^ Caglayan, GГјnhan Buddoo, Horace (September 2014), 'Tetrahedral numbers', The Mathematics Teacher, 108 (2): 92вЂ“97, doi: 10.5951/mathteacher.1, JSTOR 10.5951/mathteacher.1.Guy, Richard (1998), 'Square pyramid numbers', The Book of Numbers, Springer, pp.В 47вЂ“49, ISBN 978-3-9 ^ Babcock, Ben Van Tuyl, Adam (2013), 'Revisiting the spreading and covering numbers', The Australasian Journal of Combinatorics, 56: 77вЂ“84, arXiv: 1109.5847, MR 3097709.After incomplete proofs by Lucas and Claude-SГ©raphin Moret-Blanc, the first complete proof that no other such numbers exist was given by G. Г‰douard Lucas found the 4900-ball pyramid with a square number of balls, and in making the cannonball problem more widely known, suggested that it was the only nontrivial solution. The cannonball problem, asking whether there are any square pyramidal numbers that are also square numbers other than, is said to have developed out of this exchange. The same problem, formulated as one of counting the cannonballs in a square pyramid, was posed by Walter Raleigh to mathematician Thomas Harriot in the late 1500s, while both were on a sea voyage. The square pyramidal numbers were also one of the families of figurate numbers studied by Japanese mathematicians of the wasan period, who named them 'kirei saijo suida'. Formulas for summing consecutive squares to give a cubic polynomial, whose values are the square pyramidal numbers, are given by Archimedes, who used this sum as a lemma as part of a study of the volume of a cone, and by Fibonacci, as part of a more general solution to the problem of finding formulas for sums of progressions of squares. The pyramidal numbers were one of the few types of three-dimensional figurate numbers studied in Greek mathematics, in works by Nicomachus, Theon of Smyrna, and Iamblichus.

Bricks n balls level 285