(I know, pretty bold of me to assume this will still be my favorite when I look back years later. I look forward to the day this is challenged.)
I’ve always had a keen interest in puzzles. Less so the jigsaw type (maybe it was its repetitive nature to me), more so the math and logic type. I loved a chance to engage my mind in something different, more challenging. I also generally felt that my mind grew a bit from each puzzle with which I wrestled. So far, two puzzles have spurred the biggest growth in my mind: the first puzzle I remember ever engaging with (I’ll save this for another post), and a neat little puzzle I encountered about a decade later. This little puzzle was the first one in a book I checked out (for the kicks) from my college library.
The Puzzle
“Suppose that 1025 tennis players want to play an elimination tournament. That means: they pair up, at random, for each round; if the number of players before the round begins is odd, one of them, chosen at random, sits out that round. The winners of each round, and the odd one who sat it out (if there was an odd one), play in the next round, till finally there is only one winner, the champion. What is the total number of matches to be played altogether, in all the rounds of the tournament?“
Paul R. Halmos, Problems for Mathematicians, Young and Old
The puzzle seemed simple enough. My mind immediately began breaking up the total match count into the matches per round. Since every match is played 1v1, and only the winner (and whoever sits out that round) moves on, the 1st round must have \dfrac{1025 - 1}{2} = 512 matches; the 2nd round, \dfrac{512 + 1 - 1}{2} = 256 matches; and so on. At this point, I felt I had an efficient strategy (calculate the matches played per round, then add up all the matches) but didn’t want to actually keep going and finish the calculations, so I skipped ahead to the solutions at the back of the book to see if I were on the right track.
I still vividly remember the tingling sensation around the back of my neck as I processed Halmos’ solution. It was so simple. Since every match eliminates 1 player (this is the only way to eliminate a player), and there can only be 1 winner in this tournament, there must’ve been 1025 - 1 = 1024 losers in this tournament, or equivalently, 1024 matches played in this tournament, each match eliminating one player.
Aftermath
Since that day I first read the puzzle, I’ve given a lot of thought to this elegant solution. I think what really draws me to this solution is how it so quickly cuts right to the core of the problem. All I had to do was take a step back and think carefully about what each part of the problem really implied. It definitely would’ve saved me the trouble of doing lots of calculations (that I didn’t do anyways).
Whether this puzzle was the spark or just an enhancer, I don’t remember anymore. What I do remember is its lasting impact on how I think now. At first, I was constantly reminding myself to think through the implications of whatever I read or heard (time-consuming as it may be); after a while, this constant asking of “What are the implications of this?” became almost reflexive for me in thought, in reading, in lectures, and in conversations—possibly to the annoyance of those on the receiving end. Eventually, my habit broadened into a general questioning of how well I actually understood what was presented to me: enough to be skeptical, probe the implications, and realize the limitations, but not enough to be stuck, nor annoy too many people with constant questioning.
At least, that’s the hope.