This is an interesting puzzle type, commonly found in the form of single questions or a set in the DI-LR section. It typically involves a game of sticks in which, two players (playing a logically perfect game) have to pick sticks till the time one stick remains. The person who picks that last stick loses the game. Let’s look at how to solve these questions.
The premise looks extremely simple but as there is very less information and a lot of possibilities, unless you are clear with the logic, you will be spending a good amount of time scribbling on your rough sheet as to how could the game pan out. They keyword here is the term ‘logically perfect’ / ‘play with an intention to win’ / etc.
While most of the candidates think that there could be infinite number of variations to the gameplay and so, the outcome cannot be uniquely determined, there are indeed a few things that can be done so as to ‘control’ the game and keep your opponent under check. Although you have no control over your opponent’s move (s/he is playing equally logically and wants to beat you as badly), you can control the outcome of a pair of moves. You have to keep on doing it for all pairs of moves such that when the last move comes, the opponent has only one stick in front of him which forces a defeat and that is where the logic lies while solving these questions.
Let’s take a few cases and see if we can figure out this bit of logic.
1 or 2 sticks scenario
Probably the simplest and the most common of the question types involving a game of sticks wherein, the minimum number of sticks a player can pick is 1 and the maximum is 2. Let’s think of this scenario in reverse starting with the outcome:
If there is 1 stick left when it is your turn to pick, you would lose
If there are 2 sticks left when it is your turn to pick, you would win (pick 1 stick and the opponent will be forced to pick the last one)
If there are 3 sticks left when it is your turn to pick, you would win (pick 2 sticks and the opponent will be forced to pick the last one)
If there are 4 sticks left when it is your turn to pick, no matter whatever you do, your opponent will win (pick 1 stick, your opponent picks 2 OR pick 2 sticks and your opponent picks 1)
If there are 5 sticks left when it is your turn to pick, you would win if you pick 1 stick and the opponent has no chance of coming back – it’s either 1-2-1-1 or 1-1-2-1)
And it will continue in this manner. While CAT does not ask for explanations and this is a perfectly sound way to solve such questions, it would take a lot of time during the test. So, it’s better to see the logic behind this.
The ‘control the game’ aspect I mentioned above essentially means that irrespective of what your opponent does, you have to make sure that a particular number of sticks have been eliminated in a pair of moves commonly called a round (opponent’s move first, then yours in that order). Number of sticks that can be moved in a round can be either 2, 3 or 4. The first and the last case are dependent on what your opponent moves and so, cannot be certainly removed in a round. So, this is possible only in case you decide to move 3 sticks in a round (if your opponent picks 1, you pick 2 and if your opponent picks 2, you pick 1).
So, if you keep on eliminating 3 sticks with each round such that 1 stick is left at the end, you should be good. The question will ask you how many sticks should you pick right at the start to ensure victory. Let’s solve a few questions:
If there are 35 sticks on the table, and you have to make the first move, how many sticks should you pick to ensure victory?
As per the above logic, you should have 1 stick left at the end and exactly 3 sticks eliminated during each round. So, when your opponent has to make his/her first pick, s/he should have 3x+1 sticks in front of him/her. Now, the largest number less than 35 which is in the form of 3x+1 is 34. So, you will pick 1 stick to bring the number of sticks to 34 when it is your opponent’s turn to pick, ensuring a victory.
If there are 25 sticks on the table, how many sticks should you pick to ensure victory?
By now, it is pretty clear that whoever has to pick when there are 3x+1 sticks on the table will lose. If it’s your turn to make a pick when there are 25 sticks on the table, there is no way you can ensure victory in this scenario.
I guess we are clear about the simple 1-2 scenario. Let’s see a couple of variations.
1 or 4 sticks scenario
This is slightly different as the outcomes are not consecutive. However, the logic remains the same. In one round, you and your opponent can pick either 2, 5 or 8 sticks.
So, if there are 25 sticks on the table, it would be a good idea to 4 sticks. However, if there are 24 sticks, you will not be allowed to pick 3 sticks directly to reduce it to a 5x+1 game. So, you have to look at the alternative situations.
|Number of sticks left before your turn||Will you win/lose?|
You can see that if there are 1, 3, 6, 8, 11, 13… sticks when you are about to pick, you will lose. So, essentially if there are 5x+1 or 5x+3 sticks, you will lose. Let’s take another scenario.
1 or 3 sticks scenario
|Number of sticks left before your turn||Will you win/lose?|
If there are 4x+1 or 4x+3 sticks on the table, you will lose.
Trying it out for a few more scenarios, you get the general form of a losing position to be:
(1+n)x+1, (1+n)x+3, (1+n)x+5… where the second part will be an odd number less than (1+n). A bit difficult to follow but if you solve a few examples, you will understand it.
If there are 1 or 5 sticks to be picked, you will lose when there are 6x+1, 6x+3, 6x+5 sticks on the table.
If there are 1 or 6 sticks to be picked, you will lose when there are 7x+1, 7x+3, 7x+5 sticks on the table.
If there are 1 or 7 sticks to be picked, you will lose when there are 8x+1, 8x+3, 8x+5, 8x+7 sticks on the table.
1, 2 or 3 sticks scenario
Again, a simple scenario similar to the first type. The only outcome that can be safely assumed to come true is if the sum of sticks picked in a round is 4. So, 4x+1 you lose, rest of the time, you win.
Person who picks the last stick wins
It is essentially the same thing but in reverse. Winning situations become losing situations and conversely, the position that made you lose for sure will end up winning the game for you.
Points to remember
1. Read the question carefully. There will be three parts to the question: who is the winner, how many sticks on the table and who is the first to pick. A lot of students commit a silly mistake while noting down these three aspects and end up with the wrong bit of information (though you might find your answer among the given options).
2. The act of picking stick/s by a person is called a ‘move’ while two successive moves constitute a ’round’. Understand what terminology is being used in the question.
3. You can have control over a round only when you are picking after your opponent. If you pick 2 sticks and expect a total of 4 for the round, all you have is HOPE! which you don’t in this case.
4. As shown above, not all questions have to be solved using the (a+b)x+1 formula. The easiest way to trick a gullible candidate is by tweaking the game to a 1-3 or a 1-4 one and then just watch the world burn.
5. There will be an option to pick 1 stick as there is the last stick scenario involved. You cannot have a game of picking either 2 or 3 sticks only and then the last stick scenario.
The logic part is probably the trickiest to crack. Generally, I expect simple (1-2 or 1-2-3 type) cases to appear understanding that not many people are comfortable with these question types. If the scenarios become more dense (multiple outcomes with no logical semblance to what you know, more than two people playing the game), it is probably a good idea to either (i) leave the question and come back to it later if you have time as more than 99% of the test taking population would struggle to solve this type or (ii) write down scenarios for all the possible outcomes and try to figure out a logic. Remember that if the question looks dense, there will be some logical connect to help you if you persist. But then, persistence is not exactly an admirable quality to possess when it comes to attempting a rapid fire 60 minutes 32 questions section.
All the best!