## How to Construct Logic ProblemsWilliam T. Pelletier

A logic problem consists of a jumble of facts and relationships from which one must deduce an organized structure through the application of logical deduction and inference. To construct a logic problem, one must develop a correct set of statements (clues) from which the organized structure (answer) may be deduced. It is more difficult than it may at first appear to develop a set of clues which uniquely determine the answer, without revealing the answer too obviously. It also takes far longer to construct a logic problem than it takes to solve it.

I do not know of any published material on how to construct logic problems, but I can outline how I do it. I start with the organized structure (the answer) and work backwards to assemble a collection of statements which uniquely determine the given structure. Sherlock Holmes describes this type of process in a discussion with his friend Dr. Watson in A Study in Scarlet, 1887:

 Holmes: In solving a problem of this sort, the grand thing is to be able to reason backward. That is a very useful accomplishment, and a very easy one, but people do not practise it much. In the everyday affairs of life it is more useful to reason forward, and so the other comes to be neglected. There are fifty who can reason synthetically for one who can reason analytically. Watson: I confess that I do not quite follow you. Holmes: I hardly expected that you would. Let me see if I can make it clearer. Most people, if you describe a train of events to them will tell you what the result would be. They can put those events together in their minds, and argue from them that something will come to pass. There are few people, however, who, if you told them a result, would be able to evolve from their own inner consciousness what the steps were which led up to that result. This power is what I mean when I talk of reasoning backward, or analytically.

Below I describe the process in general and illustrate it by constructing Football Fans, the Logic Problem of the Month for October 1997. I have deliberately chosen to illustrate the basic construction process with an easy puzzle in order to avoid obscuring the basics with the complexity of a hard example. (The solution to the Football Fans logic problem appears below. One should probably work the problem first before studying the construction process.)

### Logic Problem Construction Process:

1. Decide on the general situation for the puzzle. For example, Football Fans will be about 4 Detroit Lions football fans who attend the 1998 Super Bowl to cheer the Lions on to victory.
2. Decide on the attributes to organize for the solution. Start small and add attributes later if you want to increase the complexity level. For our example, the attributes will be the names of the 4 fans, the colors of the jerseys they wore to the game, the quarter each one left his seat to visit the concession stand, the snack each brought back to eat, and the order they sat in their seats.
3. Decide on the "answer" to the logic problem. Here is the answer with which we will begin.

SEAT 1SEAT 2SEAT 3SEAT 4
Name Joseph Daniel Stephen Noah
Shirt Green Red Blue Yellow
Quarter 1 4 2 3
Snack Popcorn Pizza Hot Dog Nachos

4. Use a table and/or a grid chart to help you construct clues which eliminate every possible solution except your answer. This is an iterative process in which you may change your clues and "answer" many times. Ideas will come as you work on this part. Experience working LPs is a big help here as an experienced solver is familiar with many different types of clues that can be used. I will work through this clue development process below for this problem.
5. Write up the Introduction and Clues. Clues may be hidden in the Introduction. The Introduction should clearly specify the objective of the problem. Try to be concise and clear; avoid ambiguity. Decide on the order in which to list the clues; this should usually be different from the order in which they were developed in order to disguise the constructor's thought process. If possible, have several friends proof-read and solve the problem to check for mistakes and insure clarity. Listen to their feedback.
6. Test your problem by checking to see that the answer satisfies the introduction and all the clues. Work it from scratch to verify that the answer is unique and can be determined from the clues.

### Clue Development Process:

1. Start by arbitrarily selecting a clue which describes a relationship between the attributes in the chart.
The blue-shirted fan sat next to both the red-shirted and yellow-shirted fans.
This clue narrows the arrangement of colors to 4 possibilities:
GRBY, GYBR, RBYG, and YBRG.
Future clues should work toward eliminating the 3 incorrect possibilities and linking relationships with other attributes to fill in the remaining slots.
2. Select another relationship.
Daniel sat next to both the fan who bought a hot dog and the one who visited the concession stand in the first quarter.
Similarly, this clue gives 4 possible fan orderings:
Q1, Dan, HD, ?
?, Q1, Dan, HD
HD, Dan, Q1, ?
?, HD, Dan, Q1.
Each of the 4 possibilities from Clue 1 may be paired with one of the 4 from Clue 2 for a total of 16 possibilities at this point.
3. Link these 2 relationships together:
Daniel wore the red shirt.
This clue (Dan=R) eliminates all but the following 4 orderings for the fans:
Q1=G, Dan=R, HD=B, ?=Y
?=Y, Q1=B, Dan=R, HD=G
?=Y, HD=B, Dan=R, Q1=G
HD=G, Dan=R, Q1=B, ?=Y
4. Add another clue to eliminate some of these orderings and tie in another attribute:
The green-shirted fan sat to the left of Stephen.
This clue eliminates the 2nd and 3rd orderings resulting from the previous clue.
5. Continue eliminating possibilities:
The 4 fans are Noah, the red-shirted one, the one who got a snack during the first quarter, and the one who bought a hot dog.
From this one concludes that the ordering of the 4 fans must be:
Joseph=G, Daniel=R, Stephen=B, Noah=Y
with the 1st & 3rd fans also being Q1 & HD in some order.
Clue 3 above can now be dropped because it is a consequence of this clue and Clue 2.
6. Now add a few clues to determine Q1 & HD and tie in the remaining attributes.
The fan who bought a snack during the 2nd quarter sat between the fans who bought popcorn and nachos.
This forces the 1st fan to be Q1 and the 3rd to be HD.
7. One fan got pizza in the 4th quarter.
8. Stephen did not sit next to the popcorn muncher.

Obviously this is not the only set of clues which could be developed for this problem. This just illustrates one of many possible paths which could be chosen. At this stage one would normally study the clues looking for duplication (like Clue 3 above) and for ways to better hide the answer. If one wants a more complex puzzle, one can now add more fans and more attributes and continue adding clues to link the new attributes with the previously determined structure. The complexity one can build into a logic problem is limited only by the constructor's creativity and time.

WTP 10/10/97

Do you have good tips on constructing LPs which you think should be added to this discussion? If so, e-mail them to me for possible inclusion.