I have to admit that I am a confirmed puzzle-head. I love crosswords, acrostics, and cryptograms. But I am becoming ever more intrigued by logic problems. For one thing they teach you how to become a more attentive listener or reader to catch the nuances of language that can provide invaluable clues to their solution. For another, they teach the step-to-step process of processing information. These are skills that are valuable for nearly 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 essential elements to the final solution. I have not provided a matrix but if you are familiar with the technique you can construct one yourself from the description.
I call the problem The Wilson Elementary Subject Olympics. Ed, Bob, Susan, Anne and Wayne (in no particular order) are five bright 6th-Grade students attending Wilson School. They recently competed in the school's annual competition. The subjects were: reading, writing, arithmetic, art & poetry, and gym. For scoring purposes, the winner in each subject was awarded four points; the second place three; third, two; fourth, one; and fifth, zero. At the end of the competition the principal said that it was the closest competition ever. Each competitor was within one point of the next highest finisher. Every competitor got at least one four. From the following clues, determine the score and order of finish for each of the students. [N.B. You may want to construct two different tables, one with the names of the students and the subject, the other simply the subject and total number of points scored in each subject.
(1) Only one student got 5 different scores. Bob scored four more points than the last-place finisher. The student in second place had no zeroes.
(2) Wayne, who did not finish fourth or fifth, got a four in gym and got a higher score than (Bob) in arithmetic.
(3) Susan finished in third place in two subjects but she 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 through 4 in that order in art and poetry.
(7) Ed finished fourth in arithmetic, but second in gym. He also got identical scores in reading and writing.
(8) The third place finisher got a one in writing; the 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 our students finished within a point ahead or a point behind their competitors. If we add up 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 integers such as 11,12,13,14,15 for example. If you want to, you can sit down and experiment to see which five integers add up to fifty, but there is a simple algebraic formula that will give the number. The smallest number will be x. The next number will be x+1, then x+2, X+3 and x+4. Written out 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 turn to the clues.
Clue number one tells us that Bob had 4 more points than the last place finisher. The last place competitor scored 8 points. Bob must have scored a total of twelve, which means he finished in first place.
From Clue number two we know that Wayne did not finish 4th or 5th. Since Bob finished first we know Wayne must hsve finished 2nd or third and will have a total of 11 or 10 points.
Clue number six gives us four actual scores. Ed got a 4 in art and poetry, Susan 3, Bob 2, and Anne 1. By inference, Wayne got the zero. Since clue one tells us that the second place finisher had no zeroes, Wayne must have finished in third place with a total of ten points. We also know that he is the student who received five different scores because 4+3+2+1+0 equals 10 and clue one tells us that only student had five different scores.
Clue four tells us that Bob's best subject was writing. This means he got one four only and it was in writing. He scored 0 points in gym. Since he scored a total of 12 points, he must have gotten a total of 8 points in Reading, Arithmetic and Art& Poetry. The clue also tells us that he got the same score in two subjects. He only got one 4, so he must have gotten 2s or 3s 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 standing and all of his scores, viz, Reading 3, Writing 4, Arithmetic 2, Art and Poetry 3, Gym 0.
Clue five tells us that Anne got the four in reading and that she didn't finish last. Bob finished first, Wayne 3rd and Anne 2nd, or 4th. By the process of elimination, either Susan or Ed must have finished in last place. Please remember that the last place finisher scored a total of 8 points. Susan has been identified as having seven points so far and has at least another for her second third place finish.
Clue eight says that the third place finisher, (Wayne), got a 1 in writing We now know 8 of Wayne's total of 10 points in four subjects. This means he must have gotten a score of 2 in Reading, 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 that Ed or Ann finished in Fourth place.
Clue nine indicates that Ed got the same score in reading and writing. The only scores he could have got were ones or zeros. We know that Anne finished in fourth place, so Ed finished fifth with a total of 8 points. We already can account for 7 of them so he scored a total of 1 point in three subjects. Since he got the same score in reading and writing, these must be zeroes and his one point would be in arithmetic. By the process of elimination, we now know that Susan finished in second place with a total of 11 points. Furthermore Ed, Bob, Anne and Wayne account for 9 of the 10 points in reading, meaning Susan scored 1.
In the arithmetic column we have now accounted for all ten points without Anne's score. Thus, her score must be zero. We're almost finished.
Clue 5 reads that Anne got identical scores in writing and in gym. At this point she has a total of 5 points. The identical scores must be 2s. That leaves he last two numbers to fill in for Susan. She got a 3 in writing and a 1 in Gym.
At long last we have the standings and the scores. Bob, first, reading 3, writing 4, arithmetic 2, Art and Poetry 3 and Gym 0.
Susan, second, reading 1, writing 3, arithmetic 4, Art and Poetry 2 and Gym 1. Wayne is third with 2 in reading, 1 in writing, 3 in arithmetic, zero in art and poetry and 4 in gym. Anne, who came in fourth, has the following: 4 in reading, 2 in writing, zero in arithmetic one in Art and Poetry and 2 in gym. Last but not least Ed got a zero in Reading and writing, 1 in arithmetic. 4 in Art and Poetry and 3 in gym.
From a step by step approach, we began by finding the total number of points available from the clue about the numbers of points scored. After that we determined Bob finished first with 12 points. Each clue from that point on provided more information either by statement or inference. What seems at first to be an unintelligible mess gives way to logical analysis. If you enjoyed it, get yourself a logic book and have a ball!
