PROFESSOR LAYTON AND THE UNWOUND FUTURE IS © 2008 LEVEL-5 INC.
FIGURE 18.1 Puzzles in the Professor Layton series create breadcrumbs using the Hint Coins feature.
It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.
—JOSEPH DIE MAISTRE
Puzzles have been an essential part of games since early in the dawn of play. Much like how in today’s video-game market shooters are the genre du jour, at one time the best-selling genre was the text adventure. In text adventures, the primary antagonist was a series of puzzles. Many modern games, such as Portal or Tomb Raider, have environmental puzzles that players are required to solve to continue in the story’s progression. The iOS game The Room was at the top of the sales charts for a number of weeks in 2012, selling over a million units despite being entirely made up of a series of puzzles.
Designer Scott Kim uses this definition of what exactly qualifies as a puzzle:1
1 Scott Kim attributes this definition to Stan Isaacs: www.scottkim.com.previewc40.carrierzone.com/thinkinggames/whatisapuzzle/index.html.
A puzzle is fun and it has a right answer.
The two parts of this definition are important. The first part identifies puzzles as play, whereas the second part distinguishes puzzles from other forms of play, such as games and make-believe. “Fun” is subjective, but having a “right answer” is objective. A puzzle does not have to have exactly one right answer. It can have multiple right answers, but it must have at least one.
Scott Kim’s definition is attractive for its simplicity, but it suffers as a critical examination because of how wide open to interpretation it can be. Let’s take a look at a less succinct but perhaps more instrumental definition.
A puzzle is a type of game that requires the player to use cognitive effort to get from an unsolved state to a solved state, with some limitations:
• THE PUZZLE CANNOT BE TRIVIAL. If a puzzle is given as “Turn on the lights,” and you can just flip a switch to do so, then it requires no cognitive effort to complete and is thus trivial. In the same way, Tic-Tac-Toe is not a puzzle for any adult because the average adult can solve it and tie or win as long as he goes first. This does make the definition of a puzzle subjective for the audience it covers, but making a trivial puzzle and saying it’s for toddlers is a cop-out to avoid having to make a puzzle intellectually interesting.
• THE PUZZLE MUST INVOLVE REASONED EFFORT TO GO FROM UNSOLVED TO SOLVED. If the only way to solve a puzzle is by brute force, it is not a puzzle. “I am thinking of a number from 1 to 64. What is it?” is not a puzzle if the only feedback given is “yes” or “no.” However, if the feedback given allows the player to use reasoned intellectual effort or what Marcel Danesi calls “insight thinking” to figure out the answer, then it may be a puzzle.2
2 Danesi, M. (2002). The Puzzle Instinct: The Meaning of Puzzles in Human Life. Bloomington, IL: Indiana University Press.
• SOLVING THE PUZZLE MUST BE THE SAME AS WINNING THE GAME. Checkers is a “solved” game mathematically, but solving Checkers is a different intellectual exercise than winning the game. The goal of solving Checkers is to create a strategy that always wins, but winning Checkers is about jumping all the opponent’s pieces. In contrast, solving a jigsaw puzzle is the same as completing the jigsaw puzzle.
• THE PUZZLE CAN BE GENERATED RANDOMLY, BUT MUST BE DETERMINISTIC ONCE THE PLAYER ENCOUNTERS IT. A board of Sudoku can be generated randomly (with constraints), but once individual players start the puzzle, every player who makes the same series of moves experiences the puzzle in exactly the same way. Puzzles do not require luck or dexterity to solve. If two players encounter the same Minesweeper board and uncover the same squares in the same order, those players have identical experiences. However, if two players play tennis against each other and make the exact same movements, those players have a decidedly different game experience.
These four features and Scott Kim’s definition help to determine what a puzzle is. Is filing Federal taxes a puzzle? No, because it is not played as a game. Is Tetris a puzzle? No, because it is not deterministic. Are riddles puzzles? Yes, most are, as long as they are not trivial and do not require brute force. Is a whodunit mystery a puzzle? It can be if it meets all the criteria.
If you remember the discussion on formalism and defining games from Chapter 1, you may be surprised at the rigid structure given to what constitutes a puzzle. As always, the point of a definition is to ensure that you are applying the correct heuristics to a problem. The point of defining puzzles as I do here is to ensure that we are using the same language to examine the same elements. There very well may be puzzles that do not meet these criteria yet are still commonly understood to be puzzles. That is fine. The criteria I have been discussing here suit us for this examination of the role and design of puzzles.
