If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Find an Equivalent Rule

Occasionally, you will be asked to identify a condition that could be substituted for one of the existing rules, and which would result in exactly the same possible outcomes as the rule it's replacing.
Here's another way of putting it: the answer will restrict exactly as much as the rule it’s replacing, and no more.
Examples
Which one of the following, if substituted for the condition that Kyrie must be interviewed before LeBron, will have the same effect in determining the order of post-game interviews?
Which one of the following, if substituted for the constraint that the contra dance must be held in the town hall, would have the same effect on the assignment of events to venues?

### Step 1: Understand the rule

Go back and revisit the rule in the setup. Notice what the rule actually says. This might seem obvious, but it can be easy to get bogged down in inferences and forget what the actual rule is.
The correct answer must restrict the outcomes in the same way – it must meet the conditions outlined in the original rule.

### Step 2: Understand how it relates to the other rules

Look at how the rule you’re replacing connects to the others. What can you infer? For instance, if Rule 1 is that L is before M, and Rule 2 is that M is immediately before N, then we can infer that L is before N.
The answer will give you the same inferences, so it’s useful to know what they are. More importantly, however, these inferences can be a clue about what to look for in the answer. For instance, if M and N are a pair, then an easy substitution for the rule “L is before M” would be “L is before N”.

### Step 3: Rule out wrong choices

It’s possible that an answer will jump out at you – for instance, if one of the choices matches the example above. This is unlikely, though. With rule substitution questions, you’ll often be better off eliminating wrong choices.
So, go over the choices, and see what you can infer from them. If a choice doesn’t constrain the elements in the same way that the original rule does, eliminate it. Often, wrong choices will match an inference we made from the original rule, but won’t actually give us the initial constraints of the rule. This is something to watch out for.

### Step 4: Test the answer

It’s not essential, but if you have time, it can be a good idea to test the choice you think is right. Plug the new condition into your setup. Do you get the same inferences? If so, that’s the answer. Notice that you can switch Step 3 and Step 4. If you think you have a good idea of what the answer is, check it first! If it’s right, you can save yourself a lot of time. And if it’s wrong, you can eliminate an option.
There’s no trick to rule substitution questions. They are time consuming, and you just have to go through the choices. Fortunately, there are usually only a small number of them on each test – often they are the very last question. As always, the best preparation is to understand the setup well, and have a good grasp of inferences.
Let’s try an example:
A television programming director is scheduling a three-hour block of programs beginning at 1 P.M. The programs that are to fill this time block include an hour-long program called Generations and four half-hour programs: Roamin', Sundown, Terry, and Waterloo. The programs will be shown one after the other, each program shown exactly once. The schedule must meet the following constraints:
Generations starts on the hour rather than the half hour.
Terry starts on the half hour rather than the hour.
Roamin' is shown earlier than Sundown.
If Waterloo is shown earlier than Terry, it is shown immediately before Terry.
Which one of the following, if substituted for the constraint that Generations starts on the hour rather than the half hour, would have the same effect in determining the order in which the programs are shown?
(A) Generations is not shown immediately before Terry.
(B) Generations is either the first program or the fifth.
(C) Generations is neither the second program nor the fourth.
(D) If Generations is shown third, then Roamin' is shown first.
(E) If Generations is not shown first, then it is shown later than Terry.
Step 1: Understand the rule
First, what is the rule we’re replacing? It says that Generations starts on the hour rather than the half hour.
What does this tell us?
• We can think of the setup as ordering five programs over six half-hour spots. Generations takes up two consecutive spots.
• Since the programming starts at 1 p.m., the rule in question tells us that the GG spots are either the first two (1 p.m. and 1:30 p.m.), the third and fourth (2 p.m. and 2:30 p.m.), or the fifth and sixth (3 p.m. and 3:30 p.m.). In other words, it is either the first, third, or fifth program.
Step 2: Understand how it relates to other rules
Go over the other rules, and see how they relate to this one:
• Rule 2 is that Terry starts on the half-hour rather than the hour. This means that Terry could be second, fourth, or sixth (because it doesn’t start at 1 p.m., 2 p.m., or 3 p.m.).
• Generations takes up two spots, so it is in one of the half-hour spots, and Terry is therefore not in the half-hour spot immediately after Generations starts.
It doesn’t look like any of the other rules obviously relate to this one, so let’s move on to the choices.
Step 3: Rule out wrong choices
Let’s go through the choices, and see if any of them match the rule in the setup.
Start with (A): Generations is not immediately before Terry. Does this match the rule?
• We deduced from the rule in question that Generations is either at 1 and 1:30 p.m., 2 and 2:30 p.m., and 3 or 3:30 p.m.
• In order to be directly after Generations, Terry would have to start at 2 p.m. or 3 p.m.
• So, we can infer from the original rule that Terry is not immediately after Generations.
But does the rule match?
• Under the new rule, an acceptable outcome would be for Terry to show at 1:30 pm, and for Generations to show at 2:30 p.m.
• This violates the original rule of the setup!
So, (A) doesn’t match.
Now, try (B): Generations is either the first program or the fifth.
• We can see straightaway that this is too restrictive. According to the setup, Generations could also be third. This doesn’t match – rule it out.
What about (C)? This choice says that Generations is neither the second program or the fourth. We deduced from the rule that Generations is either first, third, or fifth, so this looks right!
If we replace the first rule of the setup with the condition that Generations is neither second nor fourth, what can we deduce? To test whether this rule matches the rule we’re replacing, we want to see if it gives us the same inferences as that rule, and also tells us that Generations starts on the half-hour rather than the hour.
What does the new rule tell us?
• There are five programs. So, if Generations is neither second nor fourth, it is first, third, or fifth.
• The programs take up six half-hour spots.
• Generations takes up two spots, but every other program takes up just one.
• The first program is at 1 p.m., so, if Generations is first, it starts on the hour.
• The new rule tells us that Generations isn’t second, so can’t start at 1:30 p.m.
• Since Generations is the only hour-long program, if Generations is third, the first two programs are at 1 p.m. and 1:30 p.m. So, if Generations is third, it starts at 2 p.m.
• Likewise, if Generations is fifth, then the first four programs are at 1, 1:30, 2, and 2:30 p.m. So, in this case, Generations starts at 3 p.m.
• In all of these cases, Generations starts on the hour.
This rule therefore restricts the setup in the same way as the original rule!

## Want to join the conversation?

• What would the diagram/setup look like for the example? • where is the diagram for this equivalent example? • Its genuinely awful there is no example of diagram and deductions. It almost is a pointless waste of time to try to comprehend. They don't even have a worked example of this question in an ordering setup. This platform does not have good AR information. • How can C be the answer? It says Generation can't be 2nd or 4th but it doesn't imply generation can't be 6th. Following the same reasoning we applied in option B, option C misses that Generation can't be 6th. So shouldn't it be ruled out as well?    