It's a Puzzlement

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Tricky Track
by John P. Robertson

Three high schools—Washington, Lincoln, and Roosevelt—competed in a track meet. Each school entered one man, and one only in each event. Susan, a student at Lincoln High, sat in the bleachers to cheer her boyfriend, the school's shot put champion. When Susan returned home later in the day, her father asked how her school had done.

"We won the shot put all right," she said, "but Washington High won the track meet. They had a final score of 22. We finished with 9. So did Roosevelt 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 all together?"

"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 might seem, this last question can be answered with only the information given. Which school won the high jump?

This puzzle is from Martin Gardner's New Mathematical Diversions from Scientific American (University of Chicago Press, 1966) and is used with permission.

Five-Card Magic

This was a trick that required two magicians working together. An audience member picks any five cards from a standard 52-card deck and hands them to one magician. This magician looks through the five cards, picks one, hands it back to the audience member, arranges the remaining four into a neat pile, and places the pile face down on a table. The second magician looks at these four cards, and announces the suit and denomination of the fifth card.

Jon Evans' solution begins by assigning numbers to each card so Ace is 1, Jack is 11, Queen is 12, King is 13, and the others are assigned their face value. Observe that among the five cards selected by the audience member, (at least) two must be of the same suit. From any such pair the first magician selects the one that, using arithmetic modulo 13 (clock arithmetic), is n greater than the other, where n is between 1 and 6. For example if an Ace and Jack were of the same suit, the first magician would select the Ace because 11 + 3 is 1 modulo 13 (and so n is 3). This card is handed back to the audience member, and the other is put on the top of the pile of four. Before starting the trick, the magicians agree on an ordering of cards in the deck, perhaps using the values assigned above, and with suits ordered Diamonds, Clubs, Hearts, Spades. The remaining three cards are then High (H), Medium (M), and Low (L), according to this ordering. The six permutations of H, M, and L can be used to encode the number n. For instance, the magicians might agree that LMH, LHM, MLH, MHL, HLM, HML correspond to 1 through 6, respectively. These three cards, in the proper order, become the bottom three cards in the deck of four. The second magician looks at the top card to get the suit, and the remaining three to determine n, which they add to the value of the top card to get the value of the card the audience member holds.

Bob Gardner reports that he and a son have amazed his son's friends with this trick.

Ed Bouchie, Bob Conger, Kevin Conley, John Hinton, Stuart Klugman, Ken Leising, Charles McClenahan, Christopher Mosbo, Tim Mosler, Richard W. Nichols, John Noble, Dave Oakden, Walter Wright, and David Uhland also submitted solutions.