Transcript from X+Y
International Mathematics Olympiad training, Taiwan
Nathan is a math genius now among peers. His social anxieties nearly paralyze his math performance. He has trouble reading the social cues of others and flinches at the slightest physical contact with another person. Martin is the English team’s coach, Zhang Mei is a Taiwanese student paired with Nathan.
So… 20 random cards are placed in a row all face-down. A move consists of turning a face-down card face-up and turning over the card immediately to the right. Show that no matter what the choice of cards to turn this sequence of moves must terminate. Nathan, hiding in the back won’t help you. Would you like to come up and show us?
Go on, Nathan.
Okay, so we need to… We need to look at the cards not as cards, but as… As numbers. We can call face-down cards… One. Face-up cards… Zero. And initially it would be a sequence of ones as the cards are all face down. But after a while it would look something like that. And, as we can see, that is a binary number. And a move that consists of turning a face-down card face up and the card immediately to the right of it could be that a one followed by a one, will turn into a zero followed by a zero. That would be like that. Or it could be a one followed by a zero turning into a zero followed by a one. In either case, we can see that the number in binary is strictly decreasing.
And that means?
Which means that the sequence must terminate.
Because you can’t keep taking away from a positive integer – without it turning negative.
No, you can’t. You definitely can’t. Good work;. Everyone. Good work.
The Anglo-Hungarian Math Connection
In X+Y, the British team has a joint training camp with the Chinese delegation. The closest analogue is the Anglo-Hungarian training camp that is held near a picturesque but secluded lake thirty miles west of Budapest. From my experience in December 2011, this was the most enjoyable of the maths camps.
The connection between the UK and Eastern Europe is rather complicated to explain, being intimately entangled with the history of the IMO. The inaugural Olympiad was held in Romania in 1959, with the competition being only open to countries under the Soviet bloc. A Hungarian mathematician, Béla Bollobás, competed in the first three Olympiads, seizing a perfect score on the third. After his PhD, Bollobás moved to Trinity College, Cambridge, to continue his research, where he fertilised Cambridge with his contributions in probabilistic and extremal combinatorics (becoming a Fellow of the Royal Society in the process). Consequently, there is a close relationship between Hungarian and Cantabrigian mathematics.