Logic Challenge #29 - The Reindeer Game

[noah]

Post your explanations here! The best one wins a free book or a gift certificate. Woo-hoo!

[Kurst]

Since this board's formatting is somewhat limited, my explanation for the reindeer game is in this PDF:

reindeer explanation and game.pdf

(48.97 KiB) Downloaded 828 times

[lsatstudy]

From Kathy Z:

Please see my attached explanation for better formatting!

MANHATTAN LSAT LOGIC GAME 29 short version.pdf

(37.4 KiB) Downloaded 357 times

The first rule involving Rudolph and Vixen gives an unmanageable number of possibilities, since there are no other rules involving Rudolph and Vixen which would help us restrict the choices. Note that 4 of the 5 rules mention Sam. Focus on him and quickly discard non-solutions.

Pick 'Sam' rules and mark options on a 3-row, 2-column chart using the initial of the deer. Rule 5 says Tony is one row behind Sam and on the same side. Four possibilities are labeled below with A to D. Rule 4 says that Vixen and Sam can't be on the same row, so the '~V' indicates 'not V.' Slashes separate left/right. The charts use Rules 4 and 5:
A B C D

S/~V .../... ~V/S .../...
T/... S/~V .../T ~V/S
.../... T/... .../... .../T

If an IF-THEN statement is correct, then its contrapositive is correct. Form it by switching the order of the If and Then parts and negating them. (If the If or Then part is already negative, then make it positive to negate it.)

Rule 3 says "If Yeller is not in the 3rd row, (then) Sam cannot be in the 2nd one." The contrapositive is: "If Sam is in the 2nd row, then Yeller is in the 3rd row." Placing Sam in the 2nd row does not have to be considered if Yeller is in the 1st or 2nd row, and also placing Yeller in the 1st or 2nd rows does not have to be considered if Sam is in the 2nd row.

Rule 2 says "If Sam is in the 1st row, Yeller must be in the 2nd." The contrapositive is "If Yeller is not in the 2nd row, then Sam is not in the 1st row."

Mark these facts on the charts from above. Where ambiguities exist, create additional charts to keep options open. Avoid writing out dead end choices, such as putting Yeller in the top row on chart A (Rule 2 requires Yeller to be in the 2nd row). The charts with Rules 2 through 5 are:
A B C D

S/~V .../... ~V/S .../...
T/Y S/~V Y/T ~V/S
.../... T/Y .../... Y/T

Rule 1 says if Rudolph is in the 2nd row, Vixen must be on the left, meaning Vixen is in the left column somewhere, regardless of the row.

For each of our 4 charts, there are 3 spaces remaining to be decided. Place one of the reindeer (I chose Rudoph) in each of the 3 spaces and see how that choice effects the placement of the others (Vixen and Prancer). The three potential positions of Rudolph in the uncompleted A charts are:
A-1 A-2 A-3

S/R~V S/~V S/~V
T/Y T/Y T/Y
.../... R/... .../R

For A-1, the bottom 2 positions could be V or P in either order, V-P or P-V, giving charts A-1-a and A-1-b. For A-2 and A-3, the top right must be P since it cannot be V. The final A chart results are:
A-1-a A-1-b A-2 A-3

S/R~V S/R~V S/P~V S/P~V
T/Y T/Y T/Y T/Y
V/P P/V R/V V/R

We have all options for chart A. Use the same process (pick a space for Rudolph and see where Vixen and Prancer could be) on the B, C, and D charts. (Rule 1 eliminates the Prancer-Vixen order in the 1st row when Rudolph is in the 2nd row in charts B-3 and D-3, requiring that the 1st row be Vixen-Prancer instead.) The charts are:
B-1 B-2 B-3

R/V V/R V/P
S/P~V S/P~V S/R~V
T/Y T/Y T/Y

C-1-a C-1-b C-2 C-3

R~V/S R~V/S R~V/S R~V/S
Y/T Y/T Y/T Y/T
V/P P/V R/V V/R

D-1 D-2 D-3

R/V V/R V/P
P~V/S P~V/S R~V/S
Y/T Y/T Y/T

Question 1, choice A gives R/P, S/Y, T/V as the rows. Scanning the 1st row of our charts, we find no R/P case, so discard choice A. (Skip an answer choice as soon as a discrepancy is found.) Question 1, choice B has V/R, S/P, T/Y, which we find in chart B-2, so choice B is correct. Skip C to E.

Question 2 asks us to scan the right. The R, S, T order exists in chart D-2, but none of the other choices so.

Question 3's answer is A, since P occurs in the third row in both some A and C charts.

For Question 4, scan for V and R in any order in the right column. Chart A-1-b has both letters in the right column. 'Adjacent' (or 'next to') means in the same row, one column above or below, or diagonally touching. In the chart, P is not adjacent to S nor R, choice A.

Question 5 is an exception question which is negatively stated. Four answers will be true and one false. Choose the false answer. The words 'could be' mean it does not have to be always true, but it is possible. So, we only need to find one of our charts with this arrangement to make that choice true and eliminate it as the right answer. 'In front of' means in the row before (above) the other reindeer. Chart B-2 (and D-1) has V in the row before P, so eliminate choice A as true since we are looking for a false answer. You will find that choice E works with our charts.

For Question 6, first identify all cases where V is on the left top corner, giving charts B-2, B-3, D-2, and D-3. R can be in the right top position or in either of the row 2 positions, give 3 options, or choice C.

Question 7 asks about determining positions. The right choice will give a unique solution. Process by finding how many charts have the required positions. If more than one, then that fact does not 'determine' uniquely and it is not the correct answer. Chart D-1 has V on the right side of the first row, directly ahead of S, and no other chart has that, so A is the correct choice.

Question 8 states the 4th rule will be substituted, or replaced, by the answer choice rules, so it should be removed. What other cases would arise?

The A charts only allowed R or P to be paired with S, so we consider S being paired with V (removing the Sam and Vixen rule) and see which additional chart possibilities would be created, and do likewise with B, C, and D charts. The additional charts are below. Notice that the -a and -b versions different only in the positions of R and P.

A-4-a B-4-a C-4-a D-4-a
S/V R/P V/S R/P
T/Y S/V Y/T V/S
R/P T/Y R/P Y/T

A-4-a B-4-a C-4-a D-4-a
S/V P/R V/S P/R
T/Y S/V Y/T V/S
P/R T/Y P/R Y/T

The question asks which of the rules in the answer choices would have the same effect (eliminate the new charts without eliminating any original ones). Apply the rules from the answers to our charts to see whether they are all disqualified when the new rule is substituted in.

Consider choice A. The eight new charts all have T one row (not two) behind V, so they are eliminated. However, that is not sufficient to make choice A the correct answer. We then must examine the original charts to see whether they follow the requirements of T 2 rows behind V or V 1 row behind T. They all do.

Is choice A correct? It would be, if it weren't for the word 'except.' With exception questions, you are looking for 4 right answers and one wrong answer, and select the 'wrong' one. Since choice A has the same effect, it does not satisfy the 'except' requirement, so it is eliminated.

For choice B, new charts would be eliminated since they all have V paired with S on the same row. The 2nd requirement is that they not eliminate any old charts. All old charts have V 2 rows behind S or 1 row ahead, so no old charts fail. Therefore, choice B has the same effect and does not satisfy the exception, so do not choose choice B.

Choice C says P and R can't be in the same row, a condition in all new charts, so they are eliminated. None of the old charts have P and R in the same row, so they will not be eliminated. therefore, choice C has the same effect and is eliminated.

Choice D: In the new charts, Y is not 1 row ahead of V, but T is always 1 row behind V. If either of the clauses of an OR statement is satisfied, then the statement is true. So the new charts all satisfy choice D. If the new charts failed to satisfy choice D and the old charts did, then this choice would be eliminated. Since that is not the case, this must be the correct answer. It is not necessary to consider choice E.

[lsatstudy]

On my (Kathy Z's) response, note that the last chart in the Q 8 answer should have 'b' instead of 'a' on the chart names, i.e., A-1-b, etc.

Sorry for the difficulty reading the charts. Extra blanks are squeezed out by the system, making them difficult to read.

[Tedlwalsh]

Ok so for this one I found there to be only 10 possible scenarios for the game in total. This is largely thanks to the last two rules.

V<--/-->S

S_ _S
T_ or _T

This means that S has to be in either row 1 or 2 and that T will be below it respectively. Also S can only be matched with R or P

The other conditional rules give us the possible placements for Y and V

R2--->V_

S1--->Y2
-Y3--->-S2 or S2--->Y3

The possibilities are as follows:
(s1)
1 S R/P
2 T Y
3 V\P/R P/R\V

(s2)
1 R/P S
2 Y T
3 V\P/R P/R\V

(s3)
1 R/V V/R
2 S P
3 T Y

(s4)
1 R/V V/R
2 P S
3 Y T

(s5)
1 V P
2 S R
3 T Y

(s6)
1 V P
2 R S
3 Y T

-the first 4 scenarios I have drawn out represent 8 of the 10
(Note that if S is paired with R than V Must be in on the left)

From this it is easy and quick to determine all of the answers...

1.B
2.D
3.A
4.C
5.E
6.C
7.A
8.D

[noah]

These were 3 very strong explanations. You all ended up doing different versions of framing out, with Ted "exhausting the possibilities." That's probably further than most folks would want to go with this game, but, hey, whatever works!

When I solved it, I used the attach diagram:

reindeer-diagram.pdf

(269.23 KiB) Downloaded 392 times

It's a bit trimmer than what y'all used. There's pros and cons to that. The advantage is the flexibility and speed, the disadvantage is that there's more work to be done at each question.

Great work, all of you. I can't really say who's got the best of these as they're all so viable. I guess, for clarity's sake, I'll go with i.kurst. But, we'll send y'all a prize!

Nicely done. - Noah

[tommyid1]

Regarding the last question, I see a problem with choice E.

If either P or R only need to be adjacent to S, then we could have a scenario where V is next to S (which isn't allowed by the original rules). Let's take a look at I.Kurst's 4th frame. In that frame, we could have the following:

RP
VS
YT

Thus, shouldn't the answer choice instead read, "Sam must be in the same row as either Prancer or Rudolph?"

[Kurst]

The game defines adjacent as "in the same row as." Recall the second sentence of the game's setup:

Each row has exactly two positions _ one on the left side and one on the right _ which are adjacent to one another.

The game does not define adjacent in any other way; therefore two reindeer who occupy different rows are not understood to be adjacent.

[noah]

Regarding the last question, I see a problem with choice E.

If either P or R only need to be adjacent to S, then we could have a scenario where V is next to S (which isn't allowed by the original rules). Let's take a look at I.Kurst's 4th frame. In that frame, we could have the following:

RP
VS
YT

Thus, shouldn't the answer choice instead read, "Sam must be in the same row as either Prancer or Rudolph?"

That's such an awesomely close read! That's great.

I'll stand behind i.kurst's response, but perhaps we should have explicitly stated the definition of "adjacent." Alas, our geekdom is in doubt!