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.

Answer 1

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.