"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?"
Susan didn't know.
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!
EVENTS: | 1 | 2 | 3 | 4 | 5 | SCORE |
---|---|---|---|---|---|---|
John Handley | 2 | 5 | 5 | 5 | 5 | 22 |
Sherando | 5 | 1 | 1 | 1 | 1 | 9 |
James Wood | 1 | 2 | 2 | 2 | 2 | 9 |
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 |