Home Page Brainteasers Guess The Photo For Geeks & Brainiacs Lateral Thinking Number Puzzles Perceptual Puzzles Riddles Spot The Difference Target Number Word Puzzles Monthly Puzzle Contest Monthly Contest Winners Monthly Newsletter Link To Us References Send Us Your Puzzle Printable Puzzles Logic Puzzles Solitaire Games

Follow us:

Share this page on:

Need a Hint?

HINT: The lockers that are closed at the end are ones that have been closed an odd number of times. Which lockers are closed an odd number of times and why (i.e. what is unique about them)?

See Answer

ANSWER: 31 lockers.

EXPLANATION: The number of times a locker is opened or closed is equal to the number of factors the locker number has. For example, locker number 21 has four factors (1, 3, 7, and 21) and will be opened and closed a total of four times. The first student will close it, the 3rd student will open it, the 7th student will close it, and finally, the 21st student will open it.

All lockers that are reversed an even number of times will end in the open position (the same position they started in). All lockers that are reversed an odd number of times will end in the closed position. So the question is how many numbers between 1 and 1000 have an odd number of factors?

If you take certain numbered lockers at random, you will notice that most have an even number of factors because the factors are "paired up". For example, the locker number 12 has six factors which can be paired up as: 1 x 12, 2 x 6, and 3 x 4. It turns out that the only numbers that have an odd number of factors are square numbers (perfect squares) such as 1, 4, 9, 16... Take locker number 16 for instance. The 1 pairs up with the 16, the 2 pairs up with the 8, but the 4 is unpaired (since it essentially pairs with itself). So locker number 16, like all square numbers (pefect squares), has an odd number of factors.

The highest square number (perfect square) that is 1000 or less is 961 (31 squared) which is of course the 31st square number (perfect square). So 31 lockers will end up closed. With a calculator, you can make a list of all the square numbers (perfect squares) between 1 and 1000. There are 31 in all: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961. Each of those 31 lockers will end up closed.

Do you have a suggestion for this puzzle (e.g. something that should be mentioned/clarified in the question or solution, bug, typo, etc.)?

<- Previous puzzle | Next puzzle ->

Click here to return to the list of puzzles

Contact Us| Bookmark Site| Terms Of Use| Privacy Policy