Hi K,
Thanks for the question! These Maximum questions can be tricky because they are based on you manipulating the variables to achieve a certain outcome. When initially approaching these questions, I like to look at two sets of variables: randoms, and what I call power variables. Let's discuss both:
Randoms
Randoms should make some sense here. If you are trying to form the largest possible group, then you'd want to include variables that have no effect on other variables. They can be quickly added because they will never cause any issues in making the group as large as possible.
Power Variables
Power variables are the ones that have the greatest effect--positive or negative--on other variables. For example, let's say you have a rule such as "If F is selected, then neither J nor K can be selected." In this rule, F is pretty powerful; its mere presence immediately knocks out two other variables. In this sense, it's a negative power variable, and clearly one you'd want to avoid when trying to form the largest group. That leads to the simple (and obvious) rule that you should typically avoid negative power variables because they do more harm than good when selected.
But, there are other variables that also fall under the power variable classification, in a more positive way. Consider the following rule: "If F is selected, then G is selected, and if G is selected, then H is selected." Under this particular scenario, we get the following chain: F
G
H. The first thought might be that we have to then include F in the group, but that's not what I would conclude from this rule. Instead, I'd look at H. H is a variable that better be in my group because without H, then via the contrapositive, I'll lose F
and G. Because those two variables depend on H, H is a powerful positive variable. If there were another rule that negatively affected H, I'd need to look at that carefully because my initial analysis of H tells me I probably want to protect it and keep it in the group.
Just as a fun (!) exercise, let's combine the two rules I've been using into a mini-game, add a new rule, and then use some of the ideas discussed above:
- Seven Variables: D F G H J K L
Rules: If F is selected, then neither J nor K can be selected.
If F is selected, then G is selected, and if G is selected, then H is selected.
H and L cannot be selected together.
Now, try to form a group of maximum size.
Starting off, D is a random, so D is immediately in the max group:
That was the easy part
Let's now move on to analyzing the power variables.
You can see that F is pretty powerful because it figures in two rules, and the same is true for H. Now I have to decide who to include and who should be knocked out:
- I'm going to start with H, because two variables rely on H. If H is in the group, then F and G could be in the group, but L cannot be in. If I was counting just those four variables (and not counting D), that's 3 in (F, G, H) and one out (L), at least so far.
If H is out of the group, then F and G are also out, but L can be in. Counting the four, that is one in (L) and three out (F, G, H). So, having H is a net positive, and since no other variable is connected to H, I'm going to put it in the group, which then punches L out. And, I can add G in since no other variables negatively affect it. Considering all variables discussed so far,w e have the following situation:
Now on to F, another power variable. If you don't select F, you can then select J and K, meaning you are plus 1 if F is out (F, J, K). On the other hand, if F is selected, you knock out J and K, which is a loss of 1 overall (F, J, K). So, I'm going to toss out F, which then allows J and K to get in the max group:
Thus, I can get a group of 5 variables as the maximum, and 5 would be the correct answer.
That's a small example, but hopefully it gives you an idea of how you could start to systematically break down what you are seeing. Viewed from one angle, you are looking at certain variables, and the sub-groups of variables they affect, and then making decisions to maximize the group size.
Please let me know if that helps. Thanks!
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