Fall 2004

Bailey Challenge

Around 2800 BC there was a great flood of the mighty Lo river in ancient China. During this flood there emerged from the river a turtle with a curious figure on its shell. It was a three by three grid with a different number of dots in each of the nine positions. The sum of the number of dots in each row, each column and both diagonals were all the same. My attractive turtle is shown in the figure and the letters a through i are the integers 1 through 9 in some order. In modern times, this grid is called a magic square. 

Bill Soudriette '43 asked me if the values of a through i could be determined without using 'guess and check'. I looked at few of the nearly two million "magic square" hits on Google without success. I was able to find some results and offer them as problems.

Prob. 1   Find a through i so that all row, column and diagonal sums are equal.  

Your solution of problem 1 leads, by symmetry and reflection, to seven additional solutions. Note that that the value of e is the same in all of these eight solutions and that  in any of them. Solve problems 2, 3 and 4 without assuming that these eight are the only solutions.

Prob. 2   Show that the common sum found in problem 1 is the same for any other solution.

Prob. 3   Show that the e found in problem 1 has the same value for any other solution.

Prob. 4   Show that the value of a cannot be 9 in any solution.

Solvers of the Summer problems are listed. You came up with a nice variety of non-calculus solutions to the last problem. They varied from the super practical: measure the distances in the Echoes figure, to the super pure: something about an affine transformation. No one noted that the maximum of a parabola is at its vertex. In 1943 at Rose Poly, we had a full semester course in analytic geometry. M. Mergy, R. Gold and M. Bailey submitted a joint solution to the problems and perhaps should get only 1/3 credit each. On the other hand team projects are the current fad, so perhaps they should be given extra credit.

Send your solutions to the Fall problems to Herb.Bailey@rose-hulman.edu or to Herb Bailey, Math. Dept., Rose-Hulman, 5500 Wabash Ave., Terre Haute IN 47803.