Which one of the following could be a complete and accurate list of the days on which the batches of each kind of coo...

Irina on October 18, 2019

@Lauren-Au,

Thank you for your comment - I will forward it to our curriculum team. The instructors cannot record a video explanation, but I will provide a written setup in the meantime.

A bakery makes three kinds of cookies - O P S. Exactly three batches of each kind of cookie are made each week and each batch is made in a single day. This tells us that we have five days to make 9 batches overall - 3 x 3 types of cookies, so we must make more than one batch on at least one day. Notice that the game has no minimum number of batches that have to be made each day so it is possible to have a day with 0 batches as well.

There are two ways we could set up this game. We could use days of the week as a base:

M T W Th F

or we could use types of cookies as a base:

O:
P:
S:

I am going to use days of the week as a base because I find it more visually intuitive, but there is no single right way and if types of cookies works better for you, feel free to use that instead:

The following rules apply:

(1) No two batches of the same kind of cookie are made on the same day.

This rule tells us that three batches of each type of cookies must be spread out over exactly three days. So let's say we could bake O on M T W or W Th F but we could not bake two batches of O on the same day.

(2) At least one batch of cookies is made on M.

This rule tells us that we must have at least one batch on M, so it is not a day with 0 batches if we have any.

M=1+

(3) The second batch of O cookies is made on the same day as the first batch of P cookies.

This rule allows us to infer that the first batch of P cookies is not on M because the earliest the second batch of O cookies could be made is T.

1+
M T W Th F
~P

O(2) =P(1)

(4) The second batch of S cookies is made on Th.

This rule allows us to infer that the third batch of S cookies must be made on F, since we cannot have the same type of cookies on the same day per rule (1).

S S
M T W Th F

Putting all the rules together:

1+ S S
M T W Th F
~P

If you used cookies as a base, you would end up with a diagram similar to this:

O: __ __ __
P: ~M
S:__ Th F

M=1+
O(2)=P(1)

This is a fairly open-ended game, so instead of trying any specific scenarios, let's go straight to the questions.

The question asks which of the following is a complete and accurate list of the days on which the batches of each cookie is made.

(A) is correct and complies with all the rules.
(B) is incorrect because S cookies are baked on Th and F
(C) is incorrect because at least one batch must be baked on M
(D) is incorrect because P first batch must be on the same day as O second batch
(E) is incorrect because P first batch must be on the same day as O second batch

The second question asks us on how many days at most two batches of cookies could be made. Let's think about it for a moment.

Could we make three batches on M?

No, because the first batch of P cannot be on M, so at most we could make O and S on M.

Could we make three batches on T?

Yes, we could if we make P-1st batch, O- 2d, and S 1st batch on T:

O
P P P
O S O S S
M T W Th F

Could we make three batches on W?

Yes, in exactly the same scenario as on T, we could apply it to W.

S O
P P P
O O S S
M T W Th F

Could we make three batches on Th?

Yes, we already know that S 2d batch is on Th, and we could have O 3d batch and S 2d batch on Th as well:
O
P P P
O S O S S
M T W Th F

Could we make three batches on F?

Yes, we could have the third batch of each cookie on F.

O
P P P
O S O S S
M T W Th F

So M is the only day where we could not make more than two batches. We could also see it from our initial diagram that M is the only day that is excluded for P cookies.

The third question asks us if the first batch of P is made on T, each of the following could be true except:
We right away can tell that if P is made on T then O first batch must be on M, and second batch on T:

O: M T
P: T
S: Th F

or represented by days of the week:

P
O O S S
M T W Th F

We can see that the only scenario that is impossible is two cookies having their second batch made on W because we know that S second batch is on Th and O 2d batch is on T under this scenario, thus at most Ps second batch could be on W (option C).

The fourth question asks us if no bookies are made on W, then which of the following must be true?

This question looks similar to our scenario in question 3 - and it is very common for logic game questions to build on each other. If no batch is cooked on W, then the first batch of P must be on M, and the second batch of P and the first batch of O on T because if we had the second batch of O/ first batch of P on Th, we would have no space left for 2d and 3d batch of P. The first batch of S must also go on M or T and could be on either day. The second and third batches of P must go on Th/ F respectively. And the third batch of O could go either on Th or F

/S /O /O
/S P P P
O O S S
M T W Th F


Looking at this setup, we can tell that at least two batches of cookies must be made on Th - P & S (option D).

The fifth question asks us if the number of batches on F is exactly one, which of the following could be true?

Well, we already know that the batch on F is S from our initial setup. And we also know that the earliest we could bake the first batch of P is on T, so the second batch of P must be on W, and the third on Th since we only have S on F. We can also infer that the first batch of O must go on M and the second batch of O must go on T.

O P
O P P S S
M T W Th F

The only free variables left are the third batch of O that could go on W or Th, and the first batch of S that could go on M/T/W. Thus, we can conclude that the only possible scenario among the answer choices is the first batch of S is baked on M (option A)

Question six asks us if one kind of cookie's first batch is made on the same day as another kind of cookie's third batch, which of the following could be false?

Well, which pair of cookies could it be?
It cannot be P and O because their relationship is 1st batch P =2 second batch O.
It cannot be P and S because the earliest first batch of P could be is on T, which puts the earliest third batch of P on Th, and S 2d batch not 1st must be on Th.
It could be O and S because it is possible for O first batch to be on M and 3d batch on W, and it is possible for S first batch to be on M/T/W, so on W as well.

/P
O O /P /P
O P S S S
M T W Th F

The only option that could be false in this scenario is (E) exactly one batch of cookies is made on F because it is possible for the third batch of P to be on F

Let me know if you have any further questions.

Lauren-Au on October 18, 2019

@Irina Amazing Thank you so much this helps a ton!