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Page 65

The general form of the Fubini principle, which will be of use later, can be

phrased in terms of

number of — number of X average number of rows

describing the total number of

why 2x3 = 3x2 unknowingly was using Fubini's principle. For 2x3 means, "Count

how many x's there ...

The general form of the Fubini principle, which will be of use later, can be

phrased in terms of

**occupied seats**in an auditorium: number of X averagenumber of — number of X average number of rows

**occupied seats**per columns**occupied seats**per row column After all, both sides of the equation are simplydescribing the total number of

**occupied seats**. Anyone who learned in schoolwhy 2x3 = 3x2 unknowingly was using Fubini's principle. For 2x3 means, "Count

how many x's there ...

Page 68

us count the

occupied in row 3? In other words, how many positive integers k,k<n, are there

such that 3 divides kl This is asking for the number of multiples of 3 between 1

and n. There are approximately n/3 of these since every third integer is a multiple

of 3. More generally, the number of

In fact, it is between n/j and (n/j) — 1.) Thus the total number of

roughly ...

us count the

**occupied seats**row by row. As an example, how many seats areoccupied in row 3? In other words, how many positive integers k,k<n, are there

such that 3 divides kl This is asking for the number of multiples of 3 between 1

and n. There are approximately n/3 of these since every third integer is a multiple

of 3. More generally, the number of

**occupied seats**in row j is approximately n/j. (In fact, it is between n/j and (n/j) — 1.) Thus the total number of

**occupied seats**isroughly ...

Page 83

To do this, count the total number of

total number of

other hand, by the Fubini principle, it is also number of columns x average

number of

row ...

To do this, count the total number of

**occupied seats**in two ways. First of all, thetotal number of

**occupied seats**is number of rows X average number of**occupied****seats**per row, that is, (n*)(average number of**occupied seats**per row). (9) On theother hand, by the Fubini principle, it is also number of columns x average

number of

**occupied seats**per column, that is, (n)(average number of**occupied****seats**per column). (10) Consider a specific column, y. When is the seat where therow ...

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Amer ANOMALOUS CANCELLATION average number barycentric coordinates base bichromatic graph blue edges blue triangle blue-empty graph cake candidate circle color columns cone conic Consider contain convergent series denote density diagonal Dirichlet principles divergent series ellipses inscribed equal equation equilateral triangle example Exercise extremal graph fact Figure formula Fubini principle G. H. Hardy geometric given harmonic series Hence infinitely inscribed in ABC isoperimetric theorem latin square least logarithmic density Math mathematical matrix midpoint ellipse midpoint triangle n-by-n square number of occupied number theory occupied seats orthogonal projection partial sum participants pentagon piece plane positive integers prime numbers probability problem proof Prove quadrilateral raindrops Ramsey number Ramsey's Theorem random reciprocals red triangles regions result Riemann hypothesis Ross Honsberger sequence set of integers singular pair six vertices solutions submatrix subsets Suppose tangent Theorem total number transversal triangle ABC vertex zeros