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 
can write down information about weapons vs. people, and about weapons 
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 
more information from the facts we already know, because of the way 
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 
answers.

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:

    Daly > Basketball > Andy

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
discover more information just based on these three inequality
sentences.

Write back if you need more help.

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