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Logic Problems

© Copyright 1998, Jim Loy

This article is about solving logic problems; in particular, the Logic Problems as they appear in Dell puzzle magazines. I'll give an example, here:

Painting Competition

Four children (including the Anders child) entered paintings in the County Fair. Determine who painted what, and what ribbon each won (1st=blue, 2nd=white, 3rd=red, 4th=yellow):

1. Bill's painting was not the still life.
2. The four paintings were (in some order) Cindy's, the Clark child's, the still life, and the one that took 2nd place.
3. The Brown child's painting finished ahead of Alice's, which finished ahead of the abstract painting.
4. The portrait finished ahead of Doug's painting (which was not the abstract).
5. Bill's painting finished just ahead of the landscape, which finished just ahead of the Davis child's painting.

The statement at the top is a little helpful, as it mentions the name Anders, which is not mentioned in the numbered clues. Also, it clears up the meanings of the ribbons, but that is not necessary here. Every once in a while, a vital clue will appear in the statement at the top. So, you should read that statement carefully. An example of that would be if that statement said that two of the children shared one of the prizes.

In the Dell magazines, we often see a diagram like this:

Let's look at the first clue, "Bill's painting was not the still life." We can put that into the diagram (above). Now, we are through with that clue. It is a simple enough clue that we never need to come back to it, it can give us no further information.

I'll go through the other four clues and put them into the diagram (below). Let's discuss each of the last four clues:

2. "The four paintings were (in some order) Cindy's, the Clark child's, the still life, and the one that took 2nd place." This clue tells us that Cindy was not named Clark, her painting was not the still life and did not take 2nd. And Clark's painting was not the still life and did not take 2nd. And, the still life did not take 2nd. And, we are not through with this clue yet. We still have not used the info that all four paintings are mentioned in this clue. That fact is not so easy to put into our diagram, just yet.

3. "The Brown child's painting finished ahead of Alice's, which finished ahead of the abstract painting." Brown is not Alice, and neither one painted the abstract. Besides that, we see that Brown's painting did not take 3rd or 4th (the clue gives two paintings which placed below this painting). And, Alice's painting did not take 1st or 4th. And the abstract did not take 1st or 2nd.

4. "The portrait finished ahead of Doug's painting (which was not the abstract)." Doug's painting was not the portrait nor the abstract. The abstract did not take 4th, and Doug's painting did not take 1st.

5. "Bill's painting finished just ahead of the landscape, which finished just ahead of the Davis child's painting." Bill's painting was not the landscape, and he wasn't named Davis. And, the landscape was not painted by Davis. Also, Bill was not 3rd or 4th. The landscape was not 1st or 4th. Davis was not 1st or 2nd.

Here they are, in the diagram:

We have not finished with any of these four clues, as they will give further information about the prizes that the paintings received.

We can often fill in more of the diagram, just by examining it. In fact, we can sometimes solve the entire problem, just by looking at the diagram. For example, pretend that we know that Clark did not take first place. In that case, we can see from the diagram that Anders and Brown took 1st & 2nd (in some order). Also, pretend that we knew that Doug took 4th, and the still life took 4th. Then we would know that Doug painted the still life. That was all hypothetical. Let's return to the actual problem.

In the actual problem, we see that Bill took 1st or 2nd, and the abstract took 3rd or 4th. So, Bill did not paint the abstract. Putting an X there, shows that Bill painted the portrait. I put an O where Bill and the portrait intersect. That gives us some more X's, because nobody else painted the portrait. Also, we know that Bill is not Davis, so Davis did not paint the portrait. And the portrait took 1st or 2nd.

We can go over the last four clues, and try to reapply each of them in a simple way (as we did originally). And we get no further information, this time.

It may be time to deal with clue #5 in a more detailed manner. We have two scenarios:

1. Bill (portrait) took 1st, landscape 2nd, Davis 3rd
2. Bill (portrait) 2nd, landscape 3rd, Davis 4th

We can draw two diagrams, one for each scenario. I won't do that. Instead, see if you follow this reasoning: In the second scenario, we deduce that Davis painted the abstract (the abstract is 3rd or 4th, and it is not the landscape). Bill taking 2nd, makes Alice 3rd, which leaves 4th for Doug. That means that Doug painted the abstract. But, we already know that Doug did not paint the abstract (clue #4). So, that eliminates the second scenario.

Incidentally, the 4th and 5th clues have been milked dry. They is no longer of use to us, as we can get no more information from them.

So, we have Bill (portrait) 1st, landscape 2nd, Davis 3rd. Furthermore, since Brown took 1st or 2nd, he did either the portrait or the landscape. Clark did not paint the landscape (He/she did not take 2nd). Cindy is either abstract or still life (they all took 3rd or 4th). But, Cindy did not do the still life, so she did the abstract. We can put these into the diagram (above).

We now have 3 scenarios, Doug took 2nd, 3rd, or 4th:

1. Bill, portrait
2. Doug, landscape
3. Alice Davis, still life
4. Cindy, abstract

1. Bill Brown, portrait
2. Alice Anders, landscape
3. Doug Davis, still life
4. Cindy Clark, abstract

1. Bill Brown, portrait
2. Alice Anders, landscape
3. Cindy Davis, abstract
4. Doug Clark, still life

We deduced earlier that Clark did not do the still life. So, the third scenario is out. And Cindy is not Clark. So, the 2nd scenario is out. That leaves the first scenario:

1. Bill, portrait
2. Doug, landscape
3. Alice Davis, still life
4. Cindy, abstract

We can tidy it up now. Clark is not Cindy nor did he/she do a landscape, so Bill is Clark. Brown is 1st or 2nd, so Brown is Doug:

1. Bill Clark, portrait
2. Doug Brown, landscape
3. Alice Davis, still life
4. Cindy Anders, abstract

Did I make a mistake? I have to check this with the clues. Yup, looks OK.

Now, the explanation in the Dell magazine does not mention the diagram. And it doesn't mention scenarios. And it is usually quite a bit shorter than the above. The diagrams and scenarios are just a tool, so you can keep track of things, and sometimes do some extra eliminating. Practice will enable you to do this more smoothly, without much wasted effort.

Note: In Dell magazines, they sometimes fail to provide a diagram like those above. They say that they found that some other diagram was of more help. Regardless of what they say, a diagram like those above is always helpful.

Addendum:

Instead of the above handy diagrams, the book Math is Fun by Joseph Derazia uses diagrams like the one that I drew on the left. You can put your X's and O's in the small parallelograms, where the varous sets of lines intersect. I prefer the above diagrams. You may prefer these instead.

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