# Some Brainteasers

The math department here at JMU has a Problem of the Week competition, and it just so happens that, this semester, I am running it. Every week I choose a problem for the consideration of all who choose to participate. (Well, I actually bribe my students to participate by offering them a bonus point for each problem they get right, but whatever.) A randomly selected winner from among the correct answers gets a five dollar gift card to Starbucks. Mostly it’s just a way to get the students thinking about amusing mathematical brainteasers outside of their regular coursework.

Anyway, I am especially fond of this week’s puzzle, even if it does stretch the definition of a math problem just a bit. It makes me smile every time I read it. It’s not terribly difficult, but I think you might enjoy working it out:

The Supreme Court today reversed its earlier ruling that let stand an appellate court’s decision to overturn a lower court’s finding that a restaurant owner had no right to fire a waiter for refusing to deny service to a male patron who was not wearing a tie and jacket. If a male patron now enters the restaurant, and if we assume that the wait staff will serve anyone as long a they are confident they will not be fired for doing so, then will the male patron be served?

There’s practically a whole genre of puzzles along these lines, in which a situation is described in language specifically intended to be confusing. Your task is to decipher what is going on. In the Problem of the Week I provide a second example as a warm-up exercise:

If you ask me on Friday what day classes start, and I truthfully reply that they start two days after the day before the day after tomorrow, then on what day do classes start?

Of course, the granddaddy of this genre is this old chestnut:

A man is looking at a painting of a certain person. You ask him who he is looking at. He points to the person in the painting and replies, “Brothers and sisters I have none, but this man’s father is my father’s son.” Who is shown in the painting?

Actually, this one provides a useful clinic for how to solve such a thing. Usually, the simplest way is to work backward from the end. If I am a man with no siblings, then my father’s son is me. Thus, the statement in the problem amounts to saying, “I am the father of the man in the painting.” So the man is looking at a picture of his son.

As a variant on the previous problem, how would the answer change if the man had said, “Brothers and sisters I have none, but this man’s son is my father’s son.”?

If that was too easy for you, then have a go at a tougher one:

Jill knocks on Jack’s door. When Jack answers, Jill says to him, “Remember that book you lent me? Well, I lent it to my mother, and she lent it to her sister, who gave it to her son-in-law, who thought his wife’s maternal grandfather would like it. He did, and lent it to his wife, who gave it to her son, John. Last night John dropped in and asked me to return it to his son. So here it is.” How are Jack and Jill related?

How about one more before I sign off?

“Hi Bill. Let’s meet at my office, okay? The building is on the North side of Main Street. After entering through the front door, turn left down the hall and take one of the elevators on your right as high it goes. Across the hall there’s another elevator. Take it to the 50th floor. When you get out, turn left and look for a door on the right that says UP. Go in, turn right up a short staircase, then turn left through the exit door at the top of the stairs. Walk down the hall that goes to your right, and my office door is on the left.” Which direction will Bill be facing when he knocks on the door?

Frankly, even if you’re not interested in trying to solve them, I think these puzzles work well just as poetry!

1. #1 Reginald Selkirk
March 20, 2013

How are Jack and Jill related?

They are accident-prone hill climbing partners who work for the same water supply company.

March 20, 2013

The last problem is tough: it requires traveling to Main Street to see which direction the front door faces π

I used to ask for volunteers in class by starting (early in the course) with “Hands up anyone who does not want to volunteer” (this gets you _lots_ of volunteers the first time, but not the second – signaling that the game is on) and then working up (as the course went on) to things like “I need a volunteer. If you don’t want to avoid volunteering, please keep your hand down.” (Delivered quickly, this tends to result in most students not knowing whether they’re volunteering or not.) The main problem was thinking of things to do with the volunteers after eliciting them π

3. #3 Jason Rosenhouse
March 20, 2013

George Carlin used to do a sketch about different ways of saying hello and goodbye. At one point he said that when people asked him how he was he would reply, “I’m not unwell, thank you,” just to enjoy the confused looks he would get.

4. #4 Lenoxus
March 20, 2013

For the Jack and Jill problem, should we assume that couples bond for life in both marriage and children-having? If not, then the best I can say is that Jack is Jill’s grandfather’s wife’s grandson (I don’t think there’s a particular term for that). If so, then Jack and Jill are either first cousins or siblings, John being Jack’s father either way. Of course, it would be strange for Jill to call her father John, but it’s already unusual for her to refer to Jack’s father that way anyway.

For the Supreme Court one, my tentative answer is: Yes, they will serve the man regardless of how he is dressed (so long as he’s got a shirt and shoes!). The Court’s ultimate finding was that waiters have the right to serve people who fail the dress code.

Classes will start in three days.

The man in the first picture is the speaker himself, and in the second it’s his father. Somewhere I’ve read a variant that’s less likely to trick younger people, because part of the “trick” involves considering female relatives, like the old puzzle about the surgeon who can’t operate on a boy whose father died because “he’s my son”.

Bill should be facing south.

5. #5 Lenoxus
March 20, 2013

Glunk! The one I know I got wrong was the one whose correct answer was explicitly explained; I just glossed over it. Yes, in the first example, it’s a picture of his son. Sheesh.

6. #6 Lenoxus
March 20, 2013

And I now realize that Jack and Jill can’t be siblings if we rule out incest. Blarg. So to reiterate, he’s her mother’s father’s wife’s son’s son, or most likely, first cousin.

7. #7 Sean T
March 21, 2013

Lenoxus,

I get a different answer for which direction Bill is facing; maybe I’m reading the problem differently than you are.

Enter building : facing N
Turn left down hall : W
Enter elevator on right: N
Exit elevator: S
Enter elevator across hall: still S
Exit elevator: N
Turn left: W
Enter door on right: N
Turn right up stairway: E
Turn left through exit door: N
Walk down hall to right: E
Turn to face door on left: N

So I find that Bill should be facing north when he knocks on the office door.

8. #8 Ted Magee
Austin, Texas
March 21, 2013

I also though the man was looking at himself in the painting. Confusing.

The one about the waiter describes the most recent patron simply as a “male patron,” with no information about how he is dressed. But I got that the Supreme Court had decided that the waiter should have kept his job, so I assume the guy would get served unless he was the judge with the appellate court.

9. #9 magster2
March 21, 2013

Sean T, I got the same answer you did. I’ve also been in an elevator that had doors on more than one side, so it may need to be explicitly spelled out in the problem that the elevator is a typical one with only one set of doors.

10. #10 CherryBombSim
March 21, 2013

Of course, the U.S. Supreme Court would never reverse its own ruling in a particular case, since it always decides en banc.

11. #11 Lenoxus
March 21, 2013

Sean T

Exit elevator: N

Ah, that’s what I forgot! I worked it out as if one never turns around inside an elevator, and only turns after leaving it (by walking backwards, I guess?) The “across the hall” part presented no issue because you turn all the way around regardless, but the second part involved a solid difference between where left or right would take you.

As it happens, exiting elevators backwards was how things were done on my home planet; I was sent here as a baby just before that world’s destruction, and I now use my powers of unique elevator orientation in the pursuit of justice.

12. #12 Lance Merlino
April 1, 2013

I always enjoyed solving the puzzles and especially brainteasers when I was in school and college. Frankly, pls post some puzzles and brainteasers blog posts so that I’ll try to solve that and enjoy. Thanks

13. #13 Lance
April 2, 2013

This post actually reminded me my high school days when I was a maths tutor and asking puzzles and brainteasers to my students to take their attention towards the subject. My students used to enjoy answering those questions.