Logic Problem Solving Guide

In case if you don’t know how to solve logic problems, here is a solving guide to help you. The examples are basically rules of these hints, but also examples of those rules.

Vocabulary:

  • Box – a square with the same amount of items from every category. It contains all of the items from one category intersecting items from another category.
  • Categories – a group of items at one sector of each side of the grid.
  • Column – vertical strip of items in each box.
  • Intersections – a box where one item from one category meets another item in another category.
  • Items – an object to match in logic problems.
  • Row – horizontal strip of items in each box.

Hint #1 – Simple Clues:

Simple clues are easy clues that can confirm a single yes or no. If a match is confirmed yes, there should be a dot in the specific intersection as all of the other intersections on the same row and same column in the same box must be crossed out. If a match is confirmed no, there should be an X in the intersection.

  • Example 1: Tom’s favorite color is red.
    • The intersection between “Tom” and “red” should have a dot. At the same time, there should be X’s between all of the other intersections between the names and “red”, as well as the colors and “Tom”.
  • Example 2: John doesn’t ride a car to school.
    • The intersection between “John” and “car” should have an X since there’s no match there.

Hint #2 – Descriptions:

Some clues have extra details within parenthesis. They tend to help the puzzle solver discover a match between the object and the description. They only correspond to the direct object.

  • Example 1: The circle (which is blue) has an 8 in it.
    • A dot should exist in the intersection between “circle” and “blue” since the circle is blue.
    • A dot should exist in the intersection between “circle” and “8” since 8 is the number within the circle.
    • A dot should exist in the intersection between “blue” and “8” since the circle is blue and has the number 8 in it.
  • Example 2: The square (which is green) doesn’t have a 5 in it.
    • A dot should exist in the intersection between “square” and “green” since the square is green.
    • An X should exist between “square” and “5” since the number 5 isn’t in the square.
    • An X should exist between “green” and “5” because the green shape is a square, which doesn’t have the number 5 in it.
  • Example 3: The triangle (which isn’t red) has a 3 in it.
    • An X should exist between “triangle” and “red” since the triangle isn’t red.
    • A dot should exist between “triangle” and “3” since the number 3 is in the triangle.
    • An X should exist between “red” and “3” since 3 is in the triangle, which isn’t red.
  • Example 4: The star (which isn’t yellow) doesn’t have a 4 in it.
    • An X should exist between “star” and “yellow” since the star isn’t yellow.
    • An X should exist between “star” and “4” since 4 isn’t the number in the star.
    • Do not put an X in the intersection between “yellow” and “4” until more information is found. Just because the star isn’t yellow nor does it have a 4 doesn’t mean the yellow shape has a number other than 4.

Hint #3 – Either/or Clues:

These clues involve one or two items that correspond to two other items. Only one of them is correct.

  • Example 1: Either Jonathan or William likes American food.
    • This clue is an example of when one item is associated with two other items from one category. It tells us that either Jonathan or William likes American food as his favorite food.
    • The rule of these clues is to cross out all intersections between one item and the names other than the names provided.
    • So any intersection between “American food” and the names other than “Jonathan” or “William” should be crossed out as none of those boys likes American food as their favorite food.
    • In the future, if one of the boys doesn’t correspond with “American food”, the other boy gets it.
  • Example 2: Bob’s math class is either his first class or the class his red folder is used.
    • Now we’re seeing one item correspond to two other items from two different categories.
    • For these types of clues, the only space we can cross out is the space that intersects between the two items corresponded to one item.
    • We don’t know which class is the math class, but we know that the red folder doesn’t contain homework and classwork from Bob’s first class, so there should be an X in the intersection between “red” and “first”.
    • In the future, if Bob’s math class isn’t his first class, his math work should be stored in the red folder, and vice versa.
  • Example 3: Matthew and Charlie like Mexican food and Italian food, in some order.
    • These types of clues are two items from one category corresponding with two items from another category.
    • What happens is that if two items go with both items, all remaining intersections with these items must be eliminated.
    • Therefore, any name other than “Matthew” and “Charlie” should be crossed out in the intersections between “Mexican food” and “Italian food”. The intersections between “Matthew” and “Charlie” and any type of food other than “Mexican food” and “Italian food” should be crossed out as well.
    • At the end, if we find out that Matthew doesn’t like Mexican food, then we know that he likes Italian food while Charlie likes Mexican food. The same works both ways.
  • Example 4: Martin and Joseph went to a rock concert and an event that took place on Monday, in some order.
    • There are two items from one category, but they are associated with two items from two different categories.
    • The rule implied in Example 4 is a combination of the rules implied in Examples 1 and 2, but the step from Example 1 is repeated.
    • So all of the intersections between “rock concert” and any name other than “Martin” or “Joseph” should be crossed out. The same is true for the intersections between “Monday” and any name but “Martin” or “Joseph”. Plus, the intersection between “rock concert” and “Monday” should be crossed out since the rock concert didn’t take place on Monday.
    • Whoever doesn’t go to the rock concert does go to the event on Monday.
  • Example 5: Cameron and the 12-year old boy likes adventure movies and cereal, in some order.
    • Now it’s dealing with four different items from four different categories, but two associated with two.
    • The intersection between A and B, as well as the intersection between C and D, must be crossed out, as we should keep in mind that whatever doesn’t match with one does match with the other.
    • So Cameron isn’t 12 years old, and the boy whose favorite movies are adventure movies doesn’t like cereal as a favorite breakfast. An X between “Cameron” and “12” exists, as well as an X between “adventure” and “cereal”.
    • If Cameron doesn’t like adventure movies, then at least the 12-year old does, while Cameron likes cereal as his favorite breakfast. It works both ways.

Hint #4 – Neither/nor Clues:

These clues are the opposite of the Either/or clues. Instead of crossing out the unnamed intersections, we should cross out the named intersections that associate with an item.

  • Example 1: Neither Sharon nor Clara went to the movie theater.
    • For clues like these, all of the intersections named should be crossed out with the other item.
    • So any intersection between “movie theater” and any name other than “Sharon” or “Clara” should be left alone as of this point. It’s “Clara” and “Sharon” that should have the intersections between “movie theater” being crossed out.
  • Example 2: Neither Linda nor the girl with the red coat played snowball fight.
    • When two items from two different categories are associated with an item from another category in a neither/nor clue, it should be stated that all three items are separate.
    • So not only that Linda or the girl with the red coat didn’t play snowball fight, but Linda isn’t the girl with a red coat. So all intersections involving “Linda”, “red”, and “snowball fight” should be crossed out.
    • This rule is a bit similar to Example 2 from the Either/or clues, but only crossing out the two items associated with another item.

Hint #5 – Order Clues:

Order clues are the most interesting clues in logic-problem solving. Order involves time, value, size, location, and rank.

  • For time clues, words include (but not limited to) “earliest”, “earlier”, “later”, “latest”, “first”, “before”, “after”, and “last”.
  • For size clues (including age), words include (but not limited to) “longer”, “shorter”, “bigger”, “smaller”, “older”, and “younger”.
  • For value clues, words include (but not limited to) “more”, “less”, “higher”, “lower”, “bigger”, and “smaller”.
  • Time usually refers to time of the day, day of the week, and month of the year.
  • Value usually refers to number of items, cash value, and other stuff like point value.
  • Location clues mostly refer to room numbers or floor numbers.

The extrema terms (such as highest, biggest, youngest, and last) count as items on their own, but comparison terms (such as lower, smaller, older, and before) make things more interesting.

  • Example 1: John had more money than Tom.
    • Now this is simple. We don’t know if John has the most money or if Tom has the least amount of money, but we know that John doesn’t have the least amount of money nor does Tom have the most money. So the intersection between “John” and the lowest monetary value must be crossed out while the intersection between the highest monetary value and “Tom” is crossed out as well.
    • If John doesn’t have the highest monetary value, then the intersection between “Tom” and the second highest monetary value should be crossed out since that implies that Tom doesn’t have anymore money than John.
    • If Tom doesn’t have the lowest monetary value, then the intersection between “John” and the second lowest monetary value should be crossed out.
  • Example 2: Mary went to the park some time before Sarah, but some time after Sally.
    • This is a common order clue. What it shows are two items from one category mark the extrema while the subject (from the same category) is in the midpoint.
    • Assuming that time refers to time of the day, we know that Mary was neither first nor last to visit the park, so the intersection between the earliest time and “Mary”, as well as the intersection between the latest time and “Mary”, should be crossed out.
    • Since Mary came before Sarah, both intersections involving “Sarah” and the two earliest times should be crossed out. Meanwhile, both intersections involving “Sally” and the two latest times should also be crossed out.
  • Example 3: Timothy finished the race later than the racer with a blue kart, but earlier than the racer with a tan seat.
    • These kind of clues are order clues using more than one category. In fact, it is using items from three categories.
    • Just like Example 2, Timothy is not in first place or last place, so both extremes intersecting with his name should be crossed out. He came some point after the blue kart, so the intersections between both the last place and second last place (or fourth place if there are five racers) AND the blue kart should be crossed out. We do the same for the intersections between both first place and second place with the tan seat.
    • Since three items came in three separate times, we know that Timothy isn’t the racer with the blue kart nor is he the racer with the tan seat. Even the blue kart didn’t have the tan seat. So all intersections that cross two of the three should be crossed out.
  • Example 4: Melvin has more cards, but fewer coins than Marvin.
    • We could use two objects from two different categories or three objects from the same category in comparison, but these kind of clues involve two different order categories.
    • Looking back at Example 1, we should cross out the intersection between “Marvin” and the highest card count, as well as “Melvin” and the lowest card count. This will prove than Melvin has more cards than Marvin.
    • In addition, we should cross out the intersection between “Marvin” and the lowest coin count, as well as “Melvin” and the highest coin count. This will prove that Marvin has more coins than Melvin.
    • If Melvin had the highest card count or if Marvin had the highest coin count, then the intersection between the highest coin count and the highest card count should be crossed out.
    • If Melvin had the lowest coin count or if Marvin had the lowest card count, then the intersection between the lowest coin count and the lowest card count should be crossed out.
    • If neither are at the extrema, then we shouldn’t focus on the intersections between coin count and card count until more clues are revealed.

Hint #6 – Multiple Item Clues:

These clues name a whole bunch of items at once, and show that they are completely different from each other. If two or more categories are named, then all intersections involving items from each category should be crossed out.

  • Example 1: The four girls are Jane, Betty, Megan, and the girl from the North.
    • This is not a very good clue since it is too easy, but it implies that Jane, Betty, and Megan aren’t from the North, so all intersections between “North” and their names should be crossed out.
  • Example 2: The four girls are Jane, Betty, the girl from the North, and the girl from the South.
    • This one is better than Example 1, but not quite like it. Neither Jane nor Betty came from the North. They didn’t come from the South either. So the intersections between “Jane” and “North”, “Betty” and “North”, “Jane” and “South”, and “Betty” and “South” should be crossed out.
  • Example 3: The four girls are Jane, Betty, the girl from the North, and the girl with the blonde hair.
    • This one is looking better, but there are four items from three categories. This implies that neither Jane nor Betty came from the North. It also implies that neither Jane nor Betty has blonde hair. Moreover, the girl from the North doesn’t have blonde hair, so the intersections between “Jane” and “North”, “Betty” and “North”, “Jane” and “blonde”, “Betty” and “blonde”, and “North” and “blonde” are crossed out.
  • Example 4: The four girls are Jane, the girl form the North, the girl with the blonde hair, and the girl with brown boots.
    • When all four categories are used to name four different items, all intersections involving two of each should be crossed out. So any intersection that involves two of the following (Jane, North, blonde, brown boots) should be crossed out, giving us six intersections crossed out (one intersection per box).

There are special clues as well. You may never know what could happen.

For further help, try reading through this sample. It is an example involving an entire logic problem.

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