About This Series
This series is planned to be ten caches. The first nine will
provide lessons to assist you in building your puzzle-solving
skills. Each will contain a lesson centering on a specific puzzle
skill, examples of how to apply that skill, an exercise to test you
on that skill, and a cache to find as a reward for your efforts.
Study the lesson, complete the exercise, and you'll find the
location of a cache.
Each of the caches will contain a piece of information you will
need to help you take the final exam (which will be the tenth and
final cache in the series). When you visit each cache, you will
need to bring something to record those clues for later ...like
paper and pen/pencil or perhaps a camera.
One final note: Each of these caches will have an unactivated
geocoin in it as a First to Find prize. Crew416's rule: IF YOU
HAVE BEEN FIRST TO FIND ON ANY CACHE IN THIS SERIES, YOU MAY NOT -
NOT - NOT FIND OR LOG ANOTHER ONE IN THE SERIES FIRST UNTIL IT HAS
ALREADY BEEN FOUND AND LOGGED BY SOMEONE ELSE!!! That means
there should be 9 different First to Find cachers on the first 9
caches in this series. The final exam will be open to all,
including the 9 first to finders on the first nine caches, and will
also contain a nice geocoin prize for the first finder. This will
give a few new puzzle cachers the opportunity to be First to Find
on a puzzle cache, perhaps for the first time ever. This should
give more cachers the thrill of solving and being there first and
score a new geocoin prize... and one experienced puzzle solver
doesn't end up with all the coins. So shall we continue to our next
lesson?
Lesson 6: Logic Problems
Introduction
Logic problems test your deductive reasoning skills. They
challenge you to take a bunch of pieces of information and to make
logical inferences from them. Academic as this sounds, they're a
lot of fun.
This a lengthy lesson, but it's an important one. Logic problem
solving skills are required to solve a variety of puzzles
...wordplay, mathematics, cryptography, lateral thinking, and other
types of puzzles incorporate many aspects of logic problems. The
skills in this lesson are some of the most universally applicable
in this series.
What Is a Logic Problem?
A traditional logic problem deals with a set of objects (such as
people or houses) that share a set of properties (such as
occupations or colors). Each of the objects usually has a unique
value of each property (ie, there's only one farmer or one blue
house). You are given a set of true statements about the objects
and their properties - from those statements, it is your job to
figure out the value of each property of each object in the
puzzle.
Logic puzzles often provide you with a solution grid to fill in
as you analyze the clues. The grid shows all possible combinations
of all values of all properties. See the exercise below for an
example of what a grid might look like. (Evil puzzle constructors
sometimes conveniently forget to supply you with the solution
grid.)
To solve the puzzle, you must populate the grid with
information. I usually mark the things that I know are true with a
dot, and the things that I know are false with an X. When you have
filled in the entire grid with dots and Xs, you're done.
Types of Clues
A positive clue makes an explicit association between two
parts of the puzzle. For instance, if you had a puzzle that
required you to figure out what color kilts a bunch of clans wore
to a feast, a positive clue for that puzzle might be: The 4 Dogs
wore lovely rubicund kilts the night of the feast. In the
puzzle grid, you can put a dot in the cell where The 4 Dogs and
rubicund kilts meet since you know that to be true. You can also
put an X in every other color kilt for The 4 Dogs, and you can put
an X in every other clan for the rubicund kilts.
A negative clue says explicitly or implicitly that two
parts of the puzzle are not related. For instance, suppose you have
this clue: The Romulan Cabinet Ministers sat across the table
from the clan in the indigo kilts. You know right away that
there is no possible way that the Romulans could be wearing indigo,
so you can put an X in the grid cell that relates the two.
(Besides, everyone already knows that Romulans would never wear
indigo under any circumstances.)
A relative clue expresses some kind of ordered
relationship between puzzle elements. For example: The clan in
indigo kilts ate less than the clan in the fuchsia kilts, but the
clan in the ochre kilts ate less than the clan in the indigo
kilts. You now know:
ochre < indigo < fuchsia
If you have a list of amounts that each clan ate, then you know
right away that fuchsia cannot be the smallest or next smallest
values - if it was, there would be no valid way to fit ochre into
the puzzle. By the same logic, you know that the ochre cannot have
the largest or next largest value, and you know that indigo cannot
have the largest or smallest values. You can put an X in each of
those cells in the grid.
An implied clue gives you indirect information about the
solution, possibly referring to information that isn't in the grid
and may not be explicitly stated in the puzzle. For instance,
consider this clue: The number of pies eaten by the David
Copperfield Worship Society is equal to the third prime number.
You'd have to go figure out or look up the prime numbers, where
you'd discover that the DCWS ate five pies.
A single clue may have multiple types of information. For
instance, consider this clue: The Romulans ate more pies than
the clan in chartreuse kilts, who ate more pies than the clan that
ate 17 pies. You can infer that the Romulans are not wearing
chartreuse, the chartreuse clan didn't eat 17 pies, and the
Romulans didn't eat 17 pies. And the Romulans and the chartreuse
clan didn't eat any fewer than 17 pies. So you can place X marks in
all of those cells in the grid.
Solving Techniques
Use a Pencil
Don't use a pen. You will mess up. You will erase. You will
erase a lot. Trust me on this one; you'll thank me
later.
Start with the Obvious
Take a first pass through all of the clues, and record the facts
that are immediately apparent.
Eliminate All But One
If you find a row or column in a block of your grid that has an
X in every cell except for one, then you know right away that cell
must have a dot in it. Place a dot in that cell, then place
Xs in the rest of the dot's row and column in that block. As you
fill in more of the grid, you may find that doing this once will
trigger a chain of iterations of filling in dots and Xs.
Substitution
Once you know that two parts of the puzzle are related, you can
replace all instances of one of those parts with the other. For
instance, if you know the Romulans are wearing fuchsia kilts, go
through the clues again and replace the phrase "Romulan Cabinet
Ministers" with "clan wearing fuchsia", then see what that tells
you. Then do it again and replace "clan wearing fuchsia" with
"Romulan Cabinet Ministers", again looking for new information.
Inferences
An inference is something you deduce logically to be true, based
upon other true statements. For instance, if your grid says that
the Romulans are wearing fuchsia kilts and the clan wearing fuchsia
ate 23 pies, then you can safely infer that the Romulans ate 23
pies. You can then mark that combination as true in your grid and
carry on. That inference can then be used to draw more inferences
and fill out more of the grid.
Hypothesis Testing
Sometimes when solving logic problems, especially more difficult
ones, you run out of immediately obvious information. You can get
to a point where you haven't finished the grid, yet you don't have
any more information to add. Your only choice is to guess.
Mark your guess in the grid in some special way - use a
different set of symbols (such as circle and square instead of dot
and X, or use a different color pencil (because there's no possible
way you'd be using a pen, right?)). Pick one cell and mark it as
true or false, then see where it leads.
If you end up solving the puzzle, your guess was correct. But
you may end up with a contradiction - for instance, you might find
that your guess implies that both the Romulans and the David
Copperfield Worship Society are wearing rubicund kilts. Since that
can't be true, your guess was wrong ... and you have to backtrack.
Erase all of the marks (circles and squares) you made as a
result of the guess (you didn't use pen for this, did you?).
Since your guess led to a contradiction, you now know that the
opposite of your guess must be true. If your guess was to put an X
in a particular cell and that guess led to a contradiction, you can
safely replace that X with a dot.
Brute Force
This is exactly what it says. Sometimes you get to a point where
you're completely out of ideas, but you only have a small number of
possibilities left. When you started the puzzle, it would have
taken too long to guess all possible combinations of values, but if
you only have a few values left, you might be able to start
guessing.
Brute force guessing means that you test all possible remaining
combinations of values to see if that combination works. If that
combination doesn't work, then move on to the next combination.
(anecdotal side note: I recently got an e-mail from a cacher who
solved one of my puzzles. He was able to solve for all of the
coordinates but one. He then went through all ten possible digits -
zero to nine- until one provided the correct answer on geochecker.
After a lot of work solving the puzzle, there were only ten
possible solutions for the one missing coordinate number. A perfect
example of a good use of brute force guessing. Had he started
guessing when there were six digits remaining unsolved, there would
have been one million possible solutions, a few too many to start
guessing with.)
Check Your Work
There's nothing more infuriating than to finish a logic problem,
look up the solution, and discover that you made a mistake
somewhere. Fortunately, there's an easy way to avoid that
situation.
When you complete the puzzle, go back and check whether the
statements you're given are true. If you've done the puzzle
correctly, every single one should be true. If you find a statement
that isn't true, you've made a mistake along the way. Chances are
that you'll have to erase and start over again (if you did the
puzzle in pencil) or throw the puzzle away (if you did the puzzle
in pen). (Have I mentioned to you that trying to solve one of these
puzzles with a pen is a bad idea?)
