Mursalin and Agnishom Draw Diagonals!

Algebra Level 5

Mursalin and Agnishom decide to play a game where they alternately draw diagonals between the vertices of a regular $n$ -gon. In each turn, Mursalin and Agnishom has to draw a diagonal that doesn't intersect any of the diagonals that have been already formed. Mursalin goes first. The person who is unable to make a move loses. For how many possible values of $n$ , does Agnishom have a winning strategy if $457\leq n\leq 1003$ ?

The answer is 274.

2 solutions

Dhairya Gupta
Mar 6, 2015

For any n-sided polygon, the number of diagonals that can be formed without intersection = (n - 3). So, a four sided polygon has 1 such diagonal, a pentagon has 2 such diagonals, a hexagon has 3 such diagonals and so on. Now, if Mursalin starts first, then he can win if the number of diagonals formed are odd. Agnishom wins if number of diagonals formed are even. From 457 to 1003 sided polygons, number of diagonals which do not intersect for each polygon will range from 454 diagonals to 1000 diagonals respectively. The even numbers from 454 to 1000(both included) is 274. As Agnishom wins only when even number of diagonals are formed, the number of winning chances that he has is 274.

Can you prove the first claim? That was the point of this problem, to prove that any $n$ -gon has exactly $n-3$ diagonals that don't intersect one another.

Mursalin Habib - 6 years, 3 months ago

It is just the definition of polygon triangulation.

Daniel Liu - 6 years, 3 months ago

I'll write down the outline of a proof.

Each new diagnol increases the number of polygons by 1 and the maximum number of polygons is $n - 2$ . Also, if the regular n-gon hasn't been divided into $n - 2$ polygons, that implies that it has at least one polygon with more than 4 sides. This can be divided into two smaller polygons by its diagnol.

Since we start with one polygon, each move increases a polygon, the maximum number of polygons is $n -2$ and this maximum can always be attained, the number of moves that can be made are $(n - 2) - 1 = n - 3$

Siddhartha Srivastava - 6 years, 3 months ago

https://brilliant.org/problems/find-the-areaonly-your-logic-can-help-you/?group=3UHxOzwinQpA&ref_id=384997

PLEASE TRY TO DO THIS AWSOME PROBLEM TOO..post a solution if you get......................i am waiting for an awesome solution that i made while creating this problem

Yash Sharma - 6 years, 3 months ago

Aakash Singh
Apr 7, 2015

the answer should be actually 273 (=no of even no b/w 457 and 1003)

Mursalin and Agnishom Draw Diagonals!

The answer is 274.

2 solutions

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