Setup and Rule Diagrams

[Jiya]

Can someone please help me with the set up for Game 3 (train makes 5 trips around a loop...)?

Thanks in advance!

[Nikki Siclunov]

Hi Jiya,

Thanks for your question. According to the scenario in this game, a train makes five trips around a loop through five stations, P, Q, R, S, and T, in that order, stopping at exactly three of the stations on each trip. To help visualize this information, let's use the 5 trips as the base (1-5), with three "stacked" spaces above each trip (to represent the fact that the train stops at three of the stations on each trip:

• _ _ _ _ _
  _ _ _ _ _
  _ _ _ _ _
  1 2 3 4 5

Now, let's take a look at the two rules in this game:

The train stops at any given station on exactly three
trips, but not on three consecutive trips.

In other words, we'll need to place the following 15 variables into our diagram, ensuring that no 3 variables of the same kind appear consecutively on it:

• P P P Q Q Q R R R S S S T T T

The train stops at any given station at least once in any two consecutive trips.

Essentially, we need to see all 5 variables appear in any two adjacent columns (remember - each column represents a separate one of the five consecutive trips). If you have difficulty understanding this rule, try a few hypotheticals to help clarify the patterns that result from its interaction with the previous rule. Let's say the train stops at P, Q, and R on the first trip. To ensure compliance with the second rule, the train must stop at S and T on the second trip, along with one of P, Q and R (let's say P). For its third trip, the train cannot stop at P (or else we'd be in violation of the first rule), but it must stop at Q and R (in compliance with the second rule). And so on.

For most test-takers, a hypothetical such as this is enough to ensure that they understand how the game works, and move on. A minority will be able to determine the patterns governing the three stations where the train stops on each trip.


The two rules produce the following set of five patterns that must appear in every solution of the game. Each pattern represents the sequence of stops at which each of the five trains must stop:

Pattern #1: 1-3-5
Pattern #2: 1-3-4
Pattern #3: 1-2-4
Pattern #4: 2-3-5
Pattern #5: 2-4-5

Detecting the patterns while setting up the game would make the questions significantly easier; nevertheless, the game can still be attacked successfully without ever understanding the patterns if you identify strongly with each individual rule. Coming up with a few hypothetical solutions before you delve into the questions is key in games of this type.

Hope this helps! Let me know.

[Jiya]

Thanks, Nikki. Why aren't the following patterns included in your list?

1-2-5
2-4-5
1-3-5

That gives me 7 patterns or am I doing something wrong?

Also - 2-3-4 would violate Rule 3, wouldn't it?

[Ron Gore]

Hi Jiya!

You ask:

Why aren't the following patterns included in your list?

1-2-5
2-4-5
1-3-5

The 1-2-5 pattern won't work because it violates the second rule, since you would have two consecutive trips in which the train did not stop at a particular station (i.e., stops 3 and 4). For the same reason, you couldn't have 1-4-5 (i.e., no stops at 2 and 3).

The 2-4-5 pattern works, and it's absence was a typo (Nikki accidentally typed 2-3-4 rather than 2-4-5). I fixed this typo in the original response to avoid confusion.

The 1-3-5 pattern works, and it actually is in Nikki's list:

Pattern #1: 1-3-5
Pattern #2: 1-3-4
Pattern #3: 1-2-4
Pattern #4: 2-3-5
Pattern #5: 2-3-4

As to your question:

Also - 2-3-4 would violate Rule 3, wouldn't it?

Great catch! That was the typo in which Nikki meant to type 2-4-5.

Please let me know if this helps.

Thanks!

Ron

[garbicll]

Hi,

Im a little confused by this set up as it seems like it violates the second rule entirely.

shouldn't each two consecutive trips stop at all 5 stations as the second rule implies? I must be missing something.

thanks!

[Robert Carroll]

garbicll,

This is a tough game to figure out! I'll try to clarify what I think your confusion is. After that, I suggest re-reading Nikki's explanation thoroughly, and if things are still confusing, reply again and let us know!

Each trip of the train stops at exactly 3 stations. The second rule requires that each of the 5 stations be in any 2 consecutive trips. But those 2 trips will include 6 "spaces". The second rule is just requiring that those 6 "spaces" include every letter in them - and, because there are 5 stations but 6 spaces, something will be used twice, everything else once, in those 2 consecutive trips.

Each set of numbers in the solution is representing the trips at which a station could be used. So "1-2-4" is a possible set of trips for any station; that station would be found at least once in every pair of consecutive trips: in trips 1 and 2, it's found twice; in 2 and 3, it's found once; in 3 and 4, it's found once; and in 4 and 5, it's found once. It's also not found in three consecutive trips, so the first rule is satisfied.

If a station is used in trips 1, 2, and 4, as in the above paragraph, no other station is used in exactly that combo - otherwise, THAT station would be used twice in, for instance, trips 1 and 2; but as I pointed out, trips 1 and 2 have only 6 spaces, but need to use all 5 variables, so only one station can be used twice in that exact pair.

So the patterns of trips for a given station can't be reused for other stations. This is what Nikki is getting at with this comment:

The two rules produce the following set of five patterns that must appear in every solution of the game. Each pattern represents the sequence of stops at which each of the five trains must stop:

Pattern #1: 1-3-5
Pattern #2: 1-3-4
Pattern #3: 1-2-4
Pattern #4: 2-3-5
Pattern #5: 2-4-5

Let us know if any other questions remain!

Robert Carroll