Puzzles are a particularly transparent use of examining flow and the fundamental game design directive. A puzzle that is too easy (“What is 1+2?”) is trivial and leads to boredom. A puzzle that is too difficult (“What is the largest prime number that, when using a Caesar number-to-letter cipher, becomes a single English-language word?”) leads to anxiety and frustration. Potential puzzle solvers give up in both cases.
Children often enjoy puzzles and games that are flawed by adult standards because a child’s motor skills and ability to reason are limited. Thus, children achieve flow at a lower level of difficulty. Things that are trivial for adults can be frustrating for children.
The key to a great puzzle, like a great game, is to get the solver to experience the flow state. This means that it must not be so hard that the solver does not feel she has the tools available to solve it, and not so easy that it requires no cognitive effort at all. Marcel Danesi, in the book The Puzzle Instinct says “[T]he aesthetic index of a puzzle, as it may be called, seems to be inversely proportional to the complexity of its solution or to the obviousness of the pattern, trap, or trick it hides. Simply put, the longer and more complicated the answer to a puzzle, or the more obvious it is, the less appealing the puzzle seems to be.” It sounds like he is arguing for flow.
One of the joys of solving a puzzle is being able to take a problem that has many possible solutions and narrow it down to the one correct solution. The set of possible solutions to a puzzle at a given point in the solving process is known as the possibility space of the problem.
As an example, let’s look at a logic puzzle:
Zack, Glo, and Rae are about to eat dinner. They can choose from Steak, Salad, and Chicken. But they have the following preferences:
1. Nobody wants the same food.
2. Glo and Rae both want to eat meat.
3. If Rae wants Chicken, then Zack wants Steak.
Who wants which food?”
When you start this problem, the possibility space is large. [Zack→Steak, Glo→Steak, Rae→Steak] is one possible answer, but at this point there are 27 possible solutions.
Now consider the hints. With Hint #1, no one wants the same food. This means the [Zack→Steak, Glo→Steak, Rae→Steak] answer is impossible. It reduces the possibility space to only six:
[Zack→Steak, Glo→Salad, Rae→Chicken]
[Zack→Steak, Glo→Chicken, Rae→Salad]
[Zack→Salad, Glo→Steak, Rae→Chicken]
[Zack→Salad, Glo→Chicken, Rae→Steak]
[Zack→Chicken, Glo→Salad, Rae→Steak]
[Zack→Chicken, Glo→Steak, Rae→Salad]
With Hint #2, you can eliminate any possibility where Glo or Rae wants salad. This reduces the possibility space to two:
[Zack→Salad, Glo→Steak, Rae→Chicken]
[Zack→Salad, Glo→Chicken, Rae→Steak]
With Hint #3, you can eliminate the first option in the possibility space. If Rae wants Chicken, then Zack has to want Steak. But you already eliminated any possibility that Zack wanted Steak. Therefore you are down to one, the solution:
[Zack→Salad, Glo→Chicken, Rae→Steak]
This reduction of the possibility space is the process of solving the puzzle. However, it’s not always so logically straightforward and transparent.
If the key to making effective puzzles is to get the solver into a flow state, how is that accomplished when different solvers have varied puzzle-solving ability? In the story of “Hansel and Gretel,” the protagonists leave a trail of breadcrumbs in the forest in order to find their way back home. This allows them to travel from crumb to crumb (because each is left in visual range of the last) and eventually the trail leads them out of the forest. If they have no breadcrumbs at all, they have to find their way out of the forest on their own. But the task of following the trail still has challenge because they have to find the next breadcrumb. Puzzle breadcrumbs are just like that: They are a series of clues that, by logical steps, lead the solver from an unsolved puzzle to a solved puzzle.
These breadcrumbs can be extrinsic to the puzzle’s structure or they can be intrinsic.
Extrinsic breadcrumbs are easier to explain. In the Professor Layton series, players are tasked with solving brain teasers. If players get stuck, they can click on a hint button to receive progressively more revealing hints. The first hint is vague, whereas the final hint is generally fairly literal.
In FIGURE 18.1, from Professor Layton and the Unwound Future, the first hint explains how to use the memo function to make notes on the puzzle itself. This is a pretty light hint, because players of any of the previous games are already aware of this function.
PROFESSOR LAYTON AND THE UNWOUND FUTURE IS © 2008 LEVEL-5 INC.
FIGURE 18.1 Puzzles in the Professor Layton series create breadcrumbs using the Hint Coins feature.
If the player is still stuck, the next hint reminds him that just because the puzzle says the answer is next to a table with a red flower does not mean that the correct answer itself is not a table with a red flower. This hint is an important reminder of logical rules that, at the same time, insinuates that the answer has to do with a red flower.
