Note that no sequence of 3-IETs is optimally mixed for consecutive steps except for the trivial transition from 𝑁 = 0 to 𝑁 = 1 in the case of a single-step time-horizon. Arriving at an optimally mixed state for 𝑁 = 2 steps requires a priori planning over a time-horizon of two steps with the intent to cut the two segments in the initial state (𝑁 = 0) into two-thirds and one-third portions using the first 3-IET (𝑁 = 1), and then perform a second 3-IET to split the remaining longest segments in half (𝑁 = 2), as shown in Fig. . Continuing that pattern would create optimally mixed configurations for 𝑁 = 3 𝑘 -1, less often than the LS method. In contrast, the LS method in Fig. (b), which has a single-step time-horizon, first splits the colors in half (𝑁 = 1) and then can only split one of these halves leaving the other untouched. Although the LS method in Fig. each color at 𝑁 = 2, they are unequal in length. The LS method, with its single-step time-horizon, is always applied to the leftmost longest segments in the domain, and it is the process that is most often optimal. For example, although there is no 3-IET to go between 𝑁 = 1 and 𝑁 = 2 optimally mixed configurations, there is a sequence of two 3-IETs that can go from optimally mixed at 𝑁 = 1 to optimally mixed at 𝑁 = 3, which is precisely the LS method in Fig.