Logic Puzzle: You Can Leave Your Hat On

Every week we feature an original logic puzzle for your brain-bending pleasure. They follow a format similar to those on the LSAT, and their difficulty ranges the full gamut. Good luck!

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?
  2. Al
  3. Bart
  4. Charlie
  5. Don

 

  1. If Charlie does not wear a hat on Sunday which of the following must be true?
  2. Don wears a hat on Friday.
  3. Al wears a hat on Monday.
  4. Bart wears a Hat on Thursday.
  5. Charlie wears a hat on Wednesday.

 

  1. If Don wears a hat on Saturday only which of the following is true?
  2. Bart may wear a hat on Thursday.
  3. Al may wear a hat on Wednesday.
  4. Charlie may wear a hat on Monday.
  5. No one may wear a hat on Thursday.

 

  1. If rule number six is removed:
  2. Don may wear a hat 4 times a week.
  3. Don may wear a hat 5 times a week.
  4. Don may wear a hat 6 times a week.
  5. Don may wear a hat 7 times a week.

 

  1. If rule number four is removed:
  2. Bart may wear a hat 1 time a week.
  3. Bart may wear a hat 3 times a week.
  4. Bart may wear a hat 5 times a week.
  5. 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.

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Al X X
Bart X X X
Charlie X X X
Don

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.

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Al X X
Bart X X X
Charlie X X X
Don

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.

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Al X X
Bart X X X
Charlie X X X
Don X
  1. 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.

  1. 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.