The easy proof is obtained applying the pigeon-hole principle to 𝛼₁x_1+…+𝛼ₙx_n(mod 1)|x_i=0,…N, or by geometry of numbers by applying Minkowski's first theorem in C.2.19 to the symmetric convex body of volume 2ⁿ⁺¹ given by |X_0+𝛼₁X_1+…𝛼ₙX_n|<N⁻ⁿ,|X_i|<N,i=1,…N.

Their fights with other members of their own kind were always preceded by loud and fierce waulings or savage snarling and snapping of teeth […]

Cards can only be exchanged between the playing deck and the sideboard on a one-to-one basis between duels or matches, and any number of cards, up to fifteen, can be exchanged at once.

...cutting a great dash in a brandnew two guinea dress suit and a burled hogsford hired for a Fursday evenin merry pawty...