Problem Solving Assignment From "Thinking Mathematically" book

Problem Solving Assignment From the book ‘Thinking Mathematically’

Group members: Meghan, Eddie, Hung Dang 

Problem “31” page # 179

Description: Two players alternately name a number from 1, 2, 3, 4 or 5. The first player to bring the combined total of all the numbers announced to 31 wins. What is the best number to announce if you go first?

Analysis:

31	30	29	28	27	26	25	24	23	22	21	20	19
W						L						L

18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
					L						L						L

Starting from the number 31, I notice that if I can force the other player announces his number when the total announced numbers at 25 then I can win the game. From 25, the maximum number he can reach is 30,  the smallest number he can reach is 26, within the range from 26 to 30, I can reach to the winning number 31 by announcing any number from 1 to 5

Using the same strategy, if I can force the other player announce his number with the total equal to 19, from 19 he can reach from 20 to 24 then I can announce my number to bring the total to 25

In general, if any player announces numbers when the total are 25, 19, 13, 7 will lose the game or in general 6n + 1 

Strategy

So the best strategy to play the game if I go first is starting with 1 because the other player can not bring the total up to 7, the maximum number he can reach is 6. And then my next move is to bring the total to 7

Modified Problem

We can modify to obtain another problem such that: "Two players play a game which has the rule as following: there are 27 balls, each player can pick up 1, 2 or 3 balls alternately. Who picks up the last ball will lose the game. How do you play to win the game?”

Comments: 

This problem is interesting and the key point to win the game is to find out numbers which are called 'losing numbers' in order to come up with the winning strategy. Those losing numbers will be written as a pattern. Those problem look complicated at first but if we understand one particular problem then we can easily solve another ones