A LOGICALLY THOUGHT OF TECHNIQUE FOR SOLVING LOGIC PROBLEMS
INTRODUCTION
One of the most popular pencil puzzle as well as one of my favorites is the Grid Logic Problem. Since I have been working these puzzles for many years, I have developed my own strategy and ways of working these puzzles more efficiently. This is going to be workshop of sorts to walk you through working logic problems and to give you a more effective was to read the clues and put them to use. I'm going to supply you with more in-depth process than the normal solving directions provide to you.
If you are planning on getting a grid logic problem book or magazine I suggest getting Penny Press Original Logic Problems magazine. Every issue contains about 100 puzzles varying in difficulty. I have worked a lot different logic problems from a lot of different books and publishers and the Penny Press Original Logic Problems has the most extensive collection of problems with the least amount of editing issues. Subscribe to it at Original Logic Problems or go to your nearest bookstore, Wal-mart, or Target to get an issue.
First we must read and understand the rules and layout of a typical logic problem. The layout is usually a grid with categories and all the objects in those categories listed on the sides of the grid. Sometimes pictures are provided to help you solve the problems that require you to figure out what position an item is in compared to another item (for example: people sitting around a table or houses in a neighborhood).
CHAPTER 1: A SIMPLE PROBLEM WITH SIMPLE RULES
Before I let you in on some of my secrets we will work a small sample puzzle so that we know the basics of how to solve a grid logic problem. We will use only the instructions that a regular logic problem book would give you to solve it. We are only using their directions so that when I introduce my way of solving a problem, you will understand the differences between the two and the advantages that my tips give you.
Basic Solving Directions: Read the clues to determine the answer. Use a dot (• ) for "yes" and an X for "no".
Sample: Five women went to the gym last week, each on a different day of the week. These women each did a different exercise. From the information provided, determine the exercise each woman did, and the day that they went to the gym.
1. Amber went to the gym on Monday, but not to do curls.
2. The person that ran for a mile (who was Heidi) went to the gym the day after Deana.
3. The person that went to the gym on Wednesday did yoga, the person that went to the gym on Friday did gymnastics (who was not Kaylee).
This sample problem is extremely broken down below, and I encourage you not to look at it until after you have already tried to solve the problem on your own and have gotten stuck somewhere.
Looking at clue 1: Put a "yes" dot in the section of the grid that connects Amber to Monday. And since Amber is Monday you know to put "X's" on the rest of the days for Amber and the rest of the women for Monday (this is called exclusion). Because of the second part of clue 1, you know that Amber did not do curls so put an "X" in the spot that connects Amber with curls. Since Amber = Monday and Amber ¹ curls, you know that Monday ¹ curls (this is called contradiction of transitivity). So put an "X" in the space that connects Monday to curls.Clue 2: Put a dot in the part of the grid that connects between Heidi and run for a mile. Same as above, Heidi is not related to any other exercise by exclusion and run for a mile cannot be any other person. By clue 2, you know that Deana went to the gym the day before Heidi. So Deana cannot be Friday because there is no day left in the week for Heidi and Heidi cannot be Monday because there is no day for Deana before Monday.
Clue 3: dots go in the section that connect Wednesday to yoga and in the section that connect Friday to gymnastics. Again, by exclusion, put "X's" in the areas that connect Friday to any other exercise, the areas that connect Wednesday to any other exercise, the spaces that connect yoga to any other day, and the spots that connect gymnastics to any other day. By clue 3, you know that Friday = gymnastics and gymnastics ¹ Kaylee, therefore Friday ¹ Kaylee. So put an "X" in the area that connects gymnastics to Kaylee and the spot that connects Friday to Kaylee.
Now it's time to go back through the grid to any areas that have a dot and transfer what you know about one item to the item that it is linked to by a "yes" dot. (for example: Amber is linked to Monday with a dot. By what we have in the grid so far about Amber we know that Amber ¹ curls and Amber ¹ run a mile, therefore Monday cannot be curls or run a mile also. by process of elimination, Monday has to be cycling. Now carry this through the rest of the grid by putting a dot with Amber and cycling.) Do this for all the dots on the grid and carry throughout the grid.The only day we have left is Tuesday. By our grid and by process of elimination Tuesday must be Kaylee. You have everything now and all you have to do is go through the grid and transfer information to the associated items with dots as we did before.
