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DIFFICULTY

CAKEWALK

PREREQUISITES

Ad Hoc, Simple Math

PROBLEM

In the game of Coin Flip there are initially N coins on the table. All these coins are either showing Heads or Tails. Let us say that the coins are arranged in a line, numbered from 1 to N.

You have to perform N operations in which you flip all the coins from position 1 to i in the i^th round. You are to report the number of Heads / Tails after all the operations are complete.

QUICK EXPLANATION

It is easy to see that at the end of the game the coins would be in alternating positions, starting with either Heads or Tails. Therefore, either

HTHTHTHT...

Or,

THTHTHTH...

EXPLANATION

First, let us prove that the situation at the end of the game is indeed, alternating Heads and Tails.

Each coin is flipped a fixed number of times.

• The first coin has been flipped N times. Once in each round.
• The second coin has been flipped N-1 times. Once in each round except the first.
• And so on..
• The last coin has been flipped once.

We use the following insights,

• A coin flipped even number of times, will show the same side, as it did initially.
• A coin flipped odd number of times, will show the opposite side, of the one it did initially.

Now Let us consider two cases.

N is even

We can use the following insights,

• All the coins at odd positions will be in their initial configuration.
• All the coins at even positions will be opposite to their initial configuration.
• There are as many even positions as there are odd positions.

Hence, no matter what the initial position is

• The number of coins facing Head and Tail are equal.

Thus, the answer will always be N/2, for the query of Heads as well as Tails.

N is odd

We can use the following insights,

• All the coins at even positions will be in their initial configuration.
• All the coins at odd positions will be opposite to their initial configuration.
• There are floor(N/2) coins in even positions, and ceil(N/2) coins in odd positions.

Hence, no matter what the initial position is

• The number of coins that match the initial configuration are equal to floor(N/2).
• The number of coins that do not match the initial configuration are equal to ceil(N/2).

Thus, assuming integer division, the answer will be N/2, for the query that matches the initial configuration. Otherwise the answer will be N/2 + 1.

This can be summerized as in the code below.

if (N % 2 == 0 || I == Q)
    print(N/2)
else
    print(N/2 + 1)

SETTERS SOLUTION

Can be found here.

TESTERS SOLUTION

Can be found here.