You will be given a set of numbers to play this game. Two players start with a pile of counters and take turns choosing numbers from their set (repeats are allowed) to remove from the pile. The first person to leave 0 counters wins. For example, if you are given numbers 1,2,3,4, we can see that starting with a number not divisible by 5, the second player can force a win by always bringing the running total to a multiple of 5.
Hint: suppose the numbers in your set were 1,3. Then work “backwards” from the lowest numbers to see what are the winning and losing positions. For example, in this case, 1 would be a winning number since that player could remove 1 counter. 2 is a losing number, since the player with that number can only remove 1, leaving a winning number for her opponent. 3 is a winning number, since that player can remove 3. And for is a losing number, since that player can remove 1 or 3, leaving a winning number for her opponent. If you continue with this example, you will begin to see that odd numbers are losing numbers, and even numbers are winning numbers. Your problem will most likely have a more complicated analysis than this though!
Your set of numbers is: 1,4,6,10
(You can only take these numbers away from the counters; you can’t take 2, 3, 5, 7, 8, or 9)
a. Find a winning strategy if the game begins with 15 counters in the pile.
b. Find and explain the winning strategy if the game begins with 35 counters in the pile.
c. Find and explain the winning strategy if the game begins with any number counters in the pile.
Hint: suppose the numbers in your set were 1,3. Then work “backwards” from the lowest numbers to see what are the winning and losing positions. For example, in this case, 1 would be a winning number since that player could remove 1 counter. 2 is a losing number, since the player with that number can only remove 1, leaving a winning number for her opponent. 3 is a winning number, since that player can remove 3. And for is a losing number, since that player can remove 1 or 3, leaving a winning number for her opponent. If you continue with this example, you will begin to see that odd numbers are losing numbers, and even numbers are winning numbers. Your problem will most likely have a more complicated analysis than this though!
Your set of numbers is: 1,4,6,10
(You can only take these numbers away from the counters; you can’t take 2, 3, 5, 7, 8, or 9)
a.Find a winning strategy if the game begins with 15 counters in the pile.
b.Find and explain the winning strategy if the game begins with 35 counters in the pile.
c.Find and explain the winning strategy if the game begins with any number counters in the pile.
To find the winning strategy for a given set of numbers and starting counter piles, we can follow a similar approach as mentioned in the hint. We need to analyze the set of numbers to determine which ones are winning and losing numbers.
Let's start by analyzing the set of numbers given: 1, 4, 6, 10.
a. Winning strategy for 15 counters:
We will go through each number in the set and determine if it is a winning or losing number.
The winning and losing numbers can be found by working backwards from the lowest numbers.
- Number 1: A player with this number can remove 1 counter, which leaves 14 counters. So 1 is a winning number.
- Number 4: The player with this number can remove 4 counters, leaving 11 counters. So 4 is also a winning number.
- Number 6: The player with this number can remove 6 counters, leaving 9 counters. So 6 is a winning number.
- Number 10: The player with this number can remove 10 counters, leaving 5 counters. So 10 is a winning number.
Now, let's analyze the pile of 15 counters.
- If the pile has 15 counters, a player can choose any number from the set.
- Since all the numbers in the set (1, 4, 6, 10) are winning numbers, no matter which number the first player chooses, they can always force a win. They should be able to remove counters in a way that ensures they leave the second player with a losing number, ultimately leading to the second player being unable to make a move and losing the game.
b. Winning strategy for 35 counters:
Following the same approach as above, let's analyze the set of numbers and determine their winning or losing status.
- Number 1: A player with this number can remove 1 counter, leaving 34 counters. So 1 is a winning number.
- Number 4: The player with this number can remove 4 counters, leaving 31 counters. So 4 is also a winning number.
- Number 6: The player with this number can remove 6 counters, leaving 29 counters. So 6 is a winning number.
- Number 10: The player with this number can remove 10 counters, leaving 25 counters. So 10 is a winning number.
Now, let's analyze the pile of 35 counters.
- If the pile has 35 counters, a player can choose any number from the set.
- Just like in the previous example, all the numbers in the set (1, 4, 6, 10) are winning numbers. Therefore, the first player can always force a win by removing counters strategically.
c. Winning strategy for any number of counters:
To find the winning strategy for any number of counters, you need to follow the same approach. Analyze the set of numbers and determine their winning or losing status. Then, based on the number of counters in the pile, determine which player has the winning strategy.
In this case, we have already determined that all the numbers in the set (1, 4, 6, 10) are winning numbers. Therefore, for any number of counters, the first player can always force a win by removing counters strategically.