CHAPTER 2: EVERY LOGIC PROBLEM'S STRUCTURE
Now let's get some of the basic rules that you used to solve the sample problem. There are four basic mathematical laws that you will use in EVERY logic problem that you encounter and there are a couple that you will use often.
Assume for the general use of categories and objects that they are labeled as such: Category A has five objects (A1, A2, A3, A4, A5) Category B has five objects (B1, B2, B3, B4, B5) and so on. In every logic problem only one object in A can link to one object in B, only one in A can link to one object in C, etcetera.
In all of the following example grids the "IF" part of the statement is shown in the color RED and the "THEN" part of the statement is shown in the color GREEN.
If an object in category A is not related to four out of five objects in category B, then it has to be related to the fifth object in category B.
LAW #1: Proven by Elimination: If A1 ¹ B1, B3, B4, and B5, then A1 = B2
If an object in category A is related to an object in category B, then it cannot be related to any of the other objects in category B.
LAW #2: Proven by Exclusion: If A1 = B2, then A1 ¹ B1, B3, B4, or B5
If an object in category A is related to an object in category B and that same object in category B is related to an object in category C, then the object in A is related to the object in C.
LAW #3: Proven by Transitivity: If A1 = B3 AND B3 = C5, then A1 = C5
If an object in A is related to an object in B and that same object in category B is not related to an object in C, then the object in A is also not related to the object in C.
LAW #4: Proven Negative by Transitivity: If A1 = B3 AND B3 ¹ C5, then A1 ¹ C5
All four of these laws were used in the small sample problem above. Sometimes another version of the laws is also helpful. The following two laws are a doubled version of the laws above. This also works with triple and quadruple, but you won't use those as much as the double.
LAW #5: Proven by Double Exclusion: If A1 = (B3 OR B4) AND A2 = (B3 OR B4), then A3, A4, and A5 ¹ (B3 OR B4)
LAW #6: Proven Negative by Double Transitivity: If A1 = (B1 OR B2) AND B1 ¹ C4 AND B2 ¹ C4, then A1 ¹ C4
CHAPTER 3: THE IMPORTANCE OF AN INTRODUCTION
Reading the introduction does not seem as though it would be much use other than a story to set up the logic problem. After all, the only important part of it is the last sentence, where it tells you what you need to determine in order to solve the puzzle. Yes there is a lot of fluff in the intro to each problem but the introduction can contain some vital information that you would need to work the problem.
Many introductions will say something like "No persons first and last name start with the same initials". This eliminates some links that you might make if you didn't read the introduction. Like if you thought that "Arthur" could have the last name "Anderson" or "Black" but you can't figure out from the clues which last name "Arthur" has, you should have read the introduction.
Finally we come to the real reason behind all of this, hints, tips, and strategies for working logic problems more efficiently and more timely than with the normal instructions given to you by logic problem books. Most of the time the names given within a puzzle are not gender specific but the clues will give gender specific answers in them. The introduction will usually give you a list of the names that are women and the names that are men and sometimes it will tell you that 3 out of the 5 people are men and 2 out of the 5 are women.
SOLVING STRATEGY: If the introduction tells you which people are men and which are women, mark it beside their names with a "F" for female and a "M" for male.
This is so that you don't have to keep going back and reading the introduction if you have a clue that says "A man wrote a poem". You would only have to mark "poem" out for everyone that has a "F" beside their name. Sometimes you will have clues that sneak in a gender by using "Mr." or "Mrs." with a last name. Keep your eye out for these uses of gender because they will all help you in the end.
CHAPTER 4: USING CLUE NUMBERS AND LAWS IN THE GRID
First and foremost, I always recommend working the clues in the order that they are written and then going to the grid to use the laws after you have retrieved all the information from the first read-through of clues. After that is done you should go back and reread the clues if needed. Follow this cycle until the problem is finished.
SOLVING STRATEGY: A tip to help you from not having to read full clues over and over is to strike through the parts of a clue that you know you are finished with. And if you are finished with the entire clue just mark through the clue number.
