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In the expansion (k-Dn (1+x+x 2 + .~ + x k-1 f = Y* [ / " ] * ' k>2, n> 0, 1=0 clearly For Ar = 2, these are the binomial coefficients and when dealing with these we shall use the usual notation: The problem of calculating sums of the following type for k = 2 was first treated by Cournot  and Ramus  and Ramus' method is outlined in [ 4 ] : N j=0 k… (More)
Graph theory almost certainly began when, in 1735, Leonhard Euler solved a popular puzzle about bridges. The East Prussian city of Königsberg (now Kalin-ingrad) occupies both banks of the River Pregel and an island, Kneiphof, which lies in the river at a point where it branches into two parts. There were seven bridges that spanned the various sections of… (More)
Gauss, in his work on quadratic reciprocity, defined in  an analogue to the binomial coefficients: n r i (x 1 1-D C x "-1-!) • • • (x n-k + 1-l) L k J (x k-l)(x k-X-!) • • • (x-1) ' n and k positive integers. In o r d e r to make the analogy to the binomial coefficients more complete, it is customary to let [!] • ' • for n = 0, 1, 2, • • • , and H-• for… (More)
The members of the winning team were Nikifor C. Bliznashki, Jason Fer-guson, and Lingren Zhang; each was awarded a prize of $600.