The Quest for High School Mathletic Glory

Chapter 23: Until proven otherwise

One week after the AIME, people no longer have this competition in mind except maybe for Geneviève and Krista, for whom there is still hope to get to the next stage. Most people that attend the team practices at this stage only care about the VMC final and consequently are there for multi-variable calculus. Like Imélie, and other people wanting to exploit the generosity of extra credit in calculus BC to reinforce their position, knowing that now extra credit comes from multi-variable calculus.

"Even in the single-variable course we're so focused on the computational aspect that we forget about the artistic aspect of the discipline. Today we'll show the other facet of mathematics that will enlighten you if you plan to declare a major containing mathematics courses in college, proof methods" Trent begins the sesson.

"Geneviève talked about how the VMC final will contain problems of the kind « Show that, if condition X holds, then we have Y »" Cory comments.

"To solve that kind of problems, what matters is the clarity of the argumentation. You have a bit of discretion to determine what the audience is supposed to know or not, a bit like in literature when you're asked to write according to the target audience. A proof is an argument to convince the audience that a mathematical statement is true (or false)" Gen continues talking.

"Gen, that's enough! During the dinner you insinuated that my mathletic playstyle didn't lend itself to a proof competition!" Marcia shouts at her teammate, visibly mad at this description.

"Very well, Marcia, here's your chance to prove otherwise. Let there be N face-down cards, and a move tuns a face-down card face-up as well as the card to its right. Show that this game ends in a finite number of moves" Gen issues a challenge to Marcia.

Marcia works on the board, while the others, who expected to see multi-variable calculus, wonder why Gen chose a problem unrelated to this material for Marcia to prove the contrary of what Gen and Krista think, initially in secret, and openly since the dinner, of her playstyle. After a few moments, Marcia obtains the following on the board:

The deck of cards can be seen as a binary number containing N digits, with 1 being a face-down card and 0 a face-up card. There are two cases to consider: the case 10 and the case 11.

In the case 10 the two digits become 01, while in the cade 11 we obtain 00. Yet, in a binary number, changing a 1 for a 0 in a higher position amounts to reduce the value of a number, which makes the resulting sequence of binary numbers strictly diminishing.

But the sequence of numbers' lower bound is 0, which represents the whole deck face up; yet a strictly diminishing sequence of integers with a positive first term K will have a finite number X of terms before the X+1-th term becomes negative, with K > X.

"Now I must admit that you're much more advanced on proofs than I thought. I was wrong to believe that your mathletic style didn't lend itself well to proof contests. I owe you apologies for what I said at the dinner"

"Thank you, Gen"

Maybe... maybe my father was right! There has to be a reason why Marcia is capable of such bursts of speed, it seems like she has a better mental « gearbox » than the other two. Not that my girlfriend is bad at math, she largely proved she had talent to my eyes, Cory thinks while looking at the solution Marcia gave to the problem, that he recognizes for having watched X+Y with Gen and Krista.

"For everyone else, the most common error is to write a proof in reverse. Starting from the conclusion to then get to the hypotheses might be good to reflect on how the proof works, but not for the proof proper. Another big mistake is to lack clarity under the form of undeclared variables or unclear statements" Trent explains to them.