Chapter 7, Problem 7.1EAA

To determine

Determine how many and which lockers will be open at the end of the student’s activities.

Explanation of Solution

First let us consider for 15 lockers and 15 students.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

2

O

C

O

C

O

C

O

C

O

C

O

C

O

C

O

3

O

C

C

C

O

O

O

C

C

C

O

O

O

C

C

4

O

C

C

O

O

O

O

O

C

C

O

C

O

C

C

5

O

C

C

O

C

O

O

O

C

O

O

C

O

C

O

6

O

C

C

O

C

C

O

O

C

O

O

O

O

C

O

7

O

C

C

O

C

C

C

O

C

O

O

O

O

O

O

8

O

C

C

O

C

C

C

C

C

O

O

O

O

O

O

9

O

C

C

O

C

C

C

C

O

O

O

O

O

O

O

10

O

C

C

O

C

C

C

C

O

C

O

O

O

O

O

11

O

C

C

O

C

C

C

C

O

C

C

O

O

O

O

12

O

C

C

O

C

C

C

C

O

C

C

C

O

O

O

13

O

C

C

O

C

C

C

C

O

C

C

C

C

O

O

14

O

C

C

O

C

C

C

C

O

C

C

C

C

C

O

15

O

C

C

O

C

C

C

C

O

C

C

C

C

C

C

Let O = Locker Open.

 C = Locker Closed.

From the above table, we can make out that there are 3 lockers open at the end of 15^th student activity.

For 15 lockers and 15 students, the highest of the perfect squares number is 9 that is 3*3(that is, below number 15, we have 1, 4 (2²), and 9 (3²) as the perfect square numbers. In that, highest is 9). The square root of the highest of the perfect squares number gives the number of lockers open, that is v9 = 3.

Similarly, for 1000 lockers and students, 961 (31*31=961) is the highest of the perfect squares number. Therefore, v961 = 31, that is 31 lockers are open at the end of 1000^th student activity.

So, in an easier way we can say, for a particular student and locker number, the square root of highest of the perfect squares number is the number of lockers that will be open.

All the perfect square numbers (1, 4, 9, 16, 25….till 961) are the lockers that will be open.

Similarly, you can try for the below student and locker numbers:

For, 20 lockers and students, the number of lockers open is 4 (since 16 = 4*4 is the highest of the perfect squares number below 20)

For, 30 lockers and students, the number of lockers open is 5 (since 25 = 5*5 is the highest of the perfect squares number below 30)

Conclusion:

There are 31 lockers open at the end of 1000^th student activity. Also, all the perfect square numbers (1, 4, 9, 16, 25.....till 961) are the lockers that will be open.