The third hint reminds the player that since the tablecloth has to be a different color than those of all the adjacent tables, he can remove from consideration any tables that have adjacent tables with tablecloths of the same color. If the player does this, he will remove every table but one. In fact, the player doesn’t even need the hint about needing to be adjacent to a red flower.
The final hint (known as the “super hint” in the Layton series) comes right out and says that the answer has a red flower and a red tablecloth.
In this situation, the answer to the puzzle is workable without any hint at all, but each hint gives the player tools to reduce the complexity of the problem.
These extrinsic hints nudge the player in the correct direction and let him reduce the possibility space. Each subsequent hint provides a larger and larger nudge. Someone adept at puzzles may never need a hint, so she stays in flow just by solving the puzzles as is. But someone poor at puzzles may need all the hints to understand the leaps in logic needed to solve them. If only the puzzles with no hints were available, the game may be too frustrating for the poor puzzle-solver. Having the hints progressively reveal more and more brings the player closer and closer to flow. By using all the hints, even the worst puzzle-solvers should be in the realm of understanding the answer.
A different way of providing hints is to make them intrinsic to the puzzle itself. If a puzzle asks for a four-letter word that means “to move around,” there can be a number of valid answers (jogs? walk? jump? roll? stir?) or the solver may just be stumped and unable to think of a single valid answer. However, if the design offers the same question as 4-across in the puzzle in FIGURE 18.2...
FIGURE 18.2 Crossword puzzles provide breadcrumbs by giving hints in the form of nearby clues the player can solve to unlock letters in other clues.
...then the player can uncover a hint for 4-across by first solving 2-down and 3-down. For instance, say that the answer to 2-down is “Too” and the answer to 3-down is “Am.” Since the last letter of 2-down is “O,” the solver knows that the second letter of 4-across is “O.” Since the last letter of 3-down is “M,” she knows the last letter of 4-across is “M.” Now instead of being faced with all the possible four-letter words that mean “to move around,” she is limited to a four-letter word that means to move around and that fits the scheme “_O_M.” Is it possible to solve 4-across now? The player certainly has a better shot than before. The answer for this puzzle is at the end of the section, but the clues to solving it are inside the puzzle itself. This intrinsic hint-giving is used for a number of great puzzles from crosswords, to logic puzzles, to Japanese puzzles such as Sudoku, Kakuro, and Nurikabe.
Note
The Japanese magazine, Nikoli, offers well-designed and culturally independent puzzles in a great variety in both print and on its website: www.nikoli.co.jp/en/puzzles.NikolihelpedpopularizeSudoku.
The extrinsic method of leaving breadcrumbs is a bit obvious to players and may hurt their pride. In most cases, a breadcrumb system is intrinsic to the puzzle, the player is able to enter flow without having to consciously make any adjustments. The same player who balks at taking a hint in Layton likely has no qualms about coming back to a clue after he has revealed some more letters in a crossword.
A type of puzzle that has few or no breadcrumbs is a riddle. In a riddle, the solver generally needs to know the answer to a question that uses a double or hidden meaning. She either knows the answer or does not. The player has no way to work toward the solution in a riddle besides examining the possible hidden meanings on a word-by-word basis. The possibility space is enormous and there are very few cues with which to reduce it. Riddles often require specialized cultural knowledge; as such, riddles rarely cause flow. Either the player knows it, and it becomes trivial, or she cannot reason it, and it becomes impossible. The most effective riddle hides intrinsic breadcrumbs within its wordplay, but most do not offer this affordance and make for generally weak puzzles. Trivia suffers from the same problem. Unless you take steps to mitigate that feature, it can make for a poor puzzle.
In 1990, a Stanford researcher did an experiment on a particularly poor puzzle.3 She had two players and assigned them in pairs as a “tapper” and a “listener.” The tapper was to take a well-known song and tap out the rhythm on a table. The listener had to guess the song. Tappers were also asked how often they thought the listener would correctly identify the song. Tappers estimated that 50 percent of listeners would correctly answer. The actual result: 3 successes out of 120 tries or a 2.5 percent success rate!
3 Newton, E. L. (1990). The Rocky Road from Actions to Intentions. Stanford, CA: Stanford University.
What went wrong? The tapper served as the designer of the puzzle. She imagined a song and because she imagined it, she had difficulty understanding what it would be like to not simultaneously imagine the song along with the tapping. In other words, she had no perspective for the listener’s point of view. The tapper could not imagine hearing only the rhythm without the selected tune.
