By K. Ribnikov

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15. Suppose that n cuts through a circular pizza form P (n) pieces, where n ≥ 2. Use Ismailescu’s theorem to show that n (n + 1) area of biggest piece > . area of smallest piece 8n − 2 16. What does the On-Line Encyclopedia of Integer Sequences say about the sequence 2, 7, 1, 8, 2, 8, 1, 8? ) 17. Let G(n) denote the maximum number of pieces of fruit formed by n plane slices through a spherical grapefruit. This problem outlines one derivation of the grapefruit-cutter’s formula G(n) = n n n n + + + .

The point (a, b) in the Cartesian plane is a lattice point provided the coordinates a and b are both integers. Each vertex in a lattice polygon is a lattice point. For a given lattice polygon, we let L = the total number of lattice points inside or on the boundary and B = the number of boundary lattice points. Our earlier work justiﬁes the approximation area of polygon ≈ L − 1 2 B. Pick’s formula gives us the exact area of a lattice polygon in terms of L and B; our approximation is always 1 unit too big.

3. The diﬀerence table for the sequence a0 , a1 , a2 , . . 5. The row of third diﬀerences consists of 0’s. Show that n n n an = a0 + b0 + c0 . 5: A diﬀerence table sequence a0 ﬁrst diﬀerences second diﬀerences a1 b0 a3 b1 c0 third diﬀerences a4 b2 c1 0 a5 b3 c2 0 · b4 · c3 0 · a6 · · · 4. Let E(n) denote the maximum number of regions formed by n ellipses in the plane. 13 shows that E(5) ≥ 42. This problem discusses three proofs of the formula E(n) = 2(n2 − n + 1) for n = 1, 2, 3, . . 13: Five ellipses and 42 regions (a) Draw some ellipses to ﬁnd E(n) for n = 1, 2, .