Written by: Alexander JL Theoharis

Six friends are out flying their dirigibles on a Sunday afternoon. On a lark, they decide to race back home (the winner gets ownership of the prized group dirigible, the Hindenburg). The friends arrive according to the following rules:
Nigel, due to the weight of his dirigible, arrives second-to-last.
Malcom arrives directly before Jasper.
Kopernicus and Ollie never arrive consecutively.
Liam never arrives after Ollie.

1. If Ollie arrives 4th, who arrives last?
a. Nigel
b. Kopernicus
c. Malcom
d. Liam

2. If Liam arrives fourth, who arrives 2nd?
a. Malcom
b. Jasper
c. Jasper or Malcom
d. Ollie

3. Assume that Nigel lightens his load by throwing his monocle collection overboard, eliminating the first rule. Who cannot arrive last?
a. Kopernicus
b. Ollie and Kopernicus
c. Jasper
d. Liam and Malcom

4. If Ollie arrives second, which of the following cannot be true:
a. Liam arrives first
b. Kopernicus arrives fourth
c. Kopernicus arrives after Liam
d. Nigel arrives after Ollie

Answers after the break.

Answer Key

1. b. Kopernicus

2. c. Japser or Malcom

3. d. Liam and Malcom

4. b. Kopernicus arrives fourth.

Written by: Jay D. Hall

There are four people: Al, Bart, Charlie and Don. They coordinate wearing hats from Sunday through Saturday one week using the following rules.

1. Charlie wears a hat three times a week.
2. Al must wear a hat on Tuesday.
3. Bart and Charlie wear hats on the same day
4. Al and Charlie never wear a hat on the same day.
5. Charlie never wears a hat two days in a row.
6. When Don wears a hat, no one else does on that day.
7. Al always wears a hat two days in a row.

1. If Bart wears a hat on Friday, who wears a hat on Monday?

a. Al

b. Bart

c. Charlie

d. Don

2. If Charlie does not wear a hat on Sunday which of the following must be true?

a. Don wears a hat on Friday.

b. Al wears a hat on Monday.

c. Bart wears a Hat on Thursday.

d. Charlie wears a hat on Wednesday.

3. If Don wears a hat on Saturday only which of the following is true?

a. Bart may wear a hat on Thursday.

b. Al may wear a hat on Wednesday.

c. Charlie may wear a hat on Monday.

d. No one may wear a hat on Thursday.

4. If rule number six is removed:

a. Don may wear a hat 4 times a week.

b. Don may wear a hat 5 times a week.

c. Don may wear a hat 6 times a week.

d. Don may wear a hat 7 times a week.

5. If rule number four is removed:

a. Bart may wear a hat 1 time a week.

b. Bart may wear a hat 3 times a week.

c. Bart may wear a hat 5 times a week.

d. Bart may wear a hat 7 times a week.

Answers after the break.

Answer Key

The first step in solving these problems is setting up a calendar starting on Sunday and ending on Saturday. The calendar should have four rows below it to accommodate each of the people. This makes it easier to understand who can wear a hat and when; most logic games lend themselves to special representation. The trick is learning how to decide to lay it out.

Building the rules into this model should be relatively simple. There are two rules that can be combined and one rule that shows us where to start.

1. Charlie wears a hat three times a week.
2. Al must wear a hat on Tuesday.
3. Bart and Charlie wear hats on the same day
4. Al and Charlie never wear a hat on the same day.
5. Charlie never wears a hat two days in a row.
6. When Don wears a hat, no one else does on that day.
7. Al always wears a hat two days in a row.

We know Al must wear a hat on Tuesday, and two days in a row, so either Monday or Wednesday. We know Bart and Charlie always wear a hat together; this rule has a transitive property to the rule that Al and Charlie never wear hats on the same day. The transitive property just means that if Bart and Charlie always wear hats together, and Al will not wear a hat with Charlie, then Al will also never wear a hat with Bart. Shorthand expression for that rule is: (BC) ≠A

Answer 1: a

If we place B on Friday, we know C also goes there, we know the two of them must both also wear a hat on Wednesday and Sunday to be in conformity with rules 1 and 5. This means A must wear a hat on Monday because he may not wear a hat on Wednesday.

Answer 2: c.