As puzzle designers, we are guilty of being tappers far too often. It is easy for us to envision the solution to the puzzle we created. It is more difficult for players with no perspective to find the solution.
Unfortunately for puzzle designers, there is no formula for the perfect puzzle. However, there are trends that make for bad puzzles that are easy to avoid.
If the player does not know what his goal is or what he can manipulate in the puzzle, he will ignore any breadcrumb you leave him. If Dorothy does not know that the Wizard is at the end of the Yellow Brick Road and that he is within walking distance, then the road itself is worthless to her. This scenario most often happens in puzzles, not because the designer leaves something out on purpose, but usually because the designer knows exactly what to do because she created the puzzle. By playtesting your puzzles with real players and seeing where they struggle, you can avoid not communicating assumptions. Having playtesters think out their puzzle-solving process aloud also clues you in on areas or cues you have missed that are essential to the player for him to successfully complete the puzzle.
Famed designer Warren Spector has a great rule of thumb for designing puzzles. He says that a key before a locked door is much less interesting than a locked door before a key. This is because seeing a locked door without having a means to open it poses a puzzle to the player: “How do I get in there? Maybe a guard has a key? Or are there key codes on that computer?” A locked door where the key has already been collected poses no alternative futures at all. The player does not have to form a hypothesis.
To be in flow, players have to be able to receive and use feedback to get them away from states of frustration. If the player cannot manipulate the puzzle itself to see if a solution is correct or not, then she finds it difficult to understand how the puzzle works and is not able to make the cognitive leaps that gets her closer and closer to solving the puzzle.
Games that don’t allow players to manipulate their pieces before the games are complete can cause players great frustration. Additionally, puzzles that allow manipulation but don’t allow the player to reset the board when he makes a mistake also work against his need to experiment because he fears moving pieces into the wrong positions or not being able to mentally remember all the puzzle’s moving parts. Finally, puzzles that penalize the player for experimenting (sending him back to earlier levels or giving him a “time out”) also work against getting him into flow.
“I’m thinking of a number from 1 to 100.
Is it 1?
No.
Is it 2?
No.
Is it 3?”
This is not fun at all. Instead of using cognitive skills to solve the problem, the player must simply exhaust the whole possibility space. This is the case for many riddles, memory games, and simple children’s puzzles. Even word searches allow the player to solve the puzzle with a kind of algorithm rather than having to check every letter from every direction.
Memory games are a special instance of brute-force puzzles. If the player has not uncovered a symbol she has already seen, then the puzzle is completely brute force since she has no way to know which tile she should choose over any other tile. Once the player has uncovered a previously seen symbol, the only puzzle element becomes apparent: “Where did I last see that symbol?” In most instances, this is a trivial question. Remember that in flow, the optimal state needs to be more challenging than trivial, but less challenging than frustrating.
Most mazes fill this criterion. In a maze, what decisions are the players facing? Usually, the player’s options are restricted to going left, going forward, or going right. If the player chooses one of those options and ends up at a dead end, he retraces his steps back to his last decision and chooses a different option. Most mazes have many junctions that require decision-making, but that decision-making is almost always blind, and thus it never requires any ingenuity. Although the maze itself looks complicated, it’s usually solved by a simple algorithm (FIGURE 18.3).
Some game puzzles work in a similar manner, surrounding the player with a lot of red herrings even though he can solve the puzzle in a simple and straightforward way. Adding all the red herrings makes the puzzle look substantial, but it doesn’t make the puzzle better.
The absolute worst puzzle is the obvious puzzle. The obvious puzzle may have a lot of features, but it always points to one and only one possible answer (FIGURE 18.4). The best puzzles lead players down a path that eventually teaches them that there are different ways to think about the problem.
Terry Cavanaugh’s now-defunct site freeindiegam.es discovered an ingenious Japanese puzzle game named Jelly no Puzzle (FIGURE 18.5).
IMAGE FROM JELLY NO PUZZLE BY QROSTAR, USED WITH PERMISSION.
FIGURE 18.5 Jelly no Puzzle has some of the finest spatial puzzles ever designed.
Note
Jelly no Puzzle is a free Windows download at http://qrostar.skr.jp/index.cgi?page=jelly&lang=en. There is a partial HTML5 port at http://martine.github.io/jelly.
In the first level, the objective is to move the blocks left or right so that all blocks of a particular color touch (in Figure 18.5, B = blue, G = green, R = red). Gravity causes any block to fall that is not being held up. Blocks of the same color that touch lock together and become rigid.
