Associated Topics || Dr. Math Home || Search Dr. Math

### Where and with What Weapon Was Each Man Murdered?

Date: 09/19/2001 at 22:18:29
From: Sarah
Subject: Word Problem

One rainy evening, five military men (a general, a captain, a
lieutenant, a sergeant, and a corporal) were murdered in the old
mansion on Willow Lane. The murders took place in the bedroom,
basement, pantry, den, and attic of the house. No two men were
murdered in the same room or with the same weapon. The weapons used
were poison, a poker, a gun, a knife, and a shovel. From the clues
given, try to determine the room in which each man was killed and the
weapon used to do him
in.

1. The murder with the shovel was not done in the den or the attic;
neither the captain nor the lieutenant was killed with the shovel,
nor was either killed in the den or the attic.

2. The captain was not murdered in the bedroom.

3. The poker was not the murder weapon used in the attic.

4. Neither the general nor the corporal was murdered with poison, a
gun, or a shovel.

5. The man murdered in the basement had just had dinner with the
corporal, the captain, the man done in with poison, and the victim
of the poker.

Date: 09/21/2001 at 13:33:15
From: Doctor Code
Subject: Re: Word Problem

Sarah,

To do these types of problems, first set up a grid like this:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |_|_|_|_|_|
captain    |_|_|_|_|_| |_|_|_|_|_|
lieutenant |_|_|_|_|_| |_|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |_|_|_|_|_|
_ _ _ _ _
shovel     |_|_|_|_|_|
poison     |_|_|_|_|_|
poker      |_|_|_|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

The columns across the top should be labeled:

Basement, Den, Attic, Pantry, Bedroom for the rooms
Shovel, Poison, Poker, Gun, Knife for the weapons

I only had room for the first letter of the column names. Notice how
the weapons appear twice, both as rows on the bottom-left and as
columns on the top-right. We'll see why we need this in a moment.

Now let's fill in the information from clue #1 into the table. First,
"the murder with the shovel was not done in the den or attic."  Put an
'O' in the shovel row and in the den and attic columns, like this:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |_|_|_|_|_|
captain    |_|_|_|_|_| |_|_|_|_|_|
lieutenant |_|_|_|_|_| |_|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |_|_|_|_|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|_|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

The 'O' signifies that something is false; for example, the shovel
wasn't used in the den, and the shovel wasn't used in the attic. Now
the 2nd part of clue #1: "Neither the captain nor the lieutenant was
killed with the shovel":

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |_|_|_|_|_|
captain    |_|_|_|_|_| |O|_|_|_|_|
lieutenant |_|_|_|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |_|_|_|_|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|_|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Now you can see why we needed the weapons in two places, so that we
vs. rooms. For the 3rd part of clue #1: "Nor was either killed in the
den or attic."

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |_|_|_|_|_|
captain    |_|O|O|_|_| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |_|_|_|_|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|_|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

That's all the information from clue #1.  Now for clue #2: "The
captain was not murdered in the bedroom."

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |_|_|_|_|_|
captain    |_|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |_|_|_|_|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|_|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Clue #3: "The poker was not the murder weapon used in the attic."

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |_|_|_|_|_|
captain    |_|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |_|_|_|_|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Clue #4 has a lot of information: "Neither the general nor the
corporal was murdered with poison, a gun, or a shovel."

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|_|O|_|
captain    |_|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |_|_|_|_|_|
corporal   |_|_|_|_|_| |O|O|_|O|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Now it gets interesting. Notice that the column for the shovel has 4
O's in it. The shovel was not used to kill the general, the captain,
the lieutenant, or the corporal. That leaves only one possibility,
that the shovel was used to kill the sergeant. We'll mark that fact
down with an X:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|_|O|_|
captain    |_|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |X|_|_|_|_|
corporal   |_|_|_|_|_| |O|O|_|O|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Since we know that sergeant was definitely killed with the shovel, we
know he wasn't killed with any other weapon, so we can mark O's in all
the other columns:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|_|O|_|
captain    |_|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |X|O|O|O|O|
corporal   |_|_|_|_|_| |O|O|_|O|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Finally we'll fill in clue #5: "The man murdered in the basement had
just had dinner with the corporal, the captain, the man done in with
poison, and the victim of the poker."  First, the captain and corporal
weren't killed in the basement, since they just had dinner with the
man who was:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|_|O|_|
captain    |O|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |X|O|O|O|O|
corporal   |O|_|_|_|_| |O|O|_|O|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |_|_|_|_|_|
poker      |_|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

