This year the Wisconsin Mathematics, Engineering, and Science Talent Search celebrates its 50th anniversary of bringing challenging, intriguing mathematics problems to Wisconsin students, and recognizing some of the state’s most talented young mathematicians. It all began in 1963 with the following problem.
Given n billiard balls numbered 1, 2,3...n, for some positive whole number n, show that there are exactly two ways to arrange them in a circle so that the difference of the numbers on two adjacent balls is 1 or 2. [Show that] further of the two arrangements, each is a mirror reflection of the other.
This is a great example of a talent search problem. It is not easily classified as being solved by some technique. It doesn’t require students to know a particular fact or formula, and such knowledge will not particularly bring students closer to a solution. Yet in the solution to this problem (see “Solving the problems” below) we can see important themes of mathematical problem solving.
The problem above and four other problems were distributed to Wisconsin students, and students mailed in their solutions to be graded at the Mathematics Department at the University of Wisconsin-Madison. Thirty-one students sent in a correct answer to the problem above. This set of five problems was the first of four problem sets mailed out in the 1963-64 school year. On the second set, one problem was solved correctly by 170 students.
Now, the Talent Search consists of five problem sets, each of five problems. The problem sets are mailed to all of the middle and high school math teachers in Wisconsin, and the teachers are asked to distribute the problems and information about the competition to interested students. The five problem sets are available online as well. Students have approximately a month to work on each problem set. They may send their solutions by email or postal mail, where they are still graded in the Mathematics Department at the University of Wisconsin-Madison.
These days, there a lot of different math competitions that students and schools may participate in. However, the Talent Search is different from most of these competitions in several respects. It requires that students provide reasoning and justification, that is, “proofs,” for their answers and the solutions are graded on the quality of these proofs. The problems are more challenging than many in math competitions because the students have a month to wrestle with the problems, which allows for problems that requires more ideas and insights than anyone can be expected to have in a timed format. Any student may participate in the Talent Search. While the Talent Search relies on teachers to help spread the word about the competition, participation does not require any further commitment from the school. A school that is too small for a math team or without the resources to travel to competitions may still have one or two students who are interested in mathematical challenges and who could participate in the Talent Search.
In 1991, the Talent Search began offering a four year scholarship to UW-Madison as the top prize. At that time the scholarship was $4,000 a year for four years. Currently, the Van Vleck Scholarship, the top prize in the Talent Search, is $6,000 a year for four years. After the five rounds of the mail-in problem sets, approximately 25 of the top scorers from Wisconsin are invited to take a proctored and timed scholarship exam. The winner or winners of this scholarship exam are awarded the Van Vleck Scholarship to UW-Madison. The top scorers and their parents are also invited to UW-Madison for an Honors Day. There the students talk to professors, attend some mathematics and science lectures, and visit a research facility on campus.
Where the problems come from
For each problem set, the organizers in the Mathematics Department at the University of Wisconsin-Madison write a list of proposed problems. They then get to enjoy the challenge of working on the others’ problems before they come together to decide which five will be best for the competition. The organizers look for problems that don’t require too much specialized knowledge, but give students the opportunity to think creatively. Ideally the problems are inviting, and give students something to play around with and puzzle out as they examine small cases and try different strategies. Some problems in each set should be more accessible to allow more students to get involved, and some should be challenging enough to give even the best students something to think about over the course of the month they have to work on them.
Solving the problems
Each problem in the talent search gives students a chance to explore a small piece of mathematics. Often, there is some surprising or interesting structure in the mathematics. The problems are designed to give students a chance to stretch their minds and develop their problem solving skills. They give students a chance to develop their mathematical skills without just going further in the school curriculum.
For example, how would a student solve the problem above, which is on the easier end of talent search problems? First, one can consider small cases. For example, if there are three balls, some playing around will quickly convince you that there are only 2 ways to put three balls in a circle--such that they are 1,2,3 in order clockwise or counterclockwise--which both satisfy the conditions of the problem and are mirror reflection of one another. However, when there are four balls, you can see that there are now more circular arrangements--6 of them in fact--but by looking at each one you can again observe that the statement of the problem holds for four balls. Though for any particular n one could in theory list the arrangements and then check each one to see if it satisfies the difference condition of the problem, such an approach will never solve the problem for all n. For this, one needs to think systematically about how to build such arrangements.