What makes Jelly no Puzzle fiendishly difficult is the vast number of seemingly possible solutions that turn out to be dead ends. In Figure 18.5 for instance, the player can try moving the top red block down on top of the other red one, but that creates a vertical pillar that forbids her from moving the rightmost blue block. Likewise, she cannot move the leftmost green block to the right or it creates a rigid green vertical pillar, also trapping the blue block to the right.
Eventually, using trial and error and thinking about the setup, the player notices that since the leftmost blue block is locked into place, she must bring the rightmost blue block to it. But this causes new problems! The blue block must get to the left, but a red block and a green block are in the way. Since there is no way to pick up a block, the player must find two “hiding spaces” for those blocks in order to get the blue block free access across the screen. One of those spaces is obvious: the gully in the middle of the screen. Where can the other one be? And what block should she put in the gully? One option is to move the leftmost green block over to the right. This creates a vertical 1×2 green block. However, this prohibits the player from moving the blocks to the right. So should she try to make a horizontal green 1×2 in the gully or put a red with the green down there?
As you can see, this puzzle offers a lot of avenues for thought. Although Jelly no Puzzle is inaccessible for a number of reasons (it is quite difficult, offers little in the way of introduction, and is currently only in Japanese), it does show the depth of thought that well-designed puzzles suggest. The possibility space of Jelly no Puzzle is large, but the designers achieve that large space with a limited number of pieces and a possibility space that can be whittled down by simple mechanics.
“Yeah, that’s a right answer, but not the right answer....”
This is a problem novices often run into when coding puzzles. Instead of checking to see if all the logical solutions cause the win state, they check only the solution they first see as being correct. Thus, the player has to get inside the designer’s head to find out the “true” solution, even if her original solution was valid. Here’s an example:
By clicking on the grid, you can fill in squares. The author here gives you the clue: “18th letter.” The 18TH letter in the alphabet is R, so the player draws an R. Which of the following is correct (FIGURE 18.6)?
To the player, all three would be correct. All are representations of the letter R. But if the designer coded for only one possible set of squares to be the “right” R, then any player not able to read the designer’s mind would be thinking that he solved the puzzle correctly given the clues, but yet would not able to continue. The player should never have to guess at what the “best” right answer is.
Generally, I avoid taxonomies because they often have arbitrarily declared boundaries and are often incomplete. However, Marcel Danesi’s work in The Puzzle Instinct provides numerous good examples of puzzles from which game designers can draw inspiration.
Danesi breaks logic puzzles into four types: deduction puzzles, truth puzzles, deception puzzles, and paradoxes.
Deduction puzzles require the solver to draw conclusions based on a given set of facts. Danesi breaks this type of puzzle into four subtypes: deductions, set-theoretical puzzles, relational puzzles, and inferential puzzles.
“Zack, Gloriana, and Christa just finished a three-player board game in which there were no ties. Zack scored one spot higher than his wife. Christa did not come in second. The oldest player came in third. Zack is older than his wife. In what place did each player finish?”
Representing the possibility space of this type of puzzle as a table is helpful; that way you can mark impossible combinations with an X and certain combinations with an O (TABLE 18.1). For instance, you know that Zack scored higher than his wife; this means it’s impossible for him to have come in third since there would be no lower place for his wife to have scored. Also, you know that Christa did not come in second, so you can put an X in that possibility as well.
Now you do not yet know who Zack’s wife is or who the oldest player is (although you know it is not Zack), so you cannot mark those off. What are the remaining possibilities? Zack could have come in first, so call this Case 1. In this case, Gloriana must be Zack’s wife because his wife would have to come in second, and you know Christa did not come in second.
The other case (Case 2) is that Zack came in second. In this case, his wife would come in third because she scored one rank lower. In Case 2, you don’t know the identity of the wife. However, you do know that the oldest player came in third. Since you know that Zack is older than his wife, his wife cannot be the oldest player, and thus she cannot be the player who came in third, because in Case 2, Zack’s wife must have come in third. Thus, you can throw out Case 2. This eliminates the possibility that Zack was second. The answer must be that Zack was first, Gloriana was second, Christa was third, and Gloriana is Zack’s wife. No other combination meets all the requirements.
Sudoku puzzles are a type of abstract deduction puzzle.
A set-theoretical puzzle focuses on the relationships between groups, often takes a form resembling syllogisms, and can be represented by Venn diagrams. Many probability problems take this form. However, the example Danesi cites from author Lewis Carroll is a perfect example of the form: “What can you conclude from the following statements?”
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.
