Math Trivia

posted by .

1) A school has 1-- closed lockers. Larry comes into the school and "toggles" every second locker, then Fred toggles every 3rd, Bob every 4th, etc, up to Zoe who toggles every 100th locker. After all this"toggling" how many lockers are open?

2) If you take any valid time from a 12 hour clock, what is the maximum sum you can obtain by adding the digits? (Eg. For 7:14, the sum is 7+1+4=12)

THANKS!

• Math Trivia -

2)
Isn't it 12:59?

• Math Trivia -

yah thanks!

i am having a hard time with number one. i know i could sit down and take a long time to do it...but i was wondering if their was a formula to it, or a quicker way to do it...

• Math Trivia -

make a list of lockers up to whatever you feel like
put c for closed under each one
starting with 2,4,6,8,...switch the c to o, for open
starting with 3,6,9,12,.. switch to c's to o's, and the o's to c's
starting with 4,8,12,16,... switch the c's to o's, and the o's to c's
continue....

you will notice that only the perfect square numbers will be c's

so which are the perfect squares ?
1,4,9,16,25,36,49,64,81,and 100

somebody actually spent time and effort to create an applet that shows this pattern.
only in this problem the first student starts by switching all the lockers, so in their case all the lockers are initially open.
(For some reason on my computer it seems to jump to the final stage almost right away)

• Math Trivia -

I don't know how to do number 1 since it doesn't say how many lockers there are initially. It says "1--," but not sure if that should be something else.

• Math Trivia -

Question 1
This has to do with the number of factor of an integer. An even number of factors will cause the door to be toggled (open/close) an even number of times, therefore they will remain closed.

For example, a prime number has two factors:
7=(1,7), 13=(1,13).

Almost all other numbers have an even number of factors:
24=(1,2,3,4,6,8,12,24),
88=(1,2,4,8,11,22,44,88)

You will notice that the product of the first and the last factors give the number itself.

Since the factors are always different, there is an even number of them... except in the case of perfect squares, where the middle factors are identical, so we count them only once, and the locker will be visited an odd number of times, leaving them open.

Examples:
9=(1,3,9)
81=(1,3,9,27,81)
100=(1,2,5,10,20,50,100), etc.

Question 2:
If we were to add all digits separately, I would suggest 9:59 which gives a sum of 23 as the highest sum.

• Math Trivia -

MathMate is obviously correct. 9:59 yields a greater sum than 12:59.

Similar Questions

1. carver

there are lockerslined up numbered 1 to 1000.They are all closed . i walk by and open all the lockers . then Al walks by and goes to every second locker starting at #2and closes it. Then Mary goes to every third locker starting with …
2. Math

A new school has exactly 1,000 lockers and 1,000 students. the student meet outside the building and decide on this plan. The first student will enter and open al the lockers. The second student will enter and colse ever locker with …
3. Math

To confuseing I need help. In honor of school spirit week, the student council decided to decorate lockers in the main lobby. The 9th graders officers stuck a decal of Smiley, the school mascot, on every third locker, starting with …
4. Math 6th

Imagine there are 1000 lockers and 1000 students at school. The 1st student who comes through the school opens every locker as they enter the school. The 2nd student walks in and closes every other locker begining with locker number …
5. math

1000 lockers 1000 students, the first student goes along and opens every other locker. The second student goes along and shuts every other locker beginning with the number 2. The third student changes the state of every 3rd locker …
6. ALEGEBRA

The new school has exactly 1000 students and 1000 lockers.On the first day all the students meet outside and agree on the following plan: the first student will enter the school and open all the lockers.The second student will then …
7. math

the 20 students in Mr. Wolf's 4th grade class are playing a game in a hallway that is lined with 20 lockers in row. the 1 student starts with the first locker and goes down the hallways and opens all lockers. the 2 student starts with …
8. Math

There are 10,000 lockers. One person opens all the lockers. The second