Here one can use another important problem solving strategy by trying to make partial progress on the problem. How about before trying to see there are exactly two allowed arrangements (where an allowed arrangement is one where the difference of the numbers on two adjacent balls is 1 or 2), you could just try to see if you can construct any allowed arrangements at all? To construct one in an organized fashion, it makes sense to start with the placement of ball 1. What can be next to 1? Only 2 or 3. Yet 1 has to have two balls adjacent to it, so they must be it in some order. If n>3, what can be next to 2? Only 1,3,or 4, and 1 and 3 are already placed, so 4 must be on the other side of 2 from 1. Then, if n>4, what can be next to 3? Only 1,2,4,5, but 1,2,4 are already placed, so 5 must be on the other side of 3 from 1. We can proceed like this. We see that at the first step, we had two choices for which side of the 1 held the 2 and which held the 3. But after this, every ball position is fixed and there are no more possible choices. What started out as an attempt to build any allowed arrangement at all has turned into a proof that there are at most two allowed arrangements!
It still remains to be seen that there are indeed these two allowed arrangements. We see that the balls are being placed in order: 1,2,3,4..., alternating on the left and then the right of a line. At each point, every ball is difference 1 or 2 from all adjacent balls. When we run out of balls, the two balls on the ends will become adjacent to form a circle. Since the last two balls placed are consecutively numbered, we see that the difference of those adjacent balls is 1. Thus, we have constructed (when n>2) two different allowed arrangements which are mirrored reflections of each other. This completes the problem.
Over the years, the winners have come from many different schools across the state of Wisconsin, and they have gone on to many different paths after the Talent Search. However, they commonly recognize the important contribution the Talent Search had in the development of their mathematical skills. David Boduch, a winner in 1981 and 1982 from Waukesha Memorial High School in Waukesha said, “The problems were harder than any others I encountered in the classroom or at math meets, and I looked forward to their challenge and to competing against some of the region’s most talented math students. Each set contained at least a couple problems that did not succumb to an hour, a day, or sometimes even a week of pondering. I was often spurred to leaf through books on geometry, combinatorics, or number theory for helpful results. Occasionally I tested hypotheses or looked for patterns by writing BASIC programs on a 16K RAM microcomputer.” After studying mathematics at Harvard and MIT, David pursued a career in finance and created one of the largest and most successful fixed-income hedge funds. David reports that “The autographed copy Prof. Laurence Young gave me on Honors Day of his "Mathematicians and Their Times" still rests on my bookshelf.”
Matt Wage from Appleton East High School was a Van Vleck Scholarship winner in 2007. He said that “The Wisconsin Talent Search was great for me because it was my first experience with really difficult math problems. “ After studying math and philosophy at Princeton, he has also gone into a career in finance.
Some Talent Search winners have gone on into academic mathematics. Daniel Kane was a Van Vleck Scholarship recipient in 2000. He said “The talent search was actually a really significant learning experience for me, as it was my first real exposure to proof-based problem solving. I also found the problems generally interesting and continued to participate for years after I had already won.” Daniel majored in math and physics at MIT, got his PhD in mathematics at Harvard, and is currently a postdoctoral scholar at Stanford.
The Loh family famously had three Van Vleck Scholarship winners: Po-Shen in 1997, Po-Ru in 1999, and Po-Ling in 2002. Po-Shen said, “I thought that the Wisconsin Talent Search was a great experience, and indeed, it was my first encounter with formal proof writing. All of my previous mathematical activities had focused on numerical answers, and I found it interesting and challenging to shift the focus from calculation to proof. “ Po-Shen is now an Assistant Professor of mathematics at Carnegie Mellon University and is deputy leader of the US International Mathematical Olympiad team. Po-Ru is a postdoctoral scholar in the Harvard School of Public Health working in statistical medical genetics. Po-Ling said, “The Wisconsin Math Talent Search played an integral role in my development as a young mathematician. Although I was never one to enjoy competition, I always looked forward to the monthly problem sets, with carefully-crafted problems that I appreciated even more as I learned more mathematics in later years.” She is pursuing a PhD in statistics at Berkeley.
The most recent Van Vleck Scholarship winner is Thomas Ulrich who is from Appleton, where he was homeschooled. Thomas will use his scholarship to attend the University of Wisconsin-Madison starting this fall.
How to get students involved
When teachers receive the Talent Search problem sets in the mail, it is wonderful if they can distribute copies of them to students they think might be interested. However, any student can get involved by getting the problems from the website https://www.math.wisc.edu/talent/
The problem sets will be available online at the start of October, November, December, January, and February and are each due a month later.