# 1372 Yahtzee How-Many-Rolls Variation

Today on the spot I made up a quick variation of Yahtzee, and one of my students played it with me.

The object of the game was to get all five dice to show the same number of dots at the same time, but instead of only being allowed to have up to three rolls, we took as many rolls as need. To take a turn, one of us would roll the dice then look to see if any of the dice were the same. Any die that didn’t match would be included in an additional roll until it did match. We counted each roll we took and got one point for each roll. The lowest score would determine the winner. The student and I played four rounds. He was elated because he won EVERY round so, of course, he was the overall winner, too.

Usually, when we play a game together the scores are much closer. Sometimes I win, sometimes he wins. Today I couldn’t believe my bad luck! Sometimes none of the dice matched after my first roll. And what about my student’s very good luck getting five of a kind in just one roll? I’m sure some good probability discussions could result from this game.

Our data might suggest that 9 rolls is the most that a person could get, but I rolled the dice for the picture included in this post, and it took me 19 rolls to get that Yahtzee! And I actually had four 4’s after just 6 rolls before I took those last 13 rolls.

You never know for sure what will happen when it comes to games of chance. If you study probability, you can have a good idea about what is most likely to happen, but you cannot guarantee it will happen. If we had taken the time to play more rounds, maybe my student would have needed 10 or more rolls to get at least one of his Yahtzees, and the game would have been more competitive. (At least, that was what I was thinking before he rolled on rounds 3 and 4.)

I’d like to encourage you to try playing this game, too. I thought it was a lot of fun even though I lost miserably.

Now I’ll write a little bit about the number 1372:

• 1372 is a composite number.
• Prime factorization: 1372 = 2 × 2 × 7 × 7 × 7 which can be written 1372 = 2² × 7³
• 1372 has at least one exponent greater than 1 in its prime factorization so √1372 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1372 = (√196)(√7) = 14√7
• The exponents in the prime factorization are 2 and 3. Adding one to each exponent and multiplying we get (2 + 1)(3 + 1) = 3 × 4 = 12. Therefore 1372 has exactly 12 factors.
• The factors of 1372 are outlined with their factor pair partners in the graphic below.

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