Denken Sie nach! (Ferdinand Georg Frobenius)
After I published my last book, Mathemagical Buffet, in January 2013, I had absolutely no plan to write another book. Still I enjoyed solving or creating puzzles because that was my most enjoyable leisure activity. Whenever I found interesting problems, I would try to solve them without reading the answers—even if it took me days or weeks. And I tried routinely to improve these problems to make them more interesting or more challenging. Before I realized it, I had accumulated piles of lovely results and I felt it was a pity not to share them with others. That is how this book came into being.
Over the years, I developed a habit of reading proofs or solutions only after I proved or solved them myself. I obtain a deeper understanding by comparing and analyzing the different proofs and solutions. And when my proof or solution is simpler, more elementary, or more natural than the published ones, it becomes material for my book. In fact, this is the way I wrote my last few books, including this one. To me this is the most enjoyable way to read math books.
Ever since I was a young boy I admired people who could find concise and lean solutions. But at the same time, I was often frustrated that I could not find any clues to their thought processes that led to those wonderful insights. How I wished authors would be kind enough to explain the fine details leading to their brilliant solutions! Consider the following well-known puzzle:
Anyone can cut a 3 × 3 × 3 cube of tofu into 27 unit cubes with six cuts. But suppose you are allowed to re-arrange the pieces between the cuts—what is the minimum number of cuts to do the job?
It is nice to impress people with an elegant answer. However, as a teacher, I think it is also educational to suggest similar but easier cases that lead to the solution. For example, if students have no clue how to tackle this puzzle, then why not suggest that they try the 2-dimensional (or even 1-dimensional) version first?
Thus, in writing I always hold on to the feeling of my youth. I hope readers will enjoy exploring the places my cruise could not reach. Bon Voyage!