Analysis of All Tired Out - Mystery Master
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Mystery Master

Analysis of All Tired Out

Michael Benson


Introduction

This in-depth analysis is for the logic puzzle "All Tired Out". Looking at the logic puzzle, you can see its title, author, and the number of stars it has, along with an image to the right. One star means the puzzle is easy, and five stars means it is very difficult, so three stars is right in the middle. Those things (aka properties) are important, but they won't help you solve a logic puzzle. You need to examine the text of the puzzle with the following questions in mind.

  1. What are the objects in the puzzle? These are the nouns.
  2. What are the relationships between the nouns? These are the verbs and links.
  3. Static clues are expressed as facts, while conditional clues are expressed as rules.

The text of a logic puzzle usually has an introduction, followed by a numbered list of clues. Sometimes, the introduction may hold clues as well. Here are the clues for our puzzle.

  1. there are no clues in the introduction
  2. Three customers were Ethan, the third person in line, and the one who bought tire chains.
  3. Lisa stood just behind the person who bought the jack.
  4. The second person in line was a woman.
  5. The person who bought whitewall tires stood three places ahead of the one who requested a wheel alignment.
  6. Marge wasn't the second of the three women in line.
  7. Jeff was just ahead of the person who had shock absorbers installed.
  8. Grace stood next to at least one man in line.

Nouns

The first thing you need to determine are the nouns in the logic puzzle, and organize the nouns by type (aka group or category). This puzzle has three types of nouns: Order, Customer, and Wanted. Usually it does not matter how the nouns are entered, but there is one important consideration:

You can see that the first type is Order, so the nouns need to be in a logical order (1st, 2nd, 3rd, 4th, 5th). You will understand why when I talk about the links.

The next thing to look for in the clues are the relationships between the nouns. The relationships in this puzzle are "just behind", "three places ahead", "just ahead", and "next to". I will explain in a moment how relationships are given by verbs and links.

Verbs

There are always three verbs for a logic puzzle, the negative verb ("is not"), the possible verb ("may be"), and the positive verb ("is"). Think of the verbs as the marks that go into a grid (explained later). The possible verb is usually represented by a blank, but it also could be a question mark. The negative verb is usually represented by a 'X'. And the positive mark is usually represented by a 'O'. The goal of solving a logic puzzle is to turn the possible verbs into negative and positive verbs.

Links

The links describe all the possible relationships between any two nouns in the clues. The default link for every puzzle is "with". Looking at clue 1, I have the fact "Ethan is not the third person in line." It's obvious that this fact has the negative verb "is not". What's less obvious is that it has the default link "with". Basically, if the relationship is not explicit, then the link is "with".

A critical part of links is understanding what nouns the link refers to. In our puzzle, the links refer to the noun type Order. A puzzle can have some links that refer to one noun type, and have other links that refer to another noun type. Defining links, along with rules, is the hardest part of understanding a logic puzzle.

As noted earlier, we have the relationships "just behind", "three places ahead", "just ahead", and "next to". If you say "A is just behind B", then that is the same as "B is just ahead of A". To minimize the number of links in this puzzle, I will use "just ahead" in place of "just behind", and switch the nouns accordingly.

The "1:1" column tells us if the relationship between two nouns is one-to-one. For the "just ahead" link, this means "A is just ahead of B. So if noun A is 2nd, then noun B is 3rd, and if noun B is 4th, then noun A is 3rd.

Though the first three links in this puzzle are 1-to-1, this is not true for all links. For the link "next to", this means "A is next to B". If noun A is 1st, then noun B must be 2nd. But if noun A is 2nd, then noun B may be 1st or 3rd. And if noun B is 4th, then noun A may be 3rd or 5th. Therefore, the "next to" link is not one-to-one.

Now that nouns, verbs, and links have been discussed, we can move onto facts and rules. A fact is a static statement, while a rule is a conditional statement. All clues must be covered by facts and/or rules.

Facts

The facts tie together the nouns, verbs, and links. A fact is a static statement, and has the form: noun1 verb link noun2. If the link is not apparent, then the link is "with". For example, the first fact for our puzzle is "Ethan was not the third person in line." Using the form I just gave, this means "Ethan is not with 3rd.", or the more English-friendly "Ethan is not 3rd."

The Checkbox column tells us if a particular fact is enabled (checked) or ignored (unchecked). The Hits column tells us how many times a fact is referenced. Please read the help article on facts for more information.

This puzzle has 8 facts, where facts 1, 2, 3, 5, and 6 are type 1 (i.e., the link is With). Fact 7 is type 3 (same noun type for each noun), while facts 4 and 8 are type 4 (both nouns have different types).

Rules

Not all puzzles have them, but this one has rules. A rule is a conditional statement. This puzzle has two rules. Rules, along with links, are the hardest part of understanding a logic puzzle. The first rule is "Marge wasn't the second of the three women in line." There are several conditions we need to consider. Marge can be the first or the last person in line. Marge can be the second or fourth person in line as long as a woman is not first or last. Marge can be the third person in line as long as the other two women are either both before (1st, 2nd) or after (4th, 5th) her. You see what I'm saying?

Now see what you need to consider for the second rule "Grace stood next to at least one man in line."

The Checkbox column tells us if a particular rule is enabled (checked) or ignored (unchecked). The Hits column tells us how many times a rule is referenced. Please read the help article on rules for more information.

Puzzle Summary

To summarize our logic puzzle, this three-star logic puzzle has 3 noun types, 5 nouns per type, 1 link, 8 facts, and 2 rules with triggers. It needs only 75 marks and 3 grids. Before solving this puzzle, let's test the rules.

Test Rules

To test the rules for this puzzle, disable triggers and laws. Here are the rules for this puzzle.

  1. Marge wasn't the second of the three women in line.
  2. Grace stood next to at least one man in line.

To violate rule 1 without violating the second rule or any facts, place Lisa second, Marge third, and Grace fourth. You will see a message indicating you violated the first rule. You will need to undo your marks.

To violate rule 2, place Grace first and Marge second. You will see a message indicating you violated the second rule. You will need to undo your marks.

Question: What should happen if Grace is 1st, but no man is 2nd?

Test Triggers

Besides checking that a mark does not contradict a clue, a rule can also check that if a certain condition exists, then a mark should be entered. A mark that is entered by a rule is called a trigger. Both of the rules for the example puzzle also contain triggers. Please enable triggers.

The first rule checks that if Grace is before the third person in line and Lisa is after the third person in line, then Grace cannot be the third person in line. Though this rule was written for this exact condition, you can write your rules as general as you want. To invoke this trigger, put Grace second and Lisa fourth.

The second rule triggers a mark based on certain conditions as well. If no man can be second in line, then Grace is not first. If no man can be fourth in line, then Grace is not last. And if no man can be second second and fourth, then Grace is not third. Adding triggers to a rule enabled this puzzle to be solved without making any assumptions. Make sure rules, triggers, and laws are all enabled before solving this puzzle.

Examining Two Facts

To solve this puzzle, the following two facts need to be examined together. Note that both links are one-to-one.

  1. The person who bought whitewall tires stood three places ahead of the one who requested a wheel alignment (clue 4).
  2. Jeff was just ahead of the person who had shock absorbers installed (clue 6).

Here are the possible positions for the nouns in both facts.

If Jeff bought the tires, he had to be 1st, which means the shocks are 2nd and the alignment is 4th.

How to Solve a Logic Puzzle

Now that we know what a logic puzzle is, we need to be able to solve it. The two most important tools for solving a puzzle are the Chart and the Grids. But first, we need to talk about the marks that go into them.

Marks

The goal of solving a logic puzzle is to turn the possible verbs (the blanks) into negative ('X') and positive ('O') verbs. For any two nouns A and B of two different noun types, their relationship is based on the verb (mark) assigned to them.

  1. The possible verb tells us that noun A may be with noun B
  2. The negative verb tells us that noun A is not with noun B
  3. The positive verb tells us that noun A is with noun B

Here are the marks when the solution is found for our example logic puzzle.

Chart

The chart displays the nouns that are with another noun. In other words, when a positive verb is assigned to noun A and noun B, this will show up in the chart. When the chart is completely filled in, then we have a solution to the puzzle.

Grids

The grids display the marks for each noun1/noun2 combination. The noun types for our example puzzle are 1) Order, 2) Customer, and 3) Wanted. To display every combination of noun types without duplication, we have the following grids.

  1. Order and Customer
  2. Order and Wanted
  3. Customer and Wanted

For a puzzle with m noun types, there are 1 + 2 + ... + (m-1) or sum(m-1) unique grids. When the grids are completely filled in, then we have a solution to the puzzle.

Conclusion

This is a challenging three-star puzzle because of the rules and triggers.