Math Delights

Two person game

By Joshua Zucker with Nancy Blachman

Object: The first player takes one turn rolling 1, 2, 3, 4, 5, or many as 15 dice so that the total number of spots on his dice is as close to 15 without going over 15. The second player can look at the results of the first player and then rolls 1, 2, 3, 4, 5, or as many as 15 dice in the hope that the total number of spots on her dice is more than the number of spots the first player obtained. The winner is the one whose total is closer to 15 but not greater than 15. If the players have the same total, the first player to reach that total wins.

Player 1: Decide how many 6-sided dice to roll. Roll the dice and compute the total number of spots.
Player 2: Decide how many 6-sided dice to roll taking into consideration what the first player rolled. Add the total number of spots on the dice you rolled.

Determine which player's total is closer to 15, without exceeding (going over) 15. If the two players obtained the same number, the first player wins.

Start the game again with the other player going first.
Play the game at least ten times.

What is the easiest way to keep track of your score?
Is there a way to win?
Are your chances of winning better if you go first or second?

Variations and Extensions

Children should decide how they will change their rules.

Play the new game to see if it works. Below are some suggestions for possible variations:

• Choose a different target, e.g., 20, 21, or 24 instead of 15.
• The winner is the person who gets closer to the target after 6 rolls.
• Subtraction may be used if the sum of the dice exceeds the target.

Strategic Dice Sum Game Facilitator's Guide

Encourage kids to argue about strategy. Discourage kids from arguing about who is winning the game.

The first player should be more conservative than the second player.
The second player's strategy should depend on the first player's total.

Josh wrote a computer program in a language called Scheme that helped him to determine the following optimal strategy:

• if the first player rolls a 1 or 2, the second player should roll 2 dice,
• if the first player rolls 3 - 9, the second player should roll 3 dice,
• if the first player rolls 10 - 14, the second player should roll 4 dice for the best possible chance of winning.

The following table from Joshua's computer program includes all possible results, listing the score of the first player followed by the number of dice the second player should roll and the probability of winning if the second player rolls that many.

(0 (2 1))
(1 (2 1))
(2 (2 35/36))
(3 (3 205/216))
(4 (3 101/108))
(5 (3 49/54))
(6 (3 31/36))
(7 (3 19/24))
(8 (3 25/36))
(9 (3 125/216))
(10 (4 655/1296))
(11 (4 551/1296))
(12 (4 71/216))
(13 (4 143/648))
(14 (4 35/324)))

If the first player rolls 1 die, based on the above results, that player has a 17/432 chance of winning, about 3.9%.
If the first player rolls 2 dice, then that player has a 10997/46656 chance, about 23.5%.
If the first player rolls 3 dice, then that player has about a 39.0% chance.
If the first player rolls 4 dice, then that player has about a 25.7% chance.
If the first player rolls 5 dice, then that player has about a 9.7% chance.

If the first player rolls more than 5 dice, the chance of busting (exceeding 15) is great; the chance of winning is small.

Here is a summary of the results if the goal is not to exceed 21:

Player 2:

• should roll 3 dice if the first player's total is 0 through 3,
• should roll 4 dice if the first player's total is 4 through 10,
• should roll 5 dice if the first player's total is 11 through 17,
• should roll 6 dice if the first player's total is 18 through 20.
• should resign gracefully if the first player's total is 21,
• and should claim victory if the first player's total is more than 21.

On the basis of this strategy for the second player, the first player should roll 3 dice. With this strategy, the first player only has about a 12.9% chance of winning.

Cube Game Sampler

from www.mheonline.com
Real Math - PTA Cube Package - Grades K-6
ISBN: 9780026744553 (cube package)
MHID: 0026744554 (booklet)
Price: $11.58

Game playing is a natural and effective way of learning. Games are lots of fun too! They give students a chance to develop and practice their mathematical skills while they are engaged in an activity that they enjoy. The Cube Games are designed to:

1. provide students with a chance to work out and practice mathematical ideas and strategies;
2. provide students with an opportunity to receive extra help; and
3. provide teachers with an opportunity to monitor student progress.

The Cube Game Sampler, which includes 6 cubes (3 with the numbers 0 - 5 and 3 with 6 - 10), is available from SRA/McGraw-Hill, www.mheonline.com. In July 2010, the Cube Game Package cost $11.58.