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Math en jeans

LES SUJETS 2018-2019

Par CATHERINE LEFRANCOIS, publié le vendredi 12 octobre 2018 10:27 - Mis à jour le vendredi 12 octobre 2018 11:00

Subject 1: Light trap

Consider a plane figure f  made of two sides AB and BC of a triangle.

In the picture you see a ray of light that slides into f   and then emerges after 3 reflections.

Problem 1: Determine a concrete example of  points A, B, C  and a ray of light , so that we obtain 4, 5, 6, …  n  reflections.

Problem 2: Is it possible, that the ray never comes out, that is to say, the number of reflections is infinite?   ─  That would mean that our simple figure f is really a light trap...

For both the above  tasks, we additionally assume that:

before and after each reflection, the light ray moves along a straight line;

during each reflection, the principle is that the angle of incidence of the ray equals the angle of its reflection;

the ray has an intersection with the interval AC sliding into f and it does not hit the point C.

Extra problem(s): If you manage to solve these problems, we may think similar problems for slightly more complex traps ...


Subject 2: „leaky choice” game

Two players alternately pull stones out of a hat which contains 7 stones (our players know that there are 7 of them). With each move, each player has only two possibilities: he draws either 1 or 3 stones (he cannot pull out 2 stones – that is why the choice is “leaky”). The one who pulls out the last stone – wins.

Problem 1 (not difficult): Does any of these players have a winning strategy? If yes – please find it.

Problem 2: Let’s generalize the problem – instead of 7 stones, let’s introduce n of them. Therefore we analyze an analogous game, but with n stones in the hat.

Please find the solution for every n natural.

Extra problem(s):  If you manage to solve these problems, we may think about similar games for slightly different “leaky choices”.


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