In 7th grade, I was a student at Fredericktown Middle School in the eponymous town northwest of Columbus, OH. Any time a lesson finished a bit early, my math teacher would call two students up to the front of the room to face off in a subtraction game: starting from $25$, the two players would alternate subtracting either $1$ or $2$, with the first player to reach $0$ being declared the winner. (This classic game goes by many non-standard names. I haven’t the foggiest recollection what that teacher called it [or what his name was, for that matter].)
I was the first person in the class to figure out the winning strategy: as first player, always subtract $1$ first, and then just do the opposite of what the opponent just did. The first player should win every time. If you’re the second player, you just mark time hoping your opponent will make a mistake.
I was dominant. I went undefeated for weeks. It’s hard to feel too proud of my prowess in a game whose outcome was more determined by my teacher deciding who went first than by anything I did, but this was middle school, and a good reputation for anything was nothing to sneeze at. One day, I was holding my position at the front of the room as the Defending Champion when my teacher asked the class for a challenger. A lone hand raised. Something in my soon-to-be-opponent’s gait was more determined and purposeful than I was expecting. As challenger, she got to choose whether to go first or second.
“First”, she chose. “24”.
Good start. But I’ve got time. She’ll make a mistake. “23”.
“21”, she countered.
Okay, she’s got the early game down, but can she keep it up? “20”
That’s three perfect moves in a row. I’d better try something to confuse her. “16”
No delay. “15”
My palms began to sweat. Is there anything to do? Can I throw her off somehow? “14”
Okay, I’ll try the other move. “10”
No, she can’t be thrown off. She can smell the finish line from here. Just try to stall…just take 1 away. “8”
Is there any hope that those first 7 moves were just luck? “5”
Nope, no chance. She knows what she’s doing. “1”, I resigned.
I have long since forgotten the name of this opponent, but I remember the feeling of losing to her quite vividly. The audience erupted cheering for their new hero as I was cast aside. That moment marked the end of my Subtraction Game career.
And then yesterday, I heard a new variant of the game. Two players start at 25 and alternate subtracting 1 and 2 until they reach 0. However, instead of the last player to move winning, the last player to have subtracted 2 wins. Under these rules, I would have won that game back in 1996 because my $3\to 1$ was the last 2 subtracted before the end on the next turn. This twist has brought new hope to my countdown playing. Any takers?