The Color Square Game (aka. Rainbow Logic) 1 “The Color Square Game is one of the best logic activities that we know. It never fails to excite the students tremendously. It can be played as a whole class, in small groups, or with a combination of both, which is the way we use it, Its an easy game to understand, but itis very challenging to play: “The reasoning that students do is quite sophisticated. They also get prac- tice in explaining their reasoning to others. The rules of the game are simple. Use a three-by- 7 three square like the one shown at right. Color the squares with three colors—red, blue, and green—with three squares of each color. Squares of the same color ‘must be contiguous (joined on a side). Do not show stu dents your colored squares but draw a blank three-by- three squate on the board or overhead projector. It is up to the students to logically figure out what is in each square. Students begin asking for the colors in particular rows or colurons. For example, a student might ask, “What is in row 12* You might reply, “Row 1 contains 2 blues and 1 green” and write “2B, 1G” next to row 1. Do not divulge the order of the 2 blues and 1 green. Itis best if you can always reply in alphabetical order. Ask if anyone knows the color ofa particular square, “Fell students not to guess, but rather figure things out logically. Always ask why the student believes the square is that color. Ifthe student can prove that a square must be a particular color, then fill n that square. We like to let the students explain theic reasoning to their group. After all squares that can be Filled in are filled in, allow the students to ask another question about another row or column. Continue the game in this fashion until the students deduce the colors ofall nine squares. They should be told that they are try- ing to find the answer using as few clues as possible. Play the three-by-three game a few times. Then graduate to the four-by- four game where there are 16 squares: four of each of four colors, This is a much better game, We don’t recommend the five-by-five game. ‘The four-by-four game can be played in a number of ways. It is a good idea to discuss with the students how the pieces can be arranged. In fact, & good exercise forthe students isto ask them to find all ofthe ways that four squares can be put together so that they are touching along at least one entire side, ‘Manipulatives will make the game and the associated thinking accessible to more students quickly, Give each group four squares of each color and have them draw a large four-by-four square on a piece of paper. The group ‘manipulates the colored squares to try out different arrangements. Someone else in the group should have a four-by-four square written down to write in their conclusions. “The introductory game is played with the teacher making the solution then asking fora student to request a column or row. That one particular a TEACHING PROBLEM SOLVING‘lue is posted and then discussed until it has been exhausted, Let students discuss possible conclusions in their groups first. Then have several students explain their group's reasoning to the class. Keep asking if anyone knows more squares than are shown. When no more squares can be deduced, let someone ask for another clue, Depending on the clue, this may be quick or it may take a few minutes. The following is a sample game, There are four squares of each color: red, yellow, blue, and green. Each one will be abbreviated in the clues with the first letter. In this game, the first clue asked for is the fourth column, This column is comprised of one yellow and three greens. The colors are not necessarily given in order or out of order. This clue in particular would generate a fair amount of discussion as to where the yellow goes. At some point, a student should be able to deduce that the yellow must go at either the top or the bottom of the row because if it were placed in the second or third row there would not be enough greens to provide a contiguous figure. The students are required to be able to prove which squares are a certain color, not just to provide possibilities As the students explore the possibilities on this square they will realize that when- ever there are three squares available and two are a given color, the middle square must be that color, The next clue asked for was the second row. The students should be able to con- clude that the middle square of the three squares remaining in the row is blue because otherwise the blues could not connect, RGD 136 ya v.36[At this point the next clue the students 136 asked for was the fourth row. Using this in conjunction with the fourth column, it . ‘was quickly evident that the last three of gg 25 7 ce the fourth row were all yellow, and that : the first one must be red. ¢ aw jal rir] er ‘That will also lead to filling the top bial squate in the fourth column as green. As the students examine this set-up more, : it's likely that they will realize thatthe red qgze| a | 8 e in the second row must appear in the first column as it cannot bein the third column 7 e and still connect to the red in the fourth row. That will also lead them toconclude gay |e | y | v | ¥ that the square in between must be red in order for the reds to connect. ‘At this point the game becomes more fe difficult, but there are actually two more 1 7 squares that can be proven. The first one is Le ¢ the remaining square for the second row. The next one is the first square inthe sec &G2B| R | B | B | G ond column. The proof of that is an indi- rect proof: By assuming in succession that the square in question is either red, yellow, or green, you find thatitcannotbeanyof 83 | R | | ¥ | Y those because in each case those colors cannot connect to that square; thus the only possible remaining color is blue. Choosing the next clue at this point has to be done carefully. Row 5 must contain the fourth yellow and could contain the fourth blue, green, or red Ifit contains the missing blue and yellow, then asking for that clue will be inconclusive as to their exact location. The second column would also be inconclusive if it shows two blues and two yellows, as either of the two squares in column 3 could go to either blue ot green. Asking for the clue for row 1 would tell us where the final red goes, but ifthe red is in that row, we will have the same problem with where the blue and the yellow are placed. So at this point, there is no guarantee that the next guess will end the game. ‘Atthis point you can let each group decide on their own what to guess next, and walk around the room giving each group their own clue. The next clue request in this game happened to be the second column. TEACHING PROBLEM 50}The game basically ends now by filling rR 136 in each of che colors in succession: the missing yellow, the missing green, and the 8 g missing blue. eer eee r|R « er ee le 28 The completed game is shown at right a 136 ‘Afeer you've played the game several : times with the whole class, let students ajalele play the game in their groups. One person makes up the solution, and the other three 6.28 solve it. The tendency for the students is to be too willing to make false conclusions, The teacher needs to set the tone by hav- ing each person who makes an assertion ell oe prove it step-by-step, In another session, have students play the game in pairs in order to ensure that each student is an active participant. Suggest students play the game at home with friends and family. ‘The game can be developed a couple of steps further, The examples below can be written on the board or on an overhead projector with the instructions to “fil in as many squeres as you can prove.” Gy « xy 2 ak 3 [ T 3y MGR RIB BIRY “The problem on the left has two squares that cannot be proved, and the one on the right can be entirely proved, v7Below is another variation: CLI wy can prove are a Fill in as many squares as yo ‘The rainbow logic game dev interaction, the concept of proof atmosphere of questioning and substantiating See lake it Simpler, by Carol Meyer and Tom for more information about Color Square. relops step-by-step 1 f by contradiction indirect proof), and an . nm siven color. easoning, sal rOUp sallee (Addison Wesley, 1985)