Other Types of Logic Problems
Logic problems come in many, many different forms - listed below
are some examples using numbers, letters, or pictures. The solving
techniques described above apply equally well to all of them.
Sudoku
Surely you've heard of this craze that's swept the nation and
the world over the past few years. All of the techniques given
above for traditional logic problems work equally well for
sudoku.
A sudoku isn't a math problem, it's a logic problem. You could
solve a sudoku with letters of the alphabet or funky font symbols
instead of numbers.
To solve a sudoku, I typically go through each square and write
all of the numbers that could be in that square (that aren't
already in the same row, column, and box). Once I get through the
entire puzzle, I go back and look at what I wrote - if a number
appears only once in a row, column, or block, then that number has
to be in that cell.
Word Fill-Ins
A word fill-in isn't a word problem, it's a logic problem that
just so happens to use words. A crossword-like framework is given
along with a list of words - your job is to put them all in the
framework so that each word is used exactly once and each cell is
filled with exactly one letter.
You can solve a fill-in in a language you don't speak - it's
just a matter of finding a way to fit the words into the given
spaces.
Paint By Numbers (aka Nonogram)
A paint by numbers puzzle consists of a grid with various
numbers across the top and down one side. The numbers in each row
or column tell you three things: how many groups of colored cells
there are in that row or column, how long each group is, and in
what order they appear. When you've finished deducing which squares
are colored and which ones are empty, you'll end up with a picture
that looks like it came from an 8-bit black-and-white video
game.
They're great puzzles, but describing them in text doesn't do
justice to them. Wikipedia has a
much better article on them, including animated illustrations to
demonstrate their solution.
Additional Resources
As always, Google has lots of excellent hits for sites,
such as this
one, that can help you solve logic problems.
There are even automated sudoku solvers, but using
one will cause you to suffer a lifetime of guilt and shame.
Exercise 6: Smorgasboard Game
I love board games and card games. I just wish I had more time
to play and more people with whom to play and more games to
play.
Five friends have just come out of a board game shop, and each
has purchased his or her favorite board game. The five friends are:
Cindy, David, Edward, Frank, and Gina. The five games they
purchased are: Risk,
Backgammon,
Ticket
to Ride, The Farming
Game, and Cosmic
Encounter. Each game costs a different amount, and each game
has been rated differently on a scale of 0 (worst) through 10
(best) by the readers of BoardGameGeek.com.
Each of these game sets may include dice, cards, or both (plus a
board, tokens, card trays, ships, soldiers, and other trinkets that
don't matter for the sake of this puzzle). If you don't know which
game is played with what items, the links above should help you
figure that out.
Using the following statements, determine what each friend's
favorite game is, how much it costs, and what its BoardGameGeek.com
rating is.
- David's favorite game costs less than Cosmic Encounter, which
costs less than the game with the 7.6 rating.
- Ticket to Ride is rated higher than the game that costs $25.89,
which is rated higher than Edward's favorite game.
- The friend who paid $22.28 for his game got a real deal since
his game has an 8.1 rating.
- Frank's favorite game isn't the top rated, but it also didn't
cost the most.
- Neither the most expensive game nor the worst rated game are
played with dice.
- Edward's favorite game costs more than The Farming Game, which
costs more than the game with the 7.5 rating.
- The game that costs $25.89 (which is not Gina's favorite) is
not rated 7.5.
- The game that costs $23.50 uses dice but not cards, unlike the
game with the 7.6 rating which uses cards but not dice.
- The person whose favorite game is rated 7.5 is male.
Friend |
Favorite Game |
Game Price |
BGG Rating |
Value
(Price * Rating) |
Cindy |
|
|
|
|
David |
|
|
|
|
Edward |
|
|
|
|
Frank |
|
|
|
|
Gina |
|
|
|
|
Once you've solved this puzzle, you're not quite through. For
each friend's game, multiply the game's price by its rating to get
its "value" (for lack of a better term). Plug the appropriate
values into the following equations to find the final location of
this cache.
North Minutes = Gina's game's value - David's game's
value |
West Minutes = Frank's game's value - Cosmic Encounter's
value |
First
to Find Honors Go To: imrgd
This cache placed by a member of
Dry Creek Geocachers