The man killed in the basement had dinner with the poison and poker
victims, so that means those weapons weren't used in the basement:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|_|O|_|
captain    |O|O|O|_|O| |O|_|_|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |X|O|O|O|O|
corporal   |O|_|_|_|_| |O|O|_|O|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |O|_|_|_|_|
poker      |O|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

The corporal and captain had dinner with the poison and poker victims
too, so they weren't killed with either of those weapons:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|_|O|_|
captain    |O|O|O|_|O| |O|O|O|_|_|
lieutenant |_|O|O|_|_| |O|_|_|_|_|
sergeant   |_|_|_|_|_| |X|O|O|O|O|
corporal   |O|_|_|_|_| |O|O|O|O|_|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |O|_|_|_|_|
poker      |O|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Now look at the people versus weapons grid. It's almost finished.
We'll add X's to the only open spot in all the rows that already have
4 O's in them, and then the same thing for columns:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|_|_| |O|O|X|O|O|
captain    |O|O|O|_|O| |O|O|O|X|O|
lieutenant |_|O|O|_|_| |O|X|O|O|O|
sergeant   |_|_|_|_|_| |X|O|O|O|O|
corporal   |O|_|_|_|_| |O|O|O|O|X|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |O|_|_|_|_|
poker      |O|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Now we know exactly who was killed with what weapon. We just have to
figure the rooms out, so we'll add X's to the people vs. rooms grid:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|O|_| |O|O|X|O|O|
captain    |O|O|O|X|O| |O|O|O|X|O|
lieutenant |_|O|O|O|_| |O|X|O|O|O|
sergeant   |_|_|_|O|_| |X|O|O|O|O|
corporal   |O|_|_|O|_| |O|O|O|O|X|
_ _ _ _ _
shovel     |_|O|O|_|_|
poison     |O|_|_|_|_|
poker      |O|_|O|_|_|
gun        |_|_|_|_|_|
knife      |_|_|_|_|_|

Here it gets a little tricky. We don't have any more columns or rows
that have 4 O's, but that's okay. We can combine some of the X's to
form new facts. For example, we know the captain was killed in the
pantry, and we know he was killed with the gun. That means the gun was
the weapon used in the pantry, and we can fill in that information in
the rooms vs. weapons grid:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|O|_| |O|O|X|O|O|
captain    |O|O|O|X|O| |O|O|O|X|O|
lieutenant |_|O|O|O|_| |O|X|O|O|O|
sergeant   |_|_|_|O|_| |X|O|O|O|O|
corporal   |O|_|_|O|_| |O|O|O|O|X|
_ _ _ _ _
shovel     |_|O|O|O|_|
poison     |O|_|_|O|_|
poker      |O|_|O|O|_|
gun        |O|O|O|X|O|
knife      |_|_|_|O|_|

We can do the same thing with one O and one X. Look at the corporal
row. We know he was killed with the knife, and we know he wasn't
killed in the basement. That means the knife wasn't used in the
basement. We'll record that fact in the weapons vs. rooms grid again:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|O|_| |O|O|X|O|O|
captain    |O|O|O|X|O| |O|O|O|X|O|
lieutenant |_|O|O|O|_| |O|X|O|O|O|
sergeant   |_|_|_|O|_| |X|O|O|O|O|
corporal   |O|_|_|O|_| |O|O|O|O|X|
_ _ _ _ _
shovel     |_|O|O|O|_|
poison     |O|_|_|O|_|
poker      |O|_|O|O|_|
gun        |O|O|O|X|O|
knife      |O|_|_|O|_|

And now we know that the shovel was used in the basement (it's the
only weapon left):

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |_|_|_|O|_| |O|O|X|O|O|
captain    |O|O|O|X|O| |O|O|O|X|O|
lieutenant |_|O|O|O|_| |O|X|O|O|O|
sergeant   |_|_|_|O|_| |X|O|O|O|O|
corporal   |O|_|_|O|_| |O|O|O|O|X|
_ _ _ _ _
shovel     |X|O|O|O|O|
poison     |O|_|_|O|_|
poker      |O|_|O|O|_|
gun        |O|O|O|X|O|
knife      |O|_|_|O|_|

