Running Out of Options

This game was a favorite for Paul Rettinger, a brilliant grad school room-mate I had at Penn State. He went from a Master's of Fisheries and Wildlife to a law degree. Amazing guy. I was lucky to ever win.

Last Letter Loses

Players take turns adding a letter to a word. The first player to be forced to spell a word (at least three letters) loses. A player can challenge the previous turn if they think there is no such word. If there is such a word, the challenger is out. If there is not, the challenged player is out.

Example:
1- A
2- A-M (doesn't lose because less than 3 letters)
1- A-M-E (P would lose, even if you're thinking of 'amplify', since amp is already a word.)
2- A-M-E-R (thinking of america)
1- (thinking they're in trouble) A-M-E-R-I
2- (happily) A-M-E-R-I-C
1- (inspiration strikes) A-M-E-R-I-C-I
2-(thinking what? that's not a word!) Challenge
1-Americium! I still would have lost but thought you might not remember that word. (It's an element.)

In that example Paul would have been player 1, snatching victory from the jaws of defeat.

(From a repository of free educational clipart, click the image.)

That inspired the following game, which I use to introduce prime number decomposition.


Last One Loses

Players take turns breaking down a number by multiplication. The first player starts with a whole number. The next player makes a string of two numbers that multiply to give the first. The next player then can break down one of the two numbers, making a three number string. The last player who can break it down is the loser of the game. Numbers chosen must be able to be broken down more than twice. A player may challenge if they disagree with a breakdown, or if they say it's the end but it's not. Players may not reuse starting numbers. 1 may not be used in the breakdown. You can't use a number that's been used this session.

Example 1:
A- 24
B- 3x8
A- 3x4x2
B- 3x2x2x2 - augh! (loses)

Example 2:
A- 112
B- 2x56
A- 2x2x28
B- 2x2x4x7
A- 2x2x2x2x7 - curses! (loses)

Instructional uses:
Have students keep track of which numbers make first player lose, and which numbers made second player lose. When the data is collected, students will see that the same number almost always has the same effect. (Although there's usually a number that was mis-factored.)

Pose the questions: what if you break the number down in a different way? Is it always the same number of steps? Is the end result always the same?

Several important ideas about the prime decomposition will come out immediately, including the idea of prime numbers! Also why 1 is not a prime. (And 1 is NOT a prime.)

I allow calculators for numbers bigger than 144. Follow up investigations can be things like: find a number that takes 6 steps and is between 500 and 1000, or trying the Sieve of Eratosthenes.