You can draw a Venn diagram to help solve this (FIGURE 18.7). First you draw a box for the despised and a box for the not despised. Because of criterion number 3, you know that illogical people are despised, so the set of all illogical people must be contained within the despised box. From criterion 1, you know that the set of all babies must be contained within the set of illogical people, so you draw a circle for babies within the circle for the illogical. Statement 2 makes you draw the set of all croc-managers inside the not-despised box.
From this, what additional information not clearly laid out in the problem can you conclude? The set of babies does not overlap anywhere with the set of crocodile handlers, thus no baby can manage crocodiles. This was Carroll’s answer. Additionally, you could answer that no croc-manager is illogical.
Relational puzzles are similar to set-theoretical puzzles in that they focus on the relationship of elements within the puzzle. The archetypical example of this is the surgeon puzzle:
A father and son are in a car crash. The father dies instantly. The son is driven to the hospital for emergency surgery. As the son is about to go under the knife, the entering surgeon exclaims “I cannot operate on him; he is my son!” How can this be?
This riddle trips up solvers because the father and son relationship is established earlier in the problem. The answer is, of course, that the surgeon is the boy’s mother. Before you decry the problem to be an artifact of cultural gender biases, you should know that a study investigating this very puzzle found that factors like an answerer’s education level, exposure to female doctors, or even a battery of tests rating their level of sexism were all irrelevant as to whether that person could solve the riddle.4
4 Wapman, M. (2014) “Riddle Me This.” Retrieved July 24, 2015, from http://mikaelawapman.com/2014/07/03/riddle-me-this/.
Inferential puzzles require the solver to use inference to figure out the solution. These puzzles can be fairly tricky. In their general form, they require the player to throw out an obvious case, which creates a new obvious case, and so on, until only one case remains. The classic example, which has spawned an entire subfield of puzzle research, is the “prisoners and hats problem” or the “King’s Wise Men”:
The King called the three wisest men in the country to his court to decide who would become his new advisor. He placed a hat on each of their heads, such that each wise man could see all the other hats, but none of them could see their own. Each hat was either white or blue. The king gave his word to the wise men that at least one of them was wearing a blue hat—in other words, there could be one, two, or three blue hats, but not zero. The king also announced that the contest would be fair to all three men. The wise men were also forbidden to speak to each other. The king declared that whichever man stood up first and announced (correctly) the color of his own hat would become his new advisor. The wise men sat for a very long time before one stood up and correctly announced the answer. What did he say, and how did he work it out?
Since the king said that there would not be zero blue hats, then the only possible cases are one blue hat, two blue hats, and three blue hats.
If it is the case that there is one blue hat, then there must be two white hats. In this case, whichever person sees two white hats can instantly stand up and say that he has a blue hat. But the other two players do not have that information, each seeing only a white hat and a blue hat, and thus they can make no deductions at all. This scenario goes against the king’s assertion that the contest is fair to everyone. Thus, there cannot be only one blue hat, and all three participants should know that.
If there are two blue hats, then any player who sees a white hat knows that he himself is not wearing a white hat. Why? Because if he sees a white hat, the only way he can be wearing a white hat is if there is only one blue hat. But the previous analysis found that there could not be only one blue hat. However, this scenario gives an advantage to the players who are wearing blue hats. The player wearing the white hat sees only blue hats and cannot know whether his hat is blue or white. Thus, it is unfair to him and it violates the king’s rule.
Therefore, the only way for there to be at least one blue hat and for the contest to be fair to all participants is for there to be three blue hats. In this case, each player can stand up and say they have a blue hat.
Truth puzzles require a special kind of deductive reasoning, which concerns the consistency of true or false statements. Danesi breaks these puzzles into two types—one of which is about identifying which statements are true, and one of which is about determining the class of participants—but the distinction is not important for the scope of this writing. These often take the form of detective stories like the following:
In a distant land, there are two tribes: liars and truth-tellers. Liars always lie and truth-tellers always tell the truth. You meet three people from this land, Mark, Zack, and Glo. Mark says: “All of us are liars.” Glo says, “No, only one of us is a truth-teller.” Zack says nothing. What are their tribes?
If Mark is telling the truth, then he is violating his own statement. Thus, Mark cannot be telling the truth. Mark is from the liars tribe. So if he is a liar, then there is at least one truth-teller.
If Glo is lying, then there cannot be exactly one truth-teller. But since you have already established that Mark is a liar, if Glo is also a liar, then that leaves Zack as the one and only truth-teller, which is contradictory, because you are assuming Glo is a liar yet is also telling the truth. And all three cannot be liars because that would make Mark’s statement true. Since Mark is a liar, this cannot be. Thus, Glo must be a truth-teller.
