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Pick up in turns riddle

Pick up in turns riddle: Who picks up the last one?

The player who picks up the last lot of matchsticks in a depleted 36 match pile wins. Each of two player picks up 1, 3 or 5 matches alternately. Who wins?

The game of pick up in turns

Ritwika spreads out 36 matchsticks on the table, "It's a two-player game. Each player may pick up 1, 3 or 5 matches from the pile on a turn. Players will have alternate turns. If you are the one who pick up the last lot of matches, you win."

Ritwika leaves the decision to pick up first to Vijay and Vijay starts the game by picking up 3 matchsticks.

Who will win the game and why?

Time to solve: 10 minutes.

Solution to the Pick up in turns riddle

Visualize what may happen after Vijay picks up 3 matches.

Matches left are 33 and now Ritwika's turn. She may pick up 1, 3 or 5 matches, three possibilities.

Each of these possibilities will create three possibilities in the next turn of Vijay.

If the game is finished by say 9 turns, the number of possibilities to analyze will be 3 multiplied by itself 9 times, a huge number.

By brute-force approach, the puzzle cannot be solved, there must be an easy rule to predict the result.

Start to end analysis already proved impracticable. Can we think of what happens in the end?

Final step must be, the winner picks up the last lot of 1, 3 or 5 matches left.

That means, before the final step, in the last but one step, 35, 33 or 31 matches have been taken out, counted from the beginning.

It occurs to you suddenly that the numbers are odd, as are the pick up numbers 1, 3 and 5. This cannot be a coincidence and must have a role to play in the solution, you decide.

On this path, you explore the result of first turn, second turn, and third turn. On his first turn, Vijay picked 3 matches, an odd number. On her next turn, Ritwika will pick up another odd number of matches and the total number of matches picked will be even.

On his third turn, Vijay will pick up again an odd number of matches and the total number of matches picked will turn to odd.

You are concentrating on whether the number of matches picked up is odd or even.

First clue: An odd number of matches will be picked up in an odd number of turns.

Who picked first? It was Vijay. So Vijay's turn will always be an odd number.

It follows,

35th, 33rd or 31st matchstick will be picked up in the last but one lot by Vijay.

Ritwika will pick up the rest in her last turn.

Ritwika wins.

Whatever be the number of matchsticks picked up by Vijay in his first turn, Ritwika will always win the game.

Not a fair game! The player who picks up first must lose.

You may of course rectify the flaw by changing the rule,

A player can pick up 2, 3 or 5 matches on a turn.

But then there will be no puzzle.

By the way, you surely can say now who will win in the original game if the number of matches to start with were 47 and Vijay picks up 3 matches in his first turn!

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