The stage magician Henry Howduzzi was demonstrating his new trick to his assistant Doris. "It uses a special pack of cards," he explained, "numbered consecutively from 1 to 15. I lay them out like this—except that in the real trick they are face down."
"This arrangement is what makes the trick work. Each card is the difference between the two cards immediately below it, to its right and left. Except for the cards in the bottom row, of course."
His assistant looked puzzled.
"For instance, Doris, the top card is 1, and those below it are 2 and 3, with a difference of 1. And the cards below 7 are 8 and 15, which have a difference of 7."
Doris still looked puzzled.
"You must take the smaller number from the bigger one to get the difference," Howduzzi said. "No negative numbers."
"But Henry," Doris said. "The cards below 11 are 15 and 10, whose difference is 5, not 11. And 13-12 is 1, not 14. And—"
"I am still perfecting the trick," said Howduzzi with dignity. "Do not mock me, Doris: remember, I am famous throughout the world for my amazing tricks."
"You are famous because they always go wrong." Doris said. "And so will this one, unless we can fix up those cards."
Can you find a triangular arrangement in which every card is the difference between the two cards immediately below it?
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The answer
There are exactly two solutions:
the one illustrated and its leftright reversal.
There are similar "difference triangles" with two, three, or four rows of cards. It has been proved that no such triangles exist with six or more rows.
The winner is Ian Hamilton