Since Glo is a truth-teller, there must be exactly one truth-teller. Since that is Glo, you know Zack is a liar.
A deception puzzle exploits the ambiguity of language to attempt to direct the solver down a nonproductive path. These puzzles are problematic for game designers because they rely on designers tricking the players. This is dangerous because it risks the player exiting a flow state when he reaches a dead end. Thus, while interesting as thought experiments, I do not see deception puzzles as workable in games.
Note
Because of the ambiguity of the riddle, though, there could be multiple answers. Maybe he met the man and his entourage at a fork and joined them on the road? They could have possibly been going to St. Ives; the riddle does not say that they were met in opposing directions.
An example of this would be the classic riddle about the man going to St. Ives:
As I was going to St. Ives,
I met a man with seven wives,
Each wife had seven sacks,
Each sack had seven cats,
Each cat had seven kits:
Kits, cats, sacks, and wives,
How many were there going to St. Ives?
The answer here is one, the narrator. The trick is in deceiving the solver to try to solve a puzzle that is not the puzzle in question.
Paradoxes are logical puzzles that lead to logical inconsistencies and thus often do not have solutions. An example would be a game designer saying that “all game designers are liars.” If that were true, then the statement “all game designers are liars” would both be true (by assumption) and false (by implication).
For game designers, this is clearly problematic because we want the player to succeed enough that she enters a flow state. If we design a puzzle we know has no clear solution, then we are eliminating any possible way for the player to reach flow by design.
Paradoxes are interesting thought experiments, but they have little direct applicability to games. That is not to say paradoxes are just curiosities. Zeno’s paradox says that for a runner to finish a race, he must first run half the distance. Then, to finish the second-half, he must run half that distance first. Then, to finish the final quarter, he must first run half that distance, and so on and so on ad infinitum. Thus, since there is always another half to run, a runner must never be able to finish a race. This boggled the minds of many great thinkers for centuries until it became one of the inspirational problems behind the invention of modern calculus.
Of course, myriad other types of puzzles exist outside Danesi’s logic puzzles. I’ll discuss some examples.
Critical path puzzles ask the solver to find a valid path through a process. Mazes, of course, are a kind of critical path puzzle, but they provide flow only to the very young. A better example of a critical path is the river-crossing puzzle in which a farmer needs to cross a river with a wolf, a sheep, and a cabbage but can carry only one item at a time. If he leaves the sheep alone with the cabbage, the cabbage will be eaten. Likewise, if he leaves the sheep alone with the wolf, the sheep will be eaten. The problem can be solved in seven specific steps as I show momentarily, but the difficulty is in understanding that the farmer must bring one of the objects back to the wrong side of the river. This step, which seems to go against the goal, is necessary to solve the problem. Video games often employ this puzzle when they ask players to find a safe route through a maze of enemies.
The farmer (F), the wolf (W), the sheep (S), and the cabbage (C) all start on the wrong side of the river:
FWSC—RIVER
1. The farmer takes the sheep over to the other side of the river. This leaves the wolf with the cabbage, which is OK.
WC—RIVER—FS
2. The farmer returns. This leaves the sheep alone, which is OK.
FWC—RIVER—S
3. The farmer takes one of the other elements over. It doesn’t matter which, so in this example, the farmer takes the wolf. This leaves the cabbage alone, which is OK because cabbage is generally shy anyway.
C—RIVER—FWS
4. The farmer cannot leave the wolf and sheep alone, so he must take one back. Since he just brought the wolf over, taking it back would return the puzzle’s state back to where it was in step 2. Instead, he takes the sheep back. This leaves the wolf alone.
FSC—RIVER—W
5. If, after returning to the wrong side of the river, the farmer turns around and takes the sheep back, this would return the puzzle to the state it was in step 4. If the farmer takes nothing, the sheep will eat the cabbage. Instead, the farmer takes the cabbage. This leaves the sheep alone.
S—RIVER—FWC
6. The farmer can leave the wolf and the cabbage alone (he already has once before on the wrong shore).
FS—RIVER—WC
7. Now, he can take the sheep across.
RIVER—FWSC
Strategy puzzles are a class of puzzle in which the player is expected to devise a strategy to solve the problem instead of simply coming up with a specific answer. A particularly clever example is the light bulbs and prisoners problem. In it, a warden tells a group of 100 prisoners that every day he’ll take one of them, at random, for questioning in a room that contains a chair and a light bulb. The light bulb will start as turned off and no one other than the prisoners can touch it. At the start of questioning the prisoner can either turn the light bulb on or off and it will remain that way until the next prisoner is questioned. At the end of each questioning session, the prisoner is allowed to ask to leave prison. If every prisoner has already been questioned, when a prisoner asks to leave, then all are released from prison. However, if any prisoners have not yet been questioned, all will be executed. The prisoners are given one day to interact in the general population before the questioning starts; when it begins, all prisoners are isolated so they cannot communicate. What strategy should the prisoners use to guarantee their release?
