Which one of the following could be an acceptable schedule for the team from June through October?

Irina on September 16, 2019

@jewan,

This is a fairly challenging game, and the setup here is the most important part.
The game involves assigning an engineering team to work for either Grayson mine (G), Krona mine (K) or company headquarters (H) over a period of 9 months March - November with 3 months spent at each of the mines, and 3 months at the headquarters.

___ ___ ___ ___ ___ ___ ___ ___ ___
M A M J J A S O N

The following rules apply:

(1) The team must work at least one month at headquarters between any two months working at different mines.

This rule tells us that any time a team switches mines, there must be at least a month or possibly more that when they work at headquarters, so the pattern could be:

G H K or G HH K or K H G

but never KG or GK because at least one H must always come in between.

(2) The team cannot work at the same mine for more than two months in a row.

This rule tells us that at most we can have two repeating letters in a row for G and K, we could have GG or KK but not GGG or KKK. Combined with rule (1), we can also infer that we can never have HHH or HH because then it would be impossible to have a month in headquarters between the months of work in either of the mines, so at most we could have one month at headquarters in a row.

The rule also allows us to infer that we must have at least two months of a row for each mine, otherwise, we will not be able to comply with rule (1). To illustrate why, consider this scenario:

G H K H G H K ? ?

We have two months left and still have 1 month left to assign to mine K and 1 month to mine G, let's say month 8 is mine K

G H K H G H K K ?

but then month 9 cannot be G because of rule (1), so the only plausible scenario is where we have one each of GG and KK sequences, for example:

G G H K K H G H K

Another interesting inference is that even though we cannot have GGG or KKK sequence, the rules only tell us that H is required every time the team switches mine but it leaves open the possibility that H could split 3 months at the same mine as well, for example:

GG H G KK H K H

Let's keep this in mind for later.

(3) The team must work at Grayson mine in August.
(4) The team must work at Krona mine in October.

___ ___ ___ ___ ___ G ___ K ___
M A M J J A S O N

These two rules tell us that since the team switches mines between August and October, they must work in headquarters in September per rule (1). We can also infer that they must work in mine K in November, otherwise, we would not be able to comply with rules (1) and (2) because there are only 2 months left to assign to H and 4 months to both mines.


___ ___ ___ ___ ___ G H K K
M A M J J A S O N


The rest of the months need to be filled with G G H H and K in some order, and we can infer that H cannot be #1 because if H is #1 there is no valid combination possible as it would necessarily violate either rule (1) or rule (2):

H K H GGG H KK (rule 2)
H G H K G... (rule 1)

So the only possible combinations are:

(1) G G H K H G H K K
M A M J J A S O N

(2) K H G G H G H K K
M A M J J A S O N

(3) G H K H G G H K K
M A M J J A S O N

(4) K H G H G G H K K
M A M J J A S O N

Notice that scenario (4) takes advantage of our inference in rule (2) that H can be used to split G H GG pattern.

For this game it is important to spend time on the setup because the rules allow us to determine every possible scenario - there are only four. If you have made all the above inferences, you would be able to answer all the questions just by looking at these four scenarios without spending any additional time.

This question asks us which of the following could be an acceptable scheduled for the team from June to October?

(A) G G G H K

Incorrect, this violates rule (2)

(B) G H G H K

Correct, this is the scenario (2) above.

(C) G H K H K

Incorrect, violates rule (3), G must go in August.

(D) H G G H G

Incorrect, violates rule (4), K must go in October.

(E) K K G H K

Incorrect, per our inference above K K combination must go in October/ November.

Let me know if this makes sense and if you have any further questions.

bb427 on September 29, 2019

Thank you. I found your characterization of this game as "fairly challenging" generous. How much time would you estimate it took to set this up first time though? Suggestions as to how to prepare for this type of game?

Ecochran on October 12, 2020

@Irina you state in your explanation that: Another interesting inference is that even though we cannot have GGG or KKK sequence, the rules only tell us that H is required every time the team switches mine but it leaves open the possibility that H could split 3 months at the same mine as well, for example:

GG H G KK H K H

But the first rule states that we must have an H between any switch between mines, so your example of GG H G KK H K H does not follow, since you put G next to K there in the middle.

Katherine on March 13, 2021

Also, why can't H go first? Example:

HGGHKGHKK

That doesn't violate any rules...G is in August, K is in October, no mine switches without H after, and no more than two consecutive months in the same mine.

Am I missing something?

Katherine on March 13, 2021

Never mind...I see what I was missing. I retract the previous question...