Summer 2002
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Summer 2002 |
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"Die ganze Zahl schuf der liebe Gott, alles Übrige ist Menschenwerk.” This quote is due to the famous German mathematician Leopold Kronecker 1823-1891, and it translates to "God made the integers, all else is the work of man." It is probably more accurate to give God only credit for the positive integers. In the seventh century, the Indian mathematician-astronomer Brahmagupta
was one of the first to give the rules for arithmetic involving zero and
negative numbers. The two problems for this issue will deal only with the
positive integers. Eight squares are arranged as shown in the figure. Two squares are said to be
touching if they have at least one vertex in common. In how many different ways
can the numbers 1,2,3,4,5,6,7,8 be assigned to the 8 squares so that each square
is assigned a different number and that no touching squares contain consecutive
numbers? PROBLEM II
Find a nine digit number N formed from the digits 1, 2, 3, 4 , 5 , 6 , 7, 8
and 9 with the following two properties. a) each digit used exactly once, b) the
two digit numbers formed by each pair of consecutive digits in N can be
expressed as a product of two single digit numbers. For example the number 126547983 satisfies condition a). It does not satisfy
b) since it contains many consecutive two digit pairs (e.g. 26) that cannot be
expressed as a product of two of single digit numbers. Send your solutions to Herb.Bailey@rose-hulman.edu or to Herb Bailey, Math. Dept., Rose-Hulman,
5500 Wabash Ave., Terre Haute IN 47803. Solution for Previous Issue
You all did well on problems 1 and 2 but some missed problem 3 which was to
find the ratio of sphere volume to cube volume for the packing shown. The body
diagonal of a box is the line segment joining 'opposite' vertices, and if the
box is a cube with sides of length s then the length of the body diagonal is s√3. Let A and B be one of the pairs of opposite vertices of the cube. Then the body diagonal AB will pass through three of the sphere centers say
CA, CO,
and CB, where
CA
is near A,
CO
is the center of the box and
CB
is near B. If r is the radius of the spheres and x the distance from A to
CA
then AB = 4r + 2x. Since x is the length of the body diagonal of a cube with edges of length r then
x = r√3, and thus AB = r(4 + 2√3). The side of the cube is then
AB/√3
and the required volume ratio is
Alumni: a. Kelsall, 1940; W. Barrick, 1941; D. Heath, 1943; W.
Soudritte, 1943; H. Payne, 1947; J. Hurt, 1948; C. Cook, 1949; A. Junker,
1950; W. Rinker, 1951; C. Hirshfield, 1954; C. Cooper, 1956; J. Chinn, 1956;
J. Moser, 1956; A. Sutton, 1956; D. Bailey, 1959; P. DeStefano, 1960; J.
Tindall, 1961; J. Snyder, 1962; R. Kraus, 1969; M. Avery, 1976; J. Slupesky,
1979; W. Atterbury, 1980; R. Orr, 1980; K. Moraga, 1981; D. Lyon, 1983; G.
Cullen, 1985; M. Lancaster, 1987; C. Abdnour, 1989; B. Burger, 1991; D.
Hector, 1992; E. Francher, 1992; K. Fredrick, 1993; J. Atkins, 1994; J. Rott,
1995; T. Adams, 1996; D. Martin, 1998; C. Miesse, 1998; M. Pilcher, 1998; G.
Lara, 2000; J. Hochstedler, 2006; D. Rotenberg, 2006; T. Cline, 2006; N.
Snyder, 2006; S. Galler, 2006; J. Hanson, 2007 Friends: L. Gaintner; D. Voltmer; F. Bernard; K. Duffy; G.
Elmer; L. Gaintner; J. McGaha; K. Smith
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