174 Game

Date: 08/16/2001 at 20:27:48
From: zhongwanting
Subject: 174 Game

Hello, Dr. Math!

I have a question. Billy, Tom and Jack are playing a game. Tom and 
Jack are the players. Billy is the judge. The rule of the game is: the 
players announce numbers in turn. The numbers they announce should be 
within 1 to 8. The judge adds the numbers they announce. If the sum 
becomes 174 (or more than that) after one player's announcement, then 
that player wins. 

Billy decides to let Tom announce first. In order to win the game, 
which number should Tom announce first?

I don't know how to start. It seems to me that perhaps every number 
from 1 to 8 is possible.

Thanks a lot!
Zhongwanting

Date: 08/17/2001 at 19:34:57
From: Doctor Terrel
Subject: Re: 174 Game

Hello,

This is a famous type of number game. There are many different 
versions, and once you know the secret, you can invent more ways to 
play.

Rather than just tell you what Tom should say, I'll explain my 
favorite version and let you solve yours by yourself.

In my version the goal to win is 100, and the numbers to add are 
anything from 1 to 10.  Here Tom would say "1" to start, and would 
guarantee a win in the following way:

If Billy says "5", Tom says "6"
If Billy says "8", Tom says "3"
If Billy says "2", Tom says "9"
If Billy says "4", Tom says "7" 
and so on.

Hint: add those two numbers in each line. Do you see a pattern?

Eventually, Billy must make the total of 99. Then Tom just adds 1, and 
wins!

Now, think hard about that game, then use the general idea to solve 
your own problem.

- Doctor Terrel, The Math Forum
  http://mathforum.org/dr.math/