In the mathematics of figurate numbers , the cannonball problem asks which numbers are both square and square pyramidal . The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid. Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.
5-428: (Redirected from Pyramid Puzzle ) Pyramid puzzle may refer to: Mathematics [ edit ] Cannonball problem , a mathematical problem Tower of Hanoi , a mathematical game Other [ edit ] Pyramid puzzle, a type of mechanical puzzle Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
10-420: A solution to the cannonball problem. The squares with side lengths from 1 to 24 have areas equal to the square with side length 70, but they cannot be arranged to tile it. A triangular-pyramid version of the cannonball problem, which is to yield a perfect square from the N Tetrahedral number, would have N = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (70 × 2 = 140 = ) 19600. This
15-437: A square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their expedition to America. Édouard Lucas formulated the cannonball problem as a Diophantine equation or Lucas conjectured that the only solutions are ( N , M ) = (0,0) , (1,1) , and (24,70) , using either 0, 1, or 4900 cannonballs. It
20-501: The title Pyramid puzzle . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Pyramid_puzzle&oldid=1022940398 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Cannonball problem When cannonballs are stacked within
25-474: Was not until 1918 that G. N. Watson found a proof for this fact, using elliptic functions . More recently, elementary proofs have been published. The solution N = 24, M = 70 can be used for constructing the Leech lattice . The result has relevance to the bosonic string theory in 26 dimensions. Although it is possible to tile a geometric square with unequal squares , it is not possible to do so with
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