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The maximum number of pieces from consecutive cuts are the numbers in the Lazy Caterer's Sequence. When a circle is cut n times to produce the maximum number of pieces, represented as p = f (n), the n th cut must be considered; the number of pieces before the last cut is f (n − 1), while the number of pieces added by the last cut is n.
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[5]: 7–8 Hazard allows the dice shooter to choose any number from five to nine as their "main" number; [6]: 168 in a pamphlet published in 1933, [7] Edward Tinker claimed that Marigny simplified the game by making the main always seven, [5] which is the mathematically optimal choice, i.e., the choice with the lowest disadvantage for the shooter.
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The solution line must cross over each "wall" exactly once, where "cross over" means to pass completely from one to the other of the two rooms that are separated by the "wall", or from a room to the area outside the drawing. This precludes "crossing" two walls at the same time by drawing the solution line through the corner at which they meet.
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The pancake number, which is the minimum number of flips required to sort any stack of n pancakes has been shown to lie between 15 / 14 n and 18 / 11 n (approximately 1.07n and 1.64n,) but the exact value remains an open problem. [10] In 1979, Bill Gates and Christos Papadimitriou [11] gave an upper bound of 5 / 3 n.
Aces around, dix or double pinochles. Score points by trick-taking and also by forming combinations of cards into melds.