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Probability Level 3

□ = □ □ + □ = □ □ + □ + □ = □ □ + □ + □ + □ = □ □ + □ + □ + □ + □ = □ □ + □ + □ + □ + □ + □ = □ □ + □ + □ + □ + □ + □ + □ = □ \begin{aligned} \square & = \square \\ \square+\square & = \square \\ \square+\square+\square & = \square \\ \square+\square+\square+\square & = \square \\ \square+\square+\square +\square+\square & = \square \\ \square+\square+\square+\square+\square+\square & = \square \\ \square+\square+\square+\square+\square+\square+\square & = \square \end{aligned}

Sharky and Ivan are playing a strange game, with Sharky going first. They then take turns to write a real number into an empty square. Sharky wins if after 35 steps all equations are true. In the other case Ivan wins.

Who has a winning strategy?

Sharky None of them Ivan

Áron Bán-Szabó by Áron Bán-Szabó , 17, Hungary

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1 solution

Sharky can only win if he has the last word in every equation. Mathematically spoken, if he forces Ivan to fill in a number when there is one degree of freedom left, he can simply make sure the equation is satisfied. Since there is a triangle of freedom of 35-2x7=21=6x7/2 (total steps minus two squares per equation or series of positive whole numbers) which is odd, Sharky will win since he takes odd turns, if Ivan were to postpone his defeat. Ofcourse, it's also possible that Ivan fills in a spot of an equation with one degree of freedom before that, then Sharky simply satisfies that equation immediately after that turn, which also makes him the winner.

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