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## #1 2009-09-12 14:56:54

bossk171
Member
Registered: 2007-07-16
Posts: 305

### High Number game, optimal solution

The other day, while I was substitute teaching a class of sixth graders, I was introduced to the High Number game. The game, designed to teach students about place value notation, goes like this:

Students take turns rolling a die. After each roll, the student gets to choose whether to place their number in the ones, tens, hundreds or thousands place. The winner is the student with the largest number after four rolls.

So, each student starts with a blank 4 digit number (_,___). Let's say Player A rolls a 4 and decides to put that in the hundred's place. Her number is now: _,4__. Player B rolls a 1 and he (of course) puts that in the ones place: _,__1.

Now player A rolls a 5 and she puts it in the thousands place: 5,4__. Player B rolls a 3 and puts it in the tens place: _,_31.

Next, Player A rolls a 6 and puts it in the tens place: 5,46_. Player B rolls another 1 and he puts it in the hundreds place: _,131.

In the final round, Player A rolls a 5 and finishes with: 5,465. Player B gets lucky, rolls a 6, and wins with: 6,131.

If both players could see the future, Player A would have won with a 6,554, beating Player B's 6,311.

So the question is, what is the optimal strategy? Obviously, anytime you roll a 6, you place it in the thousands place, and any time you roll a 1, you place it in the ones place. But what if you roll a 5, do you place it high, or hold out for a 6? Is it in your best interest to play optimistically or pessimistically?

Are there variations on these rules that might make it more interesting?

There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

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## #2 2009-09-13 01:37:06

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: High Number game, optimal solution

This is an interesting game.

It gets even harder when you consider that strategies will change depending on what the opponent has done. For example, if player B had rolled a 5 on his third turn, he couldn't put it in the thousands place even if that would normally be a good move. On the other hand, a 4 might not be a good choice to put in the thousand slot, unless your opponent has already put a 3 there.

The optimal strategy would probably be to "play the stats". If you're likely to get a better roll in subsequent turns, then don't use up the thousand slot just yet, etc.

High number -->     1          2          3         4        5         6
Turns left
|
V

1                                        0.167   0.167   0.167   0.167   0.167   0.167

2                                        0.028   0.083   0.139   0.194   0.250   0.306

3                                        0.004   0.032   0.088   0.171   0.282   0.421

4                                        0.001   0.016   0.050   0.135   0.285   0.518

So from this, it looks like you should keep a 5 from the second roll onwards, and you should keep a 4 if it was your third roll.

Why did the vector cross the road?
It wanted to be normal.

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