OBJECTIVES:

1. The visitor will identify location with two coordinates.
2. The visitor will use reason to narrow down possibilities.
3. The visitor will observe how reason narrows down possibilities.

MATERIALS:

• game boards - (provided below) two-by-two, three-by-three, four-by-four grids with rows and columns marked with numbers and letters respectively.
• colored pieces to fit onto the grids
• pennies
• the Wall - a screening device
• graphics of possible combinations (for 3x3, for 4x4)

Cartesian - Relating to René Descartes. Usually used in reference to Cartesian coordinates.

Cartesian coordinates (kar-TEE-zyun) ? A system of coordinates for locating a point on a plane by its distance from each of two perpendicular lines or in space. Often referred to as ìa graph with x and y axes.î Here, the coordinates are 4 and 2.
 

combination ? the arrangement of a number of things into various groups, as a, b, and c into ab, ac, and bc. There are 3 combinations of a,b, and c. If order is important a,b, and c form 6 permutations.
written as      , it means the number of ways r items can be selected from a collection of n items.

graph - a visual representation of abstract data, often made by plotting one variable on the horizontal axis and another variable on the vertical axis.

grid - in this lesson, a system of coordinates with rows and columns

mapping coordinates - marking points by assigning two or more values to them; with two points, one is for the horizontal axis and the other is for the vertical axis. see Cartesian coordinates.

sorting - separating by attributes, with binary sorting the separation is into two groups
              - for example: divisible by 9 / not divisible by 9

FINE TRIVIA BACKGROUND:
The idea of combinations is difficult for most children and many adults. (The success of gambling on "the numbers' may be proof of that!) For many people, the number of possible combinations will be startling new information. With this new information, they may better appreciate the power of reasoning to find the one solution out of so many possible.
Another uncommon skill is that of locating coordinates on a graph (and on maps). One cannot understand the mathematical background of the Mandelbrot set in the Beyond Numbers Fractals computer interactive unless they have a grasp of what a graph on the x-y axes looks like. The Lorenz attractor on the same interactive uses a three coordinate system! The Behind The Wall game is a pleasant way of getting people used to working with two dimensions: rows and columns.
I posed the following question on the internet to <dr.math@forum.swarthmore.edu>:

[an explanation of the game]

"....By drawing the grids, I found that the number of possible layouts for the 2 by 2 is 4, ...and for 4 by 4 ? well I got tired of drawing, but it's over 250. So my question is: Is there a formula for the number of layouts for an n by n grid with n colors where each color is connected by at least one border?"

"Okay, we've been doing some thinking. We think we've tackled the 4 by 4 question, and we've come up with a little on the general formula. It looks like finding a closed form general formula might get kind of hairy."
"Anyway, the first thing we (fellow Math Doctor Ethan Magness and I) did was to eliminate all the different rotations and color changes, and call them all essentially the same pattern. Once you figure out what the "fundamental" patterns are, you can rotate them four different ways, and then choose the colors two different ways."
"In general, with an n by n array, you'll always figure out how many different fundamental layouts there will be, and then you'll rotate it four ways and choose the colors n! (which means n(n-1)(n-2)...(2)(1)) ways. So if there are f different fundamental arrays, there are 4fn! total different layouts for your final answer."
"But there's a complication, because sometimes you'll end up duplicating a pattern by rotating it. For instance, in the 2 by 2 case, there's only one fundamental array you can make, one that has one color on n the top row and the other color on the bottom row. When you rotate this array though, you'll see that rotating it 180 degrees will produce the same array as not rotating it at all. So we've got to get rid of half of these, so we say that there are .5 different fundamental layouts, and then we multiply by 4n!, i.e. 8, and we end up with your answer, 4."
"In the 3 by 3 case, we can have three different fundamental layouts. However, if you look at the one that has three straight pieces arranged in columns, you'll notice that again, rotating it 180 degrees has the same effect as not rotating it at all. So really, we'll say we have 2.5 different fundamental arrays, and that the total is 2.5 by 4 by 6 = 60 different layouts."
"We did this for the 4 by 4 case too, and we found that there were 28.25 different fundamental arrays you could have (we simplified the process a little by noticing that some of the arrangements were just reflections of each other, and by first looking at arrangements that contained a long four-in-a-row piece across the top). So we found that you would have 28.25 by 4 by 24 = 2712 different total arrangements."

EXHIBIT REFERENCE:

There is no direct lrelationship to an exhibit component, but the skills reinforced in this activity are needed to understand the Mandelbrot set and Lorenz attractor of the Chaos computer interactive and the combinatorics behind many graph theory activities listed below:

Chaos Computer Interactive

Introduction to Graphs Electromechanical

Critical Path Manual interactive

Perfect Matching Manual interactive

Traveling Salesperson Electronic Interactives

Each time the player asks a question he receives a counter.
When the player has completed, the explainer should refer back to the possibilities chart and point out that because of their reasoning skills, the players were able to narrow down the choices and find the pattern with fewer than 60 clues. The number of counters indicates how few clues were needed.

Two-by-two grids are for younger children. Four-by-four grids are for those who want a bigger challenge.

BIBLIOGRAPHY:

Stenmark, Jean Kerr, Virginia Thompson and Ruth Cossey; Family Math, Lawrence Hall of Science, Berkeley, 1986

COPYRIGHT: 1995 Maryland Science Center

AUTHOR:

Cathy Brady, Math Specialist
Maryland Science Center

Object:
Players are to duplicate what is behind the wall without seeing the game board behind the wall.

Procedure:
Players ask the person behind the wall to tell the colors in a row OR a column by number. The explainer states the colors, but not necessarily in order. Every square must share at least one full border with another square with the same color marker.

Here are Gameboards to print out.  Below these are pictures of the possible combinations for the 2, 3 and 4 square boards.

For the two square board

For the three square board

Here are possible answers for the four square board: