Locker Problem Final

Locker Problem : Third Draft

After testing it out, I've concluded that there will be 31 lockers open in the end. The reason for that is simply because there are 31 perfect squares from 1 - 1,000. Why are the lockers who have perfect squares open? Well, the perfect square lockers have been touched an odd number of times because there is an odd amount of factors for each perfect square (See diagram 1). For example; the factors of 9 are 1, 3, and 9. Compare that to 24, whose factors are 1, 2, 3, 4, 6, 8, 12, and 24. Each number has the factors of 1, and itself. Meaning, each locker was touched by student number one, and the student with the same number as the locker number. So, each locker was touched at least an even amount of times. Since each locker was initially open, in order for it to be open again, it must be touched an odd number of times. For example; for locker number 4, its factors are 1, 2, and 4. It would be touched by students 1, 2, and 4. Student one opens it, student two closes it, and student four opens it. Since there are no more factors of four, it will remain opened.

Diagram 1:

Lockers	1	2	3	4	5	6	7	8	9	10
State	o	ox	ox	oxo	ox	oxox	ox	oxox	oxo	oxox

How did I get the answer 31? Well, my strategy was, once I’ve noticed the ‘perfect squares pattern’, I tried to figure out the perfect square closest to 1,000. (See diagram 2).

Diagram 2:

These are the perfect squares that if they are to the second power, they are equal to a number from 1 – 1,000.

12 = 1	72 =49	132 =169	192 =361	252 =625	312 =961
22 = 4	82 =64	142 =196	202 =400	262 =676	322 =1024
32 = 9	92 =81	152 =225	212 =441	272 =729
42 =16	102 =100	162 =256	222 =484	282 =784
52 =25	112 =121	172 =289	232 =529	292 =841
62 =26	122 =144	182 =324	242 =576	302 =900

As you can see in the diagram above, 31 is the greatest square root or perfect square that does not exceed the 1 – 1,000 range. 32 is shown in the diagram above to show that it goes over 1,000.