Today, Jacob is holding a livestream for his birthday. Before going live, his crew got together and decided to secretly hide several presents in 50 separate boxes numbered 1 to 50. Jacob is now about to open the boxes, which are arranged in a row in order. His crew explains that if he follows the instructions below, he will discover which boxes hold a gift. Jacob needs your help to figure out which boxes hold presents without actually carrying out the crew’s instructions. The instructions he received are: 1. Go down the line and open every box. 2. Starting with box # 2, close every other box going down the row. 3. Starting with box # 3, change every third box (open the box if it is closed and close the box if it is open). 4. Starting with box # 4, change every fourth box. 5. Starting with box # 5, change every fifth box. 6. Continue this process through box #50. Your task: a. Which boxes will contain the gifts? Follow the instructions and tell Jacob which boxes he should check. b. Examine the numbers of the boxes with presents. What patterns do you notice? c. Jacob wished the crew had hidden more presents so there would have been 200 boxes. If the crew used the same pattern to hide presents, which boxes would contain gifts this time? How many more gifts would he get? Explain how you know.
a. To determine which boxes will contain the gifts, we need to follow the given instructions step by step.
Step 1: All boxes are initially closed.
Step 2: Starting with box #2, we close every other box. This means boxes #2, #4, #6, ... all the way up to #50 will be closed.
Step 3: Starting with box #3, we change every third box. Since all boxes were closed after step 2, we will open boxes #3, #6, #9, ... up to #48.
Step 4: Starting with box #4, we change every fourth box. Boxes #4, #8, #12, ... all the way up to #48 will be opened.
Step 5: Starting with box #5, we change every fifth box. Boxes #5, #10, #15, ... up to #50 will be opened.
...
This process continues until we reach box #50.
Following these instructions, the boxes that will contain gifts are: 1, 4, 9, 16, 25, 36, and 49.
b. Examining the numbers of the boxes with presents, we can notice a pattern. The box numbers that contain gifts are all perfect squares: 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, and 7^2.
c. If there were 200 boxes arranged in the same pattern, following the above instructions would give us the boxes with gifts. Since 200 is not a perfect square, we would only check the boxes up to the square root of 200 (approximately 14.14). Hence, the boxes that would contain gifts are 1^2, 2^2, 3^2, ..., 14^2, which are 1, 4, 9, ..., 196.
To calculate how many more gifts Jacob would get, we subtract the number of gifts from the total number of boxes: 200 - 14 = 186. Therefore, Jacob would get 186 more gifts.
To determine which boxes contain gifts and to answer the additional questions, we can follow the instructions provided step by step.
Step 1: Go down the line and open every box.
In this step, Jacob will open all 50 boxes.
Step 2: Starting with box #2, close every other box going down the row.
Jacob will close box #2, skip box #3, close box #4, skip box #5, and so on, until he reaches box #50.
After this step, the even-numbered boxes will be closed (#2, #4, #6, ..., #50) and the odd-numbered boxes will remain open (#1, #3, #5, ..., #49).
Step 3: Starting with box #3, change every third box (open the box if it is closed and close the box if it is open).
Jacob will change the state (open or closed) of box #3, box #6, box #9, and so on, until he reaches box #48.
After this step, box #3 and all other boxes that are multiples of 3 (e.g., #3, #6, #9, ..., #48) will be closed, while all other boxes will remain in their current state.
Step 4: Starting with box #4, change every fourth box.
Jacob will change the state of box #4, box #8, box #12, and so on, until he reaches box #48.
After this step, box #4 and all other boxes that are multiples of 4 (e.g., #4, #8, #12, ..., #48) will be open if they were previously closed, while all other boxes will remain in their current state.
Step 5: Starting with box #5, change every fifth box.
Jacob will change the state of box #5, box #10, box #15, and so on, until he reaches box #50.
After this step, box #5 and all other boxes that are multiples of 5 (e.g., #5, #10, #15, ..., #50) will be open if they were previously closed, while all other boxes will remain in their current state.
By following these instructions, we can determine which boxes will contain gifts:
Boxes #1, #4, #9, #16, #25, #36, and #49 will contain gifts.
Now let's examine the numbers of the boxes with presents to look for any patterns:
If we observe the numbers of the boxes with gifts, we can notice that they are all perfect squares. The boxes that contain gifts have numbers that are squares of consecutive positive integers: 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, and 7^2.
Now, let's consider if there were 200 boxes following the same pattern to hide presents:
Following the same pattern, we would need to find all the perfect squares up to 200. The largest perfect square less than or equal to 200 is 14^2 = 196. Therefore, if there were 200 boxes, boxes #1, #4, #9, #16, #25, #36, #49, #64, #81, #100, #121, #144, #169, and #196 would contain gifts.
To find the additional number of gifts, we subtract the number of boxes in the original scenario (7) from the number of boxes in the new scenario (14). Thus, Jacob would get 14 - 7 = 7 more gifts compared to the original scenario.
To determine which boxes will contain gifts, we need to follow the given instructions step by step. Here's how we can do it:
1. Go down the line and open every box:
- Open all the boxes.
2. Starting with box #2, close every other box going down the row:
- Close boxes #2, #4, #6, #8, and so on, until box #50.
3. Starting with box #3, change every third box:
- Change the status (open or closed) of boxes #3, #6, #9, #12, and so on, until box #50.
4. Starting with box #4, change every fourth box:
- Change the status of boxes #4, #8, #12, #16, and so on, until box #48.
5. Starting with box #5, change every fifth box:
- Change the status of boxes #5, #10, #15, #20, and so on, until box #50.
6. Continue this process through box #50.
Now, let's examine the patterns and determine which boxes contain gifts:
a. Boxes that will contain gifts:
- Box #1 (always open since step 1)
- Box #4 (changed in steps 3 and 4)
- Box #9 (changed in steps 3 and 5)
- Box #16 (changed in step 4)
- Box #25 (changed in step 5)
- Box #36 (changed in step 4)
- Box #49 (changed in step 5)
b. Patterns for boxes with presents:
- The box numbers of the boxes with presents are perfect squares. These boxes appear to follow a pattern where the box number is equal to the square of the step number (i.e., 2^2, 3^2, 4^2, and so on).
c. If there were 200 boxes and the crew used the same pattern to hide presents, the boxes that would contain gifts can be determined. We need to find the perfect squares less than or equal to 200. By following the pattern mentioned above, we can identify the boxes with gifts.
There are 14 perfect squares less than or equal to 200: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and 196.
Therefore, if there were 200 boxes, Jacob would get 14 more gifts hidden in the boxes with these numbers.
To summarize:
a. Boxes that contain gifts are 1, 4, 9, 16, 25, 36, and 49.
b. The pattern for boxes with presents is they have box numbers that are perfect squares.
c. If there were 200 boxes, the boxes that would contain gifts are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and 196, totaling 14 more gifts.