GENIUS CHALLENGE PROBLEM OF THE MONTH 

 The Geometry Problem that Stumped the Nation

  by Gary R. Gruber, Ph.D.

 Some time ago, I wrote an article called “Are You A Genius” which was a 12 question test to determine genius IQ. Papers over the country published it. I had received hundreds of letters from readers who could not solve the last and rather simple looking geometry question. As a high school student in New York, it took me three hours to solve the problem. However, forty years later, even after being able to write 30 books on the subject of test preparation and thinking, I could not solve the problem and it was driving me crazy!  Was I getting more stupid as the years passed? So I decided to put the 12 question test out again and if someone got all 12 questions right they would be called a super genius. I was hoping someone would be able to solve the last question and then tell me how they did it. I didn’t get any response except for tons of letters asking how the 12th problem could be geometrically solved. People at the highest levels in math at major universities, government agencies, you name it--could not solve the problem. Then I got an urgent call on a Friday from the Washington Post telling me that they were getting hundreds of calls every few hours asking for a solution to the problem. I told them I could not do it. Well that wasn’t good enough for them. They said I’d better have the solution to them by Tuesday (just four days!) or else! Or else, what?--they couldn’t sue me. For forty years I couldn’t solve the problem and now I had to solve it in four days. Didn’t I have better things to do over the weekend?  Well I contacted everyone I knew who was literally a genius in math, top mathematicians in the country. I contacted fellow math students that I hadn’t talked to in thirty years, who went to school with me, and who were super math savvy and may have seen the problem. They were all eager and working on it. However Sunday rolled around and no one responded with a solution. People from NASA, from major math departments all over the country, even the top mathematicians at Educational Testing Service, the company that develops the majority of entrance, aptitude and achievement tests such as the SAT, could not solve the problem.  In fact one person completely peeved, said that they worked 10 hours straight and couldn’t solve it! Then as a very last resort, I was able, by researching and making about 20 calls, to contact my old math professor who had given me the original problem to work on. Thirty years ago. But when I called him, (he must have been somewhat senile) he told me that I was “late for class and I’d better hand in my assignments.” He kept repeating this--what a bummer!--the last person on earth  who could have given me the answer was incoherent! This was Sunday night. On Monday,  I frantically again called the whiz kids I went to school with who were working on the problem and asked for any hints they thought I might use. Each one of them told me to use my own specific math strategies that I have been writing about for years and using in all my test-preparation and thinking books.  For example, when you take geometry and want to prove that if two sides of a triangle are equal, the base angles are equal, the strategy you use is you draw a line down the triangle. This works because almost magically when you draw something extra,  you get something for it--namely the more information and an approach to the solution. I never thought about that because I thought this problem was too sophisticated!  It was Monday night now and I was working feverishly on the problem using my very own strategies. Tuesday morning came around and I just finished solving the problem. I told this to the Washington Post, they printed a full page solution and they wrote me up as “the super genius.”  

            There are actually now six ways to solve the problem and in the next installment, I’ll show the simplest one. I still get many letters from people asking how to solve the problem and have since gotten some similar solutions and two additional solution ones which now account for the total of six ways (so far) to solve the problem.  

Here’s the problem: