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?