Order of Operations

Since today is Pi Day (3/14), I thought I'd do a math-themed post. I was intrigued recently by a problem that showed up on Facebook. And then I read this Slate article about the problem with some added value of research into math's symbolology. When I first saw the problem, "6 ÷ 2(1+2) = ?", I arrived at the answer 1. But then I checked my work and decided that I had made an incorrect assumption about the proper order of operations and then arrived at the answer 9.

The author of the Slate article seems to prefer 9. I don't really wish to belabor the point but the left-right rule seems simpler than one incorporating the association of the 2 with the term inside the parentheses. However, I could imagine some inconsistent words coming out of my mouth if an expression were written "6 / 2x". Ultimately, ambiguity should probably be stamped out by the poser of the question by using parentheses.

In my most recent game, I missed several wins and wound up with a draw. I saw winning moves, but my analysis had faulty second or third or even fourth moves. Sometimes they were even moves I had seen before, but I somehow missed changing the move orders into a winning operation. I chalk it up to general tactical atrophy combined with the fatigue of a 5-hour game and clock anxiety since I had used up 140 of my 150 minutes and queens were still on the board.

 
I'm playing Black and I have an almost fantastic position with all my forces trained on the enemy king, but my own king is rather insecure. White's last move was 38.Qa1-d1, threatening both Rxh5 and Rg4 and if my queen leaves the c1-h6 diagonal, Qxd2 is also a possibility. I had been thinking of trading 2 rooks for a queen, but materially, that didn't exactly make sense and I didn't see any berserker queen follow-up.

But then I realized that the threat of Rf1+ might be better than the execution and what the position really cried out for was 38...Rf2!

My opponent sank into a big think because of the not-so-subtle threat of Qxg2#, while 39.Rg4 fails to 39...Rf1+ and the queen gobbles the rook at the end of the exchanges on f1. I'm not sure if 39.Rg4? Rf1+ 40.Qxf1 Rxf1+ 41.Kxf1 Qxg4 qualifies as a zwischenzug, since it seems to be a two-move exchange before grabbing g4. White's move was 39.g3.

 
 This move stops combinations on g2, but it also cuts off the Rh3 from defending e3. I saw this and thought what could be better than threatening a discovery with 39...Be3? But I analyzed 40.Rxe3 Qxe3?! 41.Qg4+ Ng7 42.Bh7+ and I thought I was losing control. But I overlooked my ace in the hole, the zwischenzug. 39...Be3! 40.Rxe3 Rf1+ 41.Qxf1 Qxe3+ 42.Kg2 Rxf1 43.Kxf1 Qc1+ 44.Kg2 Qxc2+ and Black wins easily.

Instead I decided to cut off the white queen's threats on g4 and h5, and also try to sac on g3, so I made the move 39...R2-f3?! White played 40.R3-h4 reviving the threat of Rg4.

 

With my queen landing on e3 with check, I no longer feared Rxe3 and Qg4, so I checked with 40...Be3+. White played 41.Kg2 fairly quickly and I went back to 41...Rf2+ thinking that I had him with either a discovery against Kg1 or a fork against Kh1 starting with Rf1+. But he surprised me with 42.Kh3.

 

Here I should have taken a breath, but it was difficult to stay calm. I still had about 9 minutes on my clock and I was sure I was overlooking all kinds of killer moves for myself and my opponent. I saw 42...Rxh2+ 43.Kxh2 Qxg3+ 44.Kh1 but couldn't find a follow-up and thought that the dangers against my king were mounting. In fact 43...Qxg3+ 44.Kh1 is a losing continuation for Black. I settled for 42...Nf4+ 43.Rexf4 R8xf4 44.Rh5 Qg7 45.Rh7 Qg5 46.Rh5 Qg7 1/2-1/2.

But Black had much, much better from the above diagram. After the game, neither of us knew exactly what went wrong, but Fritz showed me that the win was in my grasp with a mate in 5. Add all the missed wins before that and I was kicking myself most of the week. I set up my analysis board at the position above where the continuation taunted me all week.

44...Rxh2+ 45.Kxh2 Rf2+ 46.Kh1 Nxg3+ 47.Kg1 Ne2++ 48.Kh1 Qg2#.

Small consolation was that I clinched first place in my round robin. I also rationalized that perhaps I had gotten a gift of a win from a drawn position in my previous game, so my score evened out.

Sometimes a draw can feel like a loss when you miss a beautiful killer combination.