In a game with two players X and Y and given n (≥ 4) number of circles drawn on a plane paper, alternately each one of X and Y is suppose to join two different circles which were not already joined by a line. The winner is one who joins the last available pair of circles and the game ends. If X starts the game then:

In a game with two players X and Y and given n (≥ 4) number of circles drawn on a plane paper, alternately each one of X and Y is suppose to join two different circles which were not already joined by a line. The winner is one who joins the last available pair of circles and the game ends. If X starts the game then: Correct Answer Y wins whenever n is a power of 2

Explanation:

There can be many questions like this so instead of going with a general method of solving this problem, we are proceeding with reviewing options here,

We can start with an example of n = 4,

So, to connect 4 Circles we need 6 lines,

which can be stated as,

To connect n Circles we need ⁿC₂ lines, since one line can connect two circles.

So we have here,

If X starts the game with n circles,

• nC2 must be odd so that X can win.
• nC2 must be even so that Y can win.

Now, we can draw a table as,

Now from options, we can check,

• X cannot always win since nC2 not always odd.
• Y cannot always win since nC2 not always even.
• X wins whenever n is odd is also not true since when n = 5, which is odd, X does not win.
• Y wins whenever n is a power of 2 is correct since when n = 4, 8, 16, etc. nC2 becomes even and Y wins.