Answer to May 5, 2003 Problem
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"We won the shot-put all right," she said, "but John Handley High won the track meet. They had a final score of 22. We finished with 9. So did James Wood High." "How were the events scored?" her father asked. "I don't remember exactly ," Susan replied, "but there was a certain number of points for the winner of each event, a smaller number for second place and a still smaller number for third place. The numbers were the same for all events." (By "number" Susan of course meant a positive integer.) "How many events were there altogether?" "Gosh, I don't know, Dad. All I watched was the shot-put." "Was there a high jump?" asked Susan's brother. Susan nodded.
"Who won it?" Incredible as it may seem, this last question can be answered with only the information given. Which school won the high jump AND how many points were awarded for first place in each event? Explain. EXTRA CREDIT: Even more incredible, you can actually reconstruct the entire track meet from the comments above! You can determine the number of events and the place in which each school finished in each event. You can also determine the score for each event and the total scores for each school! |
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Solution to Problem: John Handley High won the high jump event in the track meet involving the three schools. Five points were awarded for each first place Three different positive integers provide points for first, second and third place in each event. The integer for first place must be at least 3. We know there are at least two events in the track meet, and that Sherando High (which won the shot-put) had a final score of 9, so the integer for first place cannot be more than 8. Can it be 8 ? N0, because then only two events could take place and there is no way that John Handley High could build up a total of 22 points. It cannot be 7 because this permits no more than three events, and three are still not sufficient to enable John Handley High to reach a score of 22. Slightly more involved arguments eliminate 6, 4 and 3 as the integer for first place. Only 5 remains as a possibility. p If 5 is the value for first place, there must be at least five events in the meet. (Fewer events are not sufficient to give John Handley a total of 22, and more than five would raise Sherando's total to more than 9.) Sherando scored 5 for the shot-put, so its four other scores must be 1. John Handley can now reach 22 in only two ways: 4, 5, 5, 5, 3 or 2, 5, 5, 5, 5. The first is eliminated because it gives James Wood a score of 17, and we know that this score is 9. The remaining possibility gives James Wood a correct final tally, so we have the unique reconstruction of the scoring shown in the following table:
John Handley High won all events except the shot-put, consequently it must have won the high jump. This problem can be solved without any calculation whatever. The necessary clue is in the last paragraph. The solution to the integer equations must indicate without ambiguity which school won the high jump. This can only be done if one school has won all the events, not counting the shot-put; otherwise the problem could not be solved with the information given, even after calculating the scoring and number of events. Since the school that won the shot-put was not the over-all winner, it is obvious that the over-all winner won the remaining events. Hence without calculation it can be said that John Handley High won the high jump. |
| 1. Jeffrey Gaither ** | Winchester, Virginia |
| 2. Richard Johnson ** | La Jolla, California |
| 3. David and Judy Dixon ** | Bennettsville, South Carolina |
| 4. Rick Jones ** | Kennett Square, Pennsylvania |
| 5. Bob Hearn ** | Winchester, Virginia |
| 6. Misty Carlisle ** | Winchester, Virginia |
| 7. Kathleen Altemose ** | Winchester, Virginia |
| 8. Michael Rodriguez ** | Great Falls, Montana |
| 9. William Funk ** | San Antonio, Texas |
| 10. Peggah Sadeghzadeh ** | Winchester, Virginia |
| 11. Dave Smith ** | Toledo, Ohio |
| 12. Walt Arrison ** | Philadelphia, Pennsylvania |