Gaming

How logic puzzles can help you become a better problem solver

I have to admit that I am an inveterate puzzler. I love crossword puzzles, acrostics, and cryptograms. But I am increasingly intrigued by problems of logic. For one thing, they teach you how to become a more attentive listener or reader to pick up nuances of language that can provide invaluable clues to your solution. On the other hand, they teach the step-by-step process of information processing. These are skills that are valuable for almost all reasoning situations.

To illustrate the process, the following is a problem I have composed that will take you step by step from recognizing the essentials to the final solution. I haven’t provided an array, but if you’re familiar with the technique, you can build one yourself from the description.

I call the problem Wilson’s Primary Subject Olympics. Ed, Bob, Susan, Anne, and Wayne (in no particular order) are five bright sixth graders who attend the Wilson School. They recently competed in the school’s annual competition. The subjects were: reading, writing, arithmetic, art and poetry, and gymnastics. For scoring purposes, the winner of each subject received four points; second place three; third, two; fourth, one; and fifth, zero. At the end of the competition, the director said that it was the closest competition in history. Each competitor was one point away from the next best ranked. Each competitor scored at least a four. From the following clues, determine the score and the order of arrival of each of the students.[NBYoumaywanttoconstructtwodifferenttablesonewiththenamesofthestudentsandthesubjecttheothersimplythesubjectandtotalnumberofpointsscoredineeachsubject[NBPuedequedeseeconstruirdostablasdiferentesunaconlosnombresdelosestudiantesylamaterialaotrasimplementelamateriayelnúmerototaldepuntosobtenidosencadamateria[NBYoumaywanttoconstructtwodifferenttablesonewiththenamesofthestudentsandthesubjecttheothersimplythesubjectandtotalnumberofpointsscoredineachsubject

(1) Only one student got 5 different scores. Bob scored four more points than the last seed. The student in second place had no zeros.

(2) Wayne, who did not finish fourth or fifth, earned a four in gym and scored higher than (Bob) in arithmetic.

(3) Susan finished fourth in two subjects but finished first in arithmetic.

(4) Bob’s best subject was writing and his worst was gym, where he got a zero.

(5) Anne got identical scores in writing and gym and a four in reading. She did not finish last.

(6) Ed, Bob, Susan, and Anne finished 1-4 in that order in art and poetry.

(7) Ed finished fourth in arithmetic, but second in gymnastics. He also had identical scores in reading and writing.

(8) Third place earned a one in writing; fourth place finisher a zero in arithmetic.

From the above we have more than enough information to solve the problem. For one thing, we know that our students finished one point ahead or one point behind their competitors. If we add the total number of possible points for each category we get 4 plus 3 plus 2 plus 1 or a total of ten. Since we have five categories with ten points in each, we have a total of 50 points. Since each student finished within a point of each other, the scores will be consecutive whole numbers, like 11,12,13,14,15, for example. If you want, you can sit back and experiment to see which five integers add up to fifty, but there is a simple algebraic formula that will give you the number. The smallest number will be x. The next number will be x+1, then x+2, X+3 and x+4. I wrote x + (x+1) + (x+2) + (x+3) + (x+4) = 50. 5x+10 = 50. 5x = 40 so x equals 8. The five integers are 8, 9, 10, 11, 12. Now let’s move on to the clues.

Clue number one tells us that Bob had 4 more points than the last qualifier. The last place competitor scored 8 points. Bob must have scored a total of twelve, which means he finished first.

From clue number two, we know that Wayne did not finish 4 or 5. Since Bob finished first, we know that Wayne must have finished second or third and will have a total of 11 or 10 points.

Clue number six gives us four true scores. Ed got a 4 in art and poetry, Susan 3, Bob 2, and Anne 1. By inference, Wayne got a zero. Since clue one tells us that the second-place finisher had no zeros, Wayne must have finished third with a total of ten points. We also know that he is the student who got five different scores because 4+3+2+1+0 equals 10 and clue one tells us that only the student got five different scores.

Clue four tells us that Bob’s best subject was writing. This means that he only got a four and it was written. He scored 0 points in gymnastics. Since he got a total of 12 points, he should have got a total of 8 points in Reading, Arithmetic, and Art and Poetry. The clue also tells us that he got the same score in two subjects. He only got a 4, so he must have gotten a 2 or 3 in the remaining subjects. The only numbers that add up to eight are 3, 3 and 2. From clue 2 we know that Wayne got a 3 in arithmetic and this was a higher score than Bob. We now know Bob’s position and all of his scores, namely Reading 3, Writing 4, Arithmetic 2, Art and Poetry 3, Gymnastics 0.

Clue five tells us that Anne got a four in reading and didn’t finish last. Bob finished first, Wayne third, and Anne second or fourth. By the process of elimination, either Susan or Ed must have finished last. Remember that the last classified obtained a total of 8 points. It has been identified that Susan has seven points so far and has at least one other for her second third place.

Clue eight says that the third place finisher, (Wayne), got a 1 on writing. We now know 8 of Wayne’s total of 10 points in four subjects. This means that he must have gotten a Reading score of 2, the only remaining blank. The rest of the clue tells us that the fourth-place finisher got a zero in arithmetic. Susan got a 4, which means either Ed or Ann finished fourth.

Clue nine indicates that Ed got the same score in reading and writing. The only scores he could have gotten were ones or zeros. We know that Anne finished fourth, so Ed finished fifth with a total of 8 points. We can already account for 7 of them, so he got a total of 1 point in three subjects. Since he got the same score in reading and writing, these must be zeros and his only score would be in arithmetic. Through the process of elimination, we now know that Susan finished second with a total of 11 points. Also, Ed, Bob, Anne, and Wayne get 9 out of 10 points in reading, which means Susan got 1.

In the arithmetic column we have now tallied up the ten points without Anne’s score. Therefore, her score must be zero. We are almost done.

Clue 5 says that Anne got identical scores in writing and gym. At this point she has a total of 5 points. Identical scores must be 2s. That leaves him two back numbers to replace Susan. She got a 3 in writing and a 1 in gymnastics.

We finally have the classification and the scores. Bob, first, reading 3, writing 4, arithmetic 2, art and poetry 3, and gymnastics 0.

Susan, second, reading 1, writing 3, arithmetic 4, art and poetry 2, and gymnastics 1. Wayne is third with 2 in reading, 1 in writing, 3 in arithmetic, zero in art and poetry, and 4 in gymnastics. Anne, who placed fourth, has the following: 4 in reading, 2 in writing, zero in arithmetic, one in art and poetry, and 2 in gymnastics. Last but not least, Ed got a zero in reading and writing, a 1 in arithmetic. 4 in Art and Poetry and 3 in gymnastics.

Taking a step-by-step approach, we start by finding the total number of available track points over the number of points earned. After that, we determine that Bob finished first with 12 points. Every clue from then on provided more information, either by statement or by inference. What at first appears to be an unintelligible mess gives way to logical analysis. If you enjoyed it, get yourself a logic book and have fun!

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