Erin Kemp
Grade 10, Germantown Academy, Fort Washington, Pennsylvania

The answer to the problem of the month is there are thirteen different ways of coloring four out of eight blocks of a sqare in, without any of them being the same when rotated of turned upside down. The way I got this answer was by starting with putting the four sections right next to eachother. I saw that there were only two ways to arrange the four consecutive sections without them being the same when rotating them. After putting the four sections together, I put three of the sections together with the last section separated from them. By putting three sections together and one apart, I found three combinations with this pattern without them being the same. After finishing this combination, I put two of the sections together and tried to find how many different ways that you could place the other two in different combinations that weren't the same any way that you rotated them. When I put the two sections together, I tried putting the two on the horizontal part of the square. I noticed that you could also put the sections togather on the corner of the square to find more combinations. When putting two sections together, There were seven combinations. The last combination that was not the same as any other, was putting one space inbetween every colored section. This was the only Combination that had the colored and noncolored sections equally spread throughout the square. By starting with the four sections together and then breaking them down by threes and two and ones, I covered all of the possible combinations.

Jen Peters
Grade 10, Germantown Academy, Fort Washington, Pennsylvania

Hi this is Jen Peters from Germantown Academy, here's my solution to the january project of the mounth. There are 13 different ways to color half of the block. How many ways is there to form half of eight? There is 4, 3&1, 2&2, 2&1&1, and 1,1,1&1. So half of the block will be colored if there are 4 adjacent sections colored, or any of the other 4 ways. From there I found how many possible ways there were to color each of the 5 combinations of numbers. I found 2 ways to color in 4 adjacent sections, 3 ways to color 3 adjacent sections and one lone section, 3 for 2&2, 4 for 2&1&1, and there is one way to color in half of eight sections were none of the sections are adjacent. sorry i dont have graphics, but i havent mastered the art of working all my weired paint/graphing programes. I'll try to send something from school.

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