We know the shovel was used in the basement, and we know the sergeant
was killed with the shovel. That means the sergeant was killed in the
basement:

B D A P B   S P P G K
_ _ _ _ _   _ _ _ _ _
general    |O|_|_|O|_| |O|O|X|O|O|
captain    |O|O|O|X|O| |O|O|O|X|O|
lieutenant |O|O|O|O|_| |O|X|O|O|O|
sergeant   |X|O|O|O|O| |X|O|O|O|O|
corporal   |O|_|_|O|_| |O|O|O|O|X|
_ _ _ _ _
shovel     |X|O|O|O|O|
poison     |O|_|_|O|_|
poker      |O|_|O|O|_|
gun        |O|O|O|X|O|
knife      |O|_|_|O|_|

Even though we ran out of clues a long time ago, we can still get lots
they were organized in the grid. Continuing in this fashion will allow
you to complete the rest of the grid, and then you will have all the

Let me know if you need more help.

- Doctor Code, The Math Forum
http://mathforum.org/dr.math/

Date: 10/08/2001 at 19:35:05
From: D. Gunter
Subject: Logic Puzzle

Hello Dr. Math.

From the following clues, can you determine the full name of each
friend (one surname is Milnes), the sport on which he or she
reported, and the number of cards that he or she has collected, as
well as the type of container in which each person stores his or her
cards?

1. The one surnamed Frede isn't Sally.
2. Peter and the one surnamed Clemont both own an even number of
cards; the person whose oral report was about hockey (who isn't
Betty) and the one who stores cards in a shoe box both have an odd
number of cards.
3. The one surnamed Daly owns 5 more cards than the one who collects
basketball cards, who has acquired more cards than Andy.
4. The one who stores cards in a manila envelope didn't give a
report on the game of soccer.
5. The one surnamed Rousse (who isn't Betty) has more cards than
Sally, who owns more cards than the collector who stores cards in a
drawer.
6. The person who stores cards in an album owns more cards than
Javier, who has aquired five more cards than the one whose favorite
sport is soccer.
7. The one who collects baseball cards has fewer than 235 cards.

I've mainly been trying to figure this out by trial and error. Some
I've gotten from looking at the clues. I have a table so that way I
can see what I am doing. Here is the information.

First names: Peter, Andy, Sally, Betty, Javier
Surnames: Milnes, Rousse, Clemont, Daly, Frede
Sports: Baseball, hockey, soccer, basketball, football
Numbers: 214, 219, 224, 229, 235
Container: drawer, tupperware, manila envelope, shoebox, album

I tried to use your method to solve my problem, but it seems to me
that it's a different kind of problem from the one shown above. For
example:

3. The one surnamed Daly owns 5 more cards than the one who collects
basketball cards, who has acquired more cards than Andy.

How could I put that on my grid? Can you help me get started?

Thanks for your time and consideration, Dr. Math!

Sincerely,
D. Gunter

Date: 10/10/2001 at 10:31:45
From: Doctor Code
Subject: Re: Logic Puzzle

Hello,

These can be tricky.  Let's look at clue #3:

3. The one surnamed Daly owns 5 more cards than the one who collects
basketball cards, who has acquired more cards than Andy.

First you can get the following three facts from the clue:

- Andy's last name is not Daly
- Daly doesn't collect basketball cards
- Andy doesn't collect basketball cards

Put O's in these spots on the grid. Next you can order the three
people mentioned in clue #3 by the number of cards they have:

Since Daly has more cards than at least two other people, he cannot
have 214 or 219 cards. He must have at least 224 in order to have more
cards than at least two other people. By this same logic, Andy cannot
have 235 or 229 cards. Basketball cannot have 235 or 214 cards.

Finally, Daly has 5 more cards than Basketball, which means Daly
cannot have 235 cards. If Daly had 235, then Basketball would have
230, but 230 is not one of the numbers available. Therefore, Daly must
have either 229 or 224. Since Basketball has 5 less than Daly, Daly
must have either 224 or 219. Since Andy has less cards than
Basketball, Andy must have either 219 or 214.

So now you've narrowed down Andy, Basketball, and Daly to two
possibilities each for their number of cards. Clues 5 and 6 give you
similar ordering relationships for the cards. Keep these three
inequality sentences written down somewhere. After you make a first
pass through all the clues, recording information as you go, you can
sentences.

Write back if you need more help.

- Doctor Code, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
High School Logic

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search