Consider the following necklace of diamonds (green) and rubies (red).
Suppose that two thieves steal this necklace and wish to cut it with the least number of cuts possible so that they both get equal number of diamonds and rubies, no matter the\r\narrangement of diamonds and rubies. This is the famous ‘Discrete Necklace Splitting Problem’. Now imagine every point in [0, 1] has a color ci , 1 ≤ i ≤ k, such that the set of points colored ci is measurable. Here, we wish to obtain a minimum collection of intervals in [0, 1] whose union captures precisely half the measure of each color. This problem is called ‘Continuous Necklace Splitting Problem’. In this talk I will give a solution to the Discrete problem using the ‘Discrete Intermediate Value Theorem (DIVP)’, and a solution to the Continuous problem using the ‘Borsuk-Ulam Theorem’.
