Pick 'n' Sum

This is a simple game of target addition, similar to many others
that are popular in the current math literature and activity books.  At
its most basic, it can be described as a game where two players take turns
adding from a set of numbers, producing  single running total.  The goal
is to be the last player able to add a value that does not cause the sum
to exceed a given number, the target.  See games #1 and #2 in "5 Number
Games" for a famous example of such a game.

This game is designed, however, to include some features different
from the majority of those which you may have already seen.  One of these
is that the players create their own set of available numbers in a semi-
random manner.  Another is that the target value is not preset, but rather
computed by the players for their particular set via a simple two-step
procedure.  This is better illustrated by showing a sample game.

Finally, the use of a calculator is assumed. This is mainly
because, in addito to providing fun for the players, another goal is to
promote the development of problem solving strategies.  And as more games
can be played more rapidly -- and accurately -- with a calculator than
without, in a given amount of time, its use is thereby justified.

Stage 1: The "Deal"

To begin the game, eleven numbers are "created", or dealt out, by following this procedure: 1. Make a list of the two-digit numbers by putting the ten's digits into the blanks shown below. __0 __1 __2 __3 __4 __5 __6 __7 __8 __9 __0 The players may take turns writing the eleven digits 1, 2, ..., 8, 9, 1, 2 anywhere in the blanks as they wish, or one player may do it all. One possible outcome is the following: 10 21 72 63 84 25 56 97 18 49 30 2. The sum of the numbers is quickly computed using a calculator. In this case it is 525. 3. The target value is then found by multiplying the sum by 0.6, yielding in this case 315.

Stage 2: The "Game"

The game is now ready to begin. The person who goes first may be decided by lot or any manner acceptable to both individuals. Player A chooses a number from the set, say 97, enters it into the calculator, and writes it on a piece of paper. (The reason for this will be ex- plained shortly.) As numbers are chosen, they may be crossed out, as they may not be used again. Then play passes to Player B, who makes a selection and adds it to the 97 already in the calculator. Here is how this game might go: Player Pick Sum A 97 97 B 56 153 A 84 237 B 63 300 A 10 310 Since there are now no numbers available that could be added and keep the sum less than or equal to 315, Player A is declared the winner! At this point the used numbers could be returned to the set and another game played, or a new "deal" could now be made for a second game.
By now it should be clear that this is a relatively simple game to play. But it is one that requires some skill and practice in order to become a consistent winner. One of the most important background skills obviously is estimation, coupled with mental math computation ability, especially as play approaches the target value. The one who developes these skills to the greater extent will normally be more successful. It is now time to explain why it was suggested that a written record be kept of the game as it progressed. As it so often occurs in games such as this one, the loser begins to ask, "What if I had chosen differently on my last turn, could I have won after all?" Then the anlysis begins. Frequently the answer will be "yes". Then one thinks, "Why wasn't I more careful?" Then the real learning begins: thinking ahead to the possible consequenses of one's decisions in a math problem, or a life problem. Observe the sample game. If B had chosen 72 on the 2nd turn, the game would have had a different outcome. Player Pick Sum A 97 97 B 56 153 A 84 237 B 72 309 Now Player A can make no reply that keeps the sum under or equal to 315. Each game produces a new "What if ...?" question to be solved. In fact, the matter may even be more complicated, depending on the maturity level of the players concerned: "Were there other ways that B (or A for that matter) could have forced a win?" The analysis can become increasingly more complicated, depending on how far they wish to go back. This allows the activity to adjust itself to the mathematical and logical abilities of the players involved. Deeper thought and analysis are factors often given too little attention in our math classrooms. This game is one additional effort to foster them.
A Final Comment To vary the number set used, a couple of techniques may be utilized. (1) Include an extra number or two, using 12 or 13 two- digit numbers. (2) The set of ten's digits could contain more repe- titions, such as "2, 2, 3, 4, 4, 5, 6, 7, 8, 8, 9". Also the goal number could be computed with a different factor, such as 0.65 or 0.7, etc. Experiment with ideas of your own.