If C does not wear a hat on Sunday it should be clear that he must on Saturday, therefore he must on Monday and Thursday along with B. Consequently A must wear his on Wednesday. Again, this leaves two days open, either of which D may wear a hat, Friday or Sunday. This table is the second of only two variations that these three are able to coordinate their hat wearing. This is as far as we can take the table so lets examine the answers. (a) is incorrect because it is phrased as if D must wear his hat on Friday when we know he may wear it on Friday, Sunday, or both. (b) is incorrect because Al cannot wear his hat on Monday without violating rule number 4. (d) is incorrect because Charlie cannot wear a hat on Wednesday without violating rules: 1,4,5, or 7. (c) is the correct choice because, Bart must wear a hat on the same day as Charlie who must wear a hat three times a week, one of which falls on Thursday because he cannot wear a hat two days in a row.

Answer 3: d.

We know by looking at the call of this question that we can use the same table we created for solving question 1 because it is the only scenario for which D may wear a hat on Saturday. Using that table we can shortcut to an answer. (a) is not true because Bart is not wearing a hat on Thursday, again B and C are linked, they follow the same rules, we know they must both wear a hat three times a week, not on concurrent days. This week, the days they wear their hat is Sunday, Wednesday and Friday. (b) is incorrect because Al must wear his hat opposite that of B and C yet next to Tuesday, since B and C have Wednesday this week A must have Monday. (c) is incorrect for the same reason (a) and (b) are incorrect. (d) is correct in this instance not because of any rule but because of the conditional form of the question. The only person able to wear a hat on Thursday given this setup would be D, however the question limits D to wearing a hat on Saturday alone, therefore (d) is the only true answer.

4. d.

Don is unrestricted, nothing anyone else does affects his choice to wear a hat, therefore he may wear a hat seven times a week.

5. b.

Bart is still constrained by the rule that he and Charlie wear hats on the same day. So too is he constrained by the rule that Charlie never wears a hat two days in a row. Therefore neither Bart, nor Charlie may wear a hat more than three times in any given week.

Written by: Alexander JL Theoharis

Derek, Eli, Forest, Gary, Henry, Ivan, and Jesse are knights sitting around a square table. Each side of the square holds one person, and each corner holds one person. There is one empty seat, since the King recently passed away.

1. Forest and Eli are not seated on a corner.
2. Forest and Jesse are seated next to Ivan.
3. Jesse is seated to the left of the empty seat, as he was the King’s left-hand man.
4. Eli is seated across from Jesse, so they can keep an eye on each other.
5. Gary and Derek are brothers and are always seated next to each other.

1. Who is seated to the right of Eli?

a. Henry

b. Gary

c. Forest

d. Derek

e. Ivan

2. For the purposes of this question, assume the empty seat is next to Derek. Who is seated to Eli’s left?

a. Henry

b. Gary

c. Forest

d. Derek

e. No one.

3. If Eli and Jesse become on better terms and Rule 4 is removed, who is seated to the right of the empty seat?

a. Eli

b. Gary

c. Derek

d. Gary or Derek

e. Gary, Derek, or Eli

Answers after the break.

Answer Key and Explanation

First we should acknowledge how the table is set up. We know that there’s eight spots around the table, and one of them, the King’s seat, is empty. We also know that half the people will be seated on corners, and the other half seated on the sides of this square table.

Building up a model based on the above rules should be relatively simple:

1. We know that F and E need to be seated on a side, not the corner of the table.

2. Forest and Jesse flank Ivan on either side.

3. The seat to the right of Jesse is empty (the King’s seat).

4. Eli needs to be seated across from Jesse. If this doesn’t work out in your diagram, try shifting Forest and Eli’s starting positions and getting to this step again.

5. We can’t be certain where each of them are sitting, but since there’s only one open slot left, we know that they need to go in the place that has two open spaces next to each other.

Question 1:

Your diagram should look like this:

By process of elimination we can work out who CAN’T be seated in the seat to the right of Eli: namely, everyone who we currently have a place for. The only one left is Henry, answer a.

Question 2:

Same layout, but now we know which order the two brothers are in, since we can place Derek definitively to the King’s right. That forced Gary to be next to Eli, so the answer is b.

Question 3:

This prevents Eli from being locked into the position across from Jesse. Thanks to this, Eli can now move to be across from Forest, the only other possible position for him (you can try other places, but they violate additional rules). This gives us this diagram:

Which means that in addition to our old two options (Gary and Derek, either of whom could have been seated next to the King) we now can add Eli. The correct answer is e.