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The Bailey Challenge
You did well on the fall problems, with many solvers being listed below. As I get older, my son, Mark, a 1976 chemical engineering alumnus and frequent Challenge solver, will be assisting me with future columns. This issue’s Spring Bonus is challenging, and I am hoping for at least eight solvers.
SPRING PROBLEM 1 Find all possible solutions of the equation: | x + 3| = 2x + 1
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SPRING PROBLEM 2 Consider the following banking transactions: Deposit $50 and make each of these withdrawals as fallows:
Withdraw $20, leaving $30 Withdraw $15, leaving $15 Withdraw $9, leaving $6 Withdraw $6, leaving $0 Total $50 Total $51 Where did the extra dollar come from? To whom does it belong?
SPRING BONUS
∆ABC is isosceles, with angles as shown. Show that BEF=30.
Hint: Show, in sequence, that the triangles, with base segments BC, CG, CF, GF, BE, and FE, are all isosceles. (Triangle with base GF is also equilateral.) Determine all angles as you proceed. F B 20 40
50
C G E
20
A
FALL BONUS SOLUTION
Problem: Let ABC be a triangle and D a point on AB. If BC = 7, BD = 5, CD=3 and AD = 6. Find AC.
Solution: Let E be the point on AB such that CE is perpendicular to AB. If x is the distance CE, then BE=BD + DE or œ 49 –x 2 = 5 + œ 9 –x 2 . After squaring, collecting and squaring again, we find x 2 = 27/4. Also (CA) 2 = x 2 + (EA) 2 , where EA = 11–BE = 11 – œ 49 –x 2 = 9/2. Thus (CA) 2 =27/4 + 81/4 = 27. Hence CA = œ 27. x C
B D E A
Send your solutions to Herb.Bailey@rose-hulman.edu or to: Herb Bailey, 8571 Robin Run Way, Avon, IN 46123. Alumni should include their class year. Congratulations to the following solvers of the summer problems: ALUMNI: J. Moser, 1956; A. Sutton, 1956; D. Bailey, 1959; J. Kirk, 1960; W. Perkins, 1960; R. Archer, 1961; R. Ireland, 1961; J. Ray, 1961; J. Tindall, 1961; A. Cleek, 1964; S. James, 1965; J. Albertine, 1969; R. Lowe, 1969; S. Jordan, 1970; E. Arnold, 1971; D. Jordan, 1971; W. Pelz, 1971; D. Hagar, 1972; G. Houghton, 1972; D. Bryant, 1973; R. Kominiarek, 1973; M. Marinko, 1973; J. Zumar, 1973; T. Rathz, 1974; D. Wheaton, 1974; P. Eck, 1975; M. Bailey, 1976; J. Schroeder, 1976; T. Greer, 1978; S. Warner, 1978; R. Priem, 1979; J. Slupesky, 1979; R. Joyner, 1980; P. Gunn, 1981; S. Nolan, 1981; M. Taylor, 1982; J. Marum, 1983; B. Downs, 1983; B. Green, 1983; R. Marchant, 1983; K. Shafer, 1983; C. Wilcox, 1985; C. Hastings, 1986; B. Wright, 1986; D. Johnson, 1987; M. Lancaster, 1987; J. Jachim, 1989; C. Abdnour, 1989; J. Allen, 1990; B. Burger, 1991; R. Hochstetler, 1991; J. Zamora, 1991; E. Geib, 1993; T. Litherland, 1993; P. Murfey, 1993; M. Pilcher, 1998; C. Ehrhart, 1999; S. Nelson, 2000; J. Askren, 2001; C. Cole, 2001; V. Roczniak, 2006; T. Homan, 2007; M. Milam, 2007; M. Trowbridge. 2008; J. Sedoff, 2009; H. Kim, 2010; D. Schoumacher, 2010; M. Schoumacher, 2010; D. Straub, 2010; M. Peterson, 2018; and R. Waite, 2019. FRIENDS: C. Cheney, T. Cutaia, A. Foulkes, L. Gainter, J. Ley, J. Marks, L. Metcalfe, L. Puetz, E. Robertson, J. Robertson, B. Schact, R. Schoumacher, A. Tyrrell, J. Walsh, J. Walter, J. Wilcox, and S. Wong
HAHN STILL SHOWING HIS MATH SOLUTIONS
Paul Hahn relishes the days when he had to show his work as a Rose-Hulman student and later as a software department manager with Rockwell Automation. So, it wasn’t surprising that he sent several pages to document his handwritten solutions to a recent Bailey Challenge.
“I thought everyone needed to show their work. If my answer is ‘long,’ then I would really like to know what the minimum acceptable answer is,” says Hahn, a 1979 mathematics graduate.
This was his first Challenge solution in more than a decade.
“I have taken stabs at the Challenges but did not submit them because I did not want to spend the time writing up my solutions. If there is a shortcut, I would really like to know it,” states Hahn, who spent his professional life in the aircraft avionics industry. “For safety and quality reasons, everything we built went through a peer review process. I am just used to having to explain how I do things.”
There’s no exact science to solving the problems, according to Hahn. “If a solution pops quickly to mind, then I will jot it down. Otherwise, I typically spread out doing one problem a day, putting in one half hour to maybe a couple of hours for each problem,” he says. “This includes the time to write up the solution for submission. Obviously, it also depends on the complexity of the problem. The real stumpers can take quite some time.”
As for his time at Dear Old Rose, Hahn remembers never having enough time to get everything done. There also was the college’s closure during the blizzard in the winter of 1978, the debate on whether students should be allowed to use calculators on tests, and “having fun studying that newfangled thing called a microprocessor.”
— Dale Long, Executive Editor