circle whose sum is at least 11.

Step-by-step explanation:

To prove that there always exists 3 consecutive numbers around the circle whose sum is at least 11, we can use the pigeonhole principle. Since there are 6 numbers positioned around the circle, if we sum up all the possible 2-number combinations (6C2 = 15 combinations), there will be at least 8 different sums. Let's assume that none of these sums are equal to or greater than 11.

In that case, the maximum sum would be 10 (when we sum up the largest two numbers). However, the remaining numbers would have to be smaller than 1, 2, 3, or 4 in order to avoid sums equal to or greater than 11.

This is not possible, because we have 6 numbers in total. Therefore, at least one of the sums must be equal to or greater than 11. This means there always exists 3 consecutive numbers around the circle whose sum is at least 11.