For example. a clue in a problem states this "1. Harry has been married 5 years longer than Tom (who is married to Gina)." After you put a dot in the area that connects Tom to Gina, you can strike through that part of the clue so that you don't have to keep reading it every time that you revisit clue 1. So then it will read "1. Harry has been married 5 years longer than Tom (who is married to Gina)."
The second tip in this chapter is going to help you figure out which clues to revisit if needed. If you mess up somewhere in the solving process, this tip will also help you to backtrack to the appropriate spot without having to erase your entire grid and start over.
Of course the following tip would not be worth the time for the easier puzzles because the simpler puzzles can be worked just as easy without using this tip. This tip, however, is very useful if you are doing a more difficult puzzle.
SOLVING STRATEGY: We are still going to use a dot (· ) for "yes", but instead of an "X" for "no" use the clue number or the law that you got the information from. For law #2 use "E" for Exclusion, for law #4 use "A" for Association, and for law #5 use a slash instead of an "X". I use "C" for anything that I find to be true by Contradicting any other possible answers. Use an "I" if you got the information from the Introduction of the problem.
On the sample problem, the "C" (contradiction method) would come into play when we try to put Heidi on Tuesday. We proved that Heidi could not have gone to the gym on Tuesday by contradicting that possibility. Therefore a "C" can be put in the space that connects Heidi to Tuesday.
Should we happen to mess up something during the solving of a puzzle, this strategy helps us to go back and erase without having to erase everything on the grid to start over. If we had a 6 clue puzzle and we made a mistake somewhere on the 5th clue, we would not erase anything that had the numbers 1 thru 4 or anything that had an "I" in it because we know that we did those four clues and the introduction correctly. This will save you a lot of time having to erase an entire problem and restart it every time that a mistake is made.
By using this strategy to work problems you know which clues to revisit without having to read through all of the clues. For example, clue 2 in the sample problem had to be revisited during solving. If we had the number "2" in the space for Deana and Friday, we would have known to go back and read clue 2 to find out more information about the day Deana came into the gym.
If I have a hard problem that I have to go back and read the clues over again (like we did for clue 2 on the sample), I will put "clue # - # of time that I reread it" in the box. For example, the second time we read clue 2 for Heidi and Deana I would put "2-2" in the box. I only use this method if I am working a large puzzle because for the smaller puzzles it can get confusing.
Both of the strategies introduced in this chapter are basic hints to help you along with ANY logic problem that you solve.
CHAPTER 5: DIVIDE AND CONQUER IN A NEW LIGHT
At times you may know that an object has to be one of two options, but you are unable to determine which one it is. For example, you know that Andy is either the one that washed his car of the one that washed his windows but you just can't figure out by any of the clues which one he did. In an instance such as this you would try one of the options and see where it leads you. Say that Andy is the one who washed his car, and by working through some more of the puzzle you find that another part of the puzzle could not possibly work. In this case then you have proven that Andy did not wash his car and Andy had to of washed his windows. So you divided the two options and worked the puzzle until you came to a contradiction.
SOLVING STRATEGY: If there is an object A1 that has the possibility of being linked to more than one object in category B (B2 OR B4) and you cannot prove by using the clues which object (B2 OR B4) it is, then try one of the options (make A1 = B2) to see if you reach a contradiction. If you do contradict that A1 = B2, then you know that A1 must equal B4.
Sometimes the objects within a category are labeled into smaller categories that are not a part of the ending solution. For instance a puzzle might say something like this, "Out of the 15 teachers, 4 of them are Science teachers, 5 are Mathematics teachers, and 6 are English teachers." In order to solve the problem you are not required to find out which teachers are science teachers, math teachers, and English teachers but that information might be helpful if you get a clue that connects certain objects to science, math, or English.
SOLVING STRATEGY: Use differing markings around objects that you know are related by another underlying category. I use circles, boxes, underlines, etc.
The following problem was scanned out of a Penny Press Original Logic Problems magazine. It depicts the same aspect as presented above but instead of teachers, this problem is gender related. The first thing I did was go through all the clues and divide the items that I knew were related to females from the items that I knew were related to males. I have circled the items that are female and put a box around the items that I know to be male. This way I can quickly determine that anything with a circle around it is NOT related to anything with a box around it.
CHAPTER 6: MAPPING OBJECTS OUT
Sometimes it is necessary to map out objects that have a relation with variables. A lot of logic problems deal with time, order (like when the sample problem said that Heidi came into the gym the day after Deana), or some sort of other variable. I find it easiest to write the clues that connect these objects by a variable in a different way, so that it is easier for me to read and easier for me to solve the problem.
SOLVING STRATEGY: Write the objects that are separated by a variable out in a map-like manner. Use a dash to show that two objects are beside each other in the mapping. Use an arrow to show that the object is "somewhere" to a particular side of another object. And use blank underline to show where there is an object in the mapping, but you are unsure of what that specific object is yet.
For example, a problem using the variable of when people go into a bank. When a clue says "There were five visitors at the bank today. Bobby came into the bank right after Jimmy, but at some point before Tom. Paul came into the bank exactly three people after Logan." you would write a mapping like the one below.
Jimmy - Bobby àTom
Logan - ___ - ___ - Paul
Now it is much easier to figure out the order that the people came into the bank. It's pretty easy to figure out the order just by looking at the mapping, but I've broken it down for you below anyway.
You know from the clue that there are only five time slots. you know then that Logan can only be 1st or 2nd and that Paul would be 4th or 5th respectively. Using that same reasoning you know that Jimmy can only be 1st, 2nd, or 3rd. Which would make Bobby 2nd, 3rd, or 4th respectively. Now we will take notes from chapter 5 and divide and conquer this map. So let's try Logan as 2nd. If Logan is 2nd then Paul is 5th.
Jimmy - Bobby à Tom
2 5
Logan - ___ - ___ - Paul
So then Jimmy cannot be 2nd because we already have Logan as 2nd. Jimmy cannot be 1st because Bobby would be 2nd, but we already have Logan as 2nd. Thus, in this scenario Jimmy must be 3rd which makes Bobby 4th and Tom 5th. But again we have contradicted this scenario with Tom being 5th because we already have Paul as 5th. Therefore every option contradicts Logan being 2nd. Then we know that Logan is 1st and Paul is 4th.
Jimmy - Bobby à Tom
1 4
Logan - ___ - ___ - Paul
Now you know that Jimmy cannot be 1st because Logan is 1st. And you know that Jimmy cannot be 3rd (that would make Bobby 4th, and Paul is already 4th). Therefore Jimmy must be 2nd, Bobby is 3rd, and that makes Tom 5th (the only one left).
SOLVING STRATEGY: You may also need to map out objects by using mathematical equations.
A problem may need to be mapped out using equations if you have something like this... "Carla withdrew twice the amount of money than David and David withdrew 10 more dollars than Chuck". Then you want to map it out using this equation.
Chuck + 10 = David x 2 = Carla
SOLVING STRATEGY: the same thing would be done with a picture logic problem. If you have a picture of objects and you know the directions of one object compared to another, map out what you know to the side of the picture problem.
For example there are six houses in the picture problem below and a clue says "Jon lives somewhere to the South of Billy. Billy lives on the East side of the road." You would draw a mapping beside the picture to help you visualize it.
CHAPTER 7: NOW YOU'RE READY
Now using everything you've learned from this workshop, here is a logic problem designed by me for you to solve... ENJOY!
The Kemper Family has game night every Saturday night. This week they each hosted a game with one of the other four family members (two of which - Chari and Janice - are women and three of which - Destiny, Matt, and Rex - are men) as a guest. They each hosted a different game (one was "Clue") and then played another game as a guest. In addition, each game was played a different number of times (1 thru 5). From the information provided, can you determine each family members guest, the game they played, and how many times they played.
1. Chari's guest had more letters in their name than Chari. Rex either hosted the game that was played 1 time or he was the guest at the game that was played 1 time.
2. Destiny (who's guest was a man) hosted the game played exactly one less time than the pair that played Sorry, but exactly two more times than the pair that Janice was a guest in.
3. Matt did not play Scrabble as host or as guest, Rex was a guest for Sorry. Matt hosted Janice.
4. Clue was played 4 times. Chari hosted Monopoly.
