Colorado Smith and the temple of gloom
Colorado Smith pointed with his whip at a partially uncovered tiled floor. "Just as I thought: the remains of an Inctec temple. Here is the final clue I need to locate the Golden Orchard of the Hesperides."
"We don't have much time," his sidekick Brunnhilde said. "I can hear the howls of the war-party coming through the jungle."
Smith hacked at a growth of vines with his machete, uncovering a stone stela with ancient inscriptions. He translated as he read. "Enter not the temple of Glu'm the god of gloom… then a bit's fallen off… once made a single triangle, in lines, each containing one more tile than the former… until the final line itself contained a square number of tiles. Then King Catawaula commanded that the tiles be rearranged into a square. None were wasted."
"What does that mean?" Brunnhilde asked, ducking a poison-tipped dart.
"Well, the final line of the triangle might contain four tiles, that's a square number. If so, the triangle contains 1+2+3+4=10 tiles. Those could make a 3x3 square, but—"
"There's one left over," said Brunnhilde. "So that's not the answer. I know—one tile!"
"No—the stela states that there are more than 1,000 tiles, but fewer than a million," said Smith, dodging a gigantic boulder as it plummeted from the sky. "We are seeking two consecutive numbers, the first being a square and the second twice a square. The number of tiles is half their product."
"Of course! I've got it! The answer is—"
"Yes?" asked Smith, felling ten burly warriors with one blow.
"It is—aargh!" Brunnhilde fell, transfixed by a spear. Smith swung himself into the treetops with his whip and fled, wondering what number she had been about to say.
What is the correct number of tiles?
Scroll down for the answer
The answer
The temple floor contains 1,225 tiles, which can form a triangle with a longest side of 49 (72) tiles and also a square with sides of 35 tiles.
The number of tiles in the triangle is 1 + 2 + … + m2, for some whole number m. By algebra, this equals [m2(m2+1)]÷2. But this must itself be square. Therefore m2 is a square and the next number m2+1 is twice a square (so m is odd). The first solution (aside from m=1) is m=7, for which m2=49=72 and m2+1=50=2x52. This is the only solution within the stated range, because (by trying successive odd numbers, or more sophisticated methods) the next solution is m=41, for which 1,681=412 and 1682=2x292. But the number of tiles would be 1,1892 = 1,413,721, more than a million.
The winner was Jens Knocke from Cardigan
Colorado Smith pointed with his whip at a partially uncovered tiled floor. "Just as I thought: the remains of an Inctec temple. Here is the final clue I need to locate the Golden Orchard of the Hesperides."
"We don't have much time," his sidekick Brunnhilde said. "I can hear the howls of the war-party coming through the jungle."
Smith hacked at a growth of vines with his machete, uncovering a stone stela with ancient inscriptions. He translated as he read. "Enter not the temple of Glu'm the god of gloom… then a bit's fallen off… once made a single triangle, in lines, each containing one more tile than the former… until the final line itself contained a square number of tiles. Then King Catawaula commanded that the tiles be rearranged into a square. None were wasted."
"What does that mean?" Brunnhilde asked, ducking a poison-tipped dart.
"Well, the final line of the triangle might contain four tiles, that's a square number. If so, the triangle contains 1+2+3+4=10 tiles. Those could make a 3x3 square, but—"
"There's one left over," said Brunnhilde. "So that's not the answer. I know—one tile!"
"No—the stela states that there are more than 1,000 tiles, but fewer than a million," said Smith, dodging a gigantic boulder as it plummeted from the sky. "We are seeking two consecutive numbers, the first being a square and the second twice a square. The number of tiles is half their product."
"Of course! I've got it! The answer is—"
"Yes?" asked Smith, felling ten burly warriors with one blow.
"It is—aargh!" Brunnhilde fell, transfixed by a spear. Smith swung himself into the treetops with his whip and fled, wondering what number she had been about to say.
What is the correct number of tiles?
Scroll down for the answer
The answer
The temple floor contains 1,225 tiles, which can form a triangle with a longest side of 49 (72) tiles and also a square with sides of 35 tiles.
The number of tiles in the triangle is 1 + 2 + … + m2, for some whole number m. By algebra, this equals [m2(m2+1)]÷2. But this must itself be square. Therefore m2 is a square and the next number m2+1 is twice a square (so m is odd). The first solution (aside from m=1) is m=7, for which m2=49=72 and m2+1=50=2x52. This is the only solution within the stated range, because (by trying successive odd numbers, or more sophisticated methods) the next solution is m=41, for which 1,681=412 and 1682=2x292. But the number of tiles would be 1,1892 = 1,413,721, more than a million.
The winner was Jens Knocke from Cardigan
