John is passing out RED, WHITE, and BLUE construction paper for an art project. There are (12) people in his group. John, who likes to show off, tells everyone to watch closely as he hands out the paper. The first person gets a RED paper. John puts the second paper at the bottom of the stack. John gives the second person a WHITE paper. John puts the fourth paper at the bottom of the stack and gives the third person a BLUE paper. John continues this pattern until there is no more paper. How did he have to stack the colored paper so that he could pass it out this way. (Thank you for any help this has stumped me for hours)
Here's how he distributed the papers. The vertical bar separates the paper that have been distributed and those remaining.
The last line shows the papers in the original order which was distributed.
All we need to do is to put in the desired order of colours, RED, WHITE and BLUE.
1 2 3 4 5 6 7 8 9 10 11 12
1 | 3 4 5 6 7 8 9 10 11 12 2
1 3 | 5 6 7 8 9 10 11 12 2 4
1 3 5 | 7 8 9 10 11 12 2 4 6
1 3 5 7 | 9 10 11 12 2 4 6 8
1 3 5 7 9 | 11 12 2 4 6 8 10
1 3 5 7 9 11 | 2 4 6 8 10 12
1 3 5 7 9 11 2 | 6 8 10 12 4
1 3 5 7 9 11 2 6 | 10 12 4 8
1 3 5 7 9 11 2 6 10 4 | 8 10
1 3 5 7 9 11 2 6 10 4 8 | 10
1 3 5 7 9 11 2 6 10 4 8 10
1=RED
3=WHITE
5=BLUE
7=RED
9=WHITE
11=BLUE
2=RED
6=WHITE
...
To figure out how John stacked the colored paper in order to pass it out in the specified pattern, let's follow the steps:
1. John starts by giving the first person a RED paper.
2. Then, he puts the second paper at the bottom of the stack.
3. John gives the second person a WHITE paper.
4. Next, he puts the fourth paper (which is the second WHITE paper) at the bottom of the stack.
5. John gives the third person a BLUE paper.
6. Then, he puts the sixth paper (which is the second BLUE paper) at the bottom of the stack.
7. John continues this pattern until there is no more paper.
Based on this pattern, you can observe that John is giving out the colors in the following order: RED, WHITE, BLUE, WHITE, RED, BLUE, WHITE, BLUE, RED, WHITE, BLUE, RED.
To figure out how he stacked the colored paper, we need to reverse the pattern and figure out the appropriate position for each color. We know that there were 12 people, so there should be 12 colored papers. Let's assign numbers to the colored papers as they should be stacked:
1. RED
2. WHITE
3. BLUE
4. WHITE
5. RED
6. BLUE
7. WHITE
8. BLUE
9. RED
10. WHITE
11. BLUE
12. RED
Now, we need to find the corresponding positions in the stack. After each step, we put the paper at the bottom of the stack. Based on this, we can determine the stacking order:
1. Place the RED paper on top.
2. Add the WHITE paper on top.
3. Place the BLUE paper on top.
4. Add the second WHITE paper to the bottom.
5. Place the second RED paper on top.
6. Add the second BLUE paper to the bottom.
7. Place the third WHITE paper on top.
8. Add the third BLUE paper to the bottom.
9. Place the fourth RED paper on top.
10. Add the fourth WHITE paper to the bottom.
11. Place the fifth BLUE paper on top.
12. Add the final RED paper to the bottom.
So, the stacked order of the colored construction paper for John to pass them out in the specified pattern is:
1. RED
2. WHITE
3. BLUE
4. WHITE
5. RED
6. BLUE
7. WHITE
8. BLUE
9. RED
10. WHITE
11. BLUE
12. RED