@The-Blade-Dancer For the mini-question, you are right, the wording is a bit difficult to understand here. It's really asking, "Count the ways to color the top six nodes. Now count the ways to have the lower two nodes (with arrows) the same color. Divide the \(2^{\text{nd}}\) number by the \(1^{\text{st}}\) number. What fraction do you get?"

M3W3D12-y-part-1-seven-nodes-solution2-10-percent.png

Another way to state this question is, "Suppose you want the lower two nodes, with arrows pointing on them, to be the same color. What is the probability that out of all possible colorings of this graph, you get what you want?"

@spaceblastxy1428 This is interesting. 🙂 There are twice as many ways for the two second-to-bottom nodes to be different colors rather than the same color, so the ratio of ways for them to be different to ways for them to be the same color is \( 2 : 1.\) The question asks for the fraction of the total ways that have the second-to-bottom nodes the same color, so the answer is \( \boxed{ \frac{1}{3}}.\)

@divinedolphin Yes, you're right, the question is based on and follows the same specifications as the main Your Turn question. I'll make a note to clarify this in the question statement. Thanks so much for the suggestion! 🎉 🙂