There are many solutions to this puzzle, but one is that one prisoner should be assigned to be the leader. When any prisoner is questioned, if it is her first time being questioned, she turns on the light. Otherwise, she does nothing. When the leader is questioned, if the light is on, he turns it off and makes a note of it. When he turns the light off 99 times, he knows that every other prisoner has been questioned and he can then ask to be free to go. This works because the light is only on if a new prisoner has been questioned. If a new prisoner has been questioned 99 times, then everyone but the leader has been questioned.
Although this puzzle is more complicated than most found in games, games often dynamically form strategy puzzles for players. The game-design danger in this is that once a strategy is deduced, if it can always work, then the player can apply the solution by rote and will no longer be challenged, which kicks the player out of the flow state.
Algebraic puzzles require a careful application of algebra to solve. Although these are easy to create, they must be carefully balanced against a player’s symbolic reasoning skills. These problems are difficult for inducing flow, perhaps because even when they are designed well, they remind players of bad school experiences. The best algebraic puzzles still require that shift in perspective that makes the player say, “Aha!” rather than a simple arrangement of variables.
Here’s an example:
In ten years, Zack will be twice as old as he was 12 years ago. How old is Zack today?
Z + 10 = 2 * (Z - 12)
Z + 10 = 2Z - 24
Z + 34 = 2Z
34 = Z
Physical manipulation puzzles are a wide class that includes tangram puzzles, Rubik’s Cubes, and jigsaw puzzles. They involve spatial reasoning to manipulate a physical structure into a certain form. One example is the class of problems known as matchstick puzzles.
In the matchstick puzzle in FIGURE 18.8, move only three matchsticks in the image of the fish swimming to the left to create one of a fish swimming to the right.
The answer to this one is particularly challenging because of the pattern awareness of the solver. The solution requires turning what was once imagined to be the tail of the fish into a new fish’s body (FIGURE 18.9).
One of the most often-used spatial manipulation puzzles is the Tower of Hanoi.5 In it, a player must move discs on three separate pillars of differing sizes one at a time so that no disc lies on top of a smaller disc, and the end goal is to get all of the discs in the correct order on the last peg. The three-disc, three-peg version is illustrated in FIGURE 18.10.
5 Tower of Hanoi. (2013). Retrieved July 24, 2015, from www.giantbomb.com/tower-ofhanoi/3015-5744.
This puzzle can be solved in just seven steps. Let the smallest disc be named 1, the middle disc be named 2, and the largest disc be named 3.
1. Move 1 to C.
2. Move 2 to B.
3. Move 1 to B.
Now 1 and 2 are on B.
4. Move 3 to C.
5. Move 1 to A.
Now all three are on different pegs with 3 on the correct spot.
6. Move 2 to C.
7. Move 1 to C.
With three discs, the solution is fairly simple. However, the complexity scales exponentially with the number of discs.6 With four discs, the minimum number of steps is 15. With five, 31. Perhaps because of its ease of implementation, Tower of Hanoi has been seen in many games. The website Giant Bomb lists at least 15 known examples. Since a solved puzzle is generally rote and uninteresting to players, you should be able to bank on a good portion of your audience having seen Tower of Hanoi before, making it a poor choice for a puzzle implementation.
6 Petkovic, M. (2009). Famous Puzzles of Great Mathematicians. Providence, RI: American Mathematical Society.
• Puzzles require a particular devotion to the fundamental game design directive because the different skill levels of players require different approaches to presenting the puzzle.
• The possibility space of a puzzle is the set of all possible answers at a particular step in reasoning.
• Breadcrumbs are hints that may be internal to the puzzle’s structure or applied from outside the structure of the puzzle used in order to nudge a solver toward either reducing the possibility space of the puzzle or making the leap in reasoning that will allow them to solve the puzzle.
• Problems in the structure of puzzles largely boil down to either solvers being unable to fairly reduce the possibility space of the puzzle to a single solution or solvers not finding the process of reducing the possibility space engaging.
• Puzzles come in a vast number of forms. Understanding the mental leaps in solving classic puzzles can help when you design your own puzzles.
