Five Nontrivial Number Games

[This piece is a shorter version of an article of mine that appeared in The ARITHMETIC TEACHER, Nov. 1972, pp. 558-560. (Reprinted in Games and Puzzles for Elementary and Middle School Mathematics: Readings from the ARITHMETIC TEACHER, NCTM, 1975, pp.148-150)]
Number games can provide elementary students with a good opportunity to test their observation skills for problem solving, and at the same time they will be practicing such drill skills as adding and subtracting. Here are five trivial number games guaranteed to provide hours of fascination and lots of hidden drill. The word trivial is used here with two meanings in mind. The games are trivial in the usual mathematical sense -- namely, once the secret is known to both players, the game's winner is predetermined, based on the starting number and who begins play. The games are nontrivial from the point of view that they are valuable teaching aids for motivating student thinking. Here are the rules. Can you find the winning strategies?
Game 1. The first player selects any integer from 1 to 10. Then the two players alternately add any integer from 1 to 10 to the sum left by the opponent. Play continues until one player can make an addition giving a grand total of 100. That player is thereby declared the winner. Game 2. The rules of this game are the same as those for game 1, except that now the winner is the player who forces his opponent to make a total of 100 or more. Game 3. The name of this game is "Aliquot." It was devised by David L. Silverman and appeared in the problem section of the Journal of Recreational Mathematics [October 1970]. Here is Mr. Silverman's own description from that journal: Two players start with a positive integer and alternately subtract any aliquot part (factor) with the exception of the number itself from the number left by the opponent. Winner is the last player able to perform such a subtraction. By way of example, if the original number is 12, first player may subtract either 1, 2, 3, 4, or 6 (but not 12). If he subtracts 2, leaving 10, second player may subtract 1, 2, or 5. The objective is to leave your opponent without a move. This can only be done by leaving him a 1, since it is the only positive integer with no aliquot part other than itself. Game 4. This game, called "Proper Aliquot", was also devised by Mr. Silverman. In the same issue of JRM, he gives the rules thus: The rules are the same as those of Aliquot with the exception that only proper divisors may be subtracted. Consider 1 as an improper divisor. Game 5. The first player selects any reasonable large number. Then the two players alternately subtract any number they choose from the number left by the opponent, provided the chosen number meets one requirement: the number subtracted must not exceed twice the value of the number subtracted by the opponent on the previous play. For example, if player A subtracts 4, then player B can select any integer from 1 to 8. The game is won when one player can take it all, that is, leave 0.
Another reason why these games should be considered as nontrivial is that until the players have discovered the secret strategy for winning, even weaker students have a chance to score some wins over a stronger adversary. So the main pedagogical aim for the teacher is that the students do indeed find those strategies. But if they don't, nothing really is lost. The play is, after all, the thing, isn't it? And if you really want to know the strategies, you can always send me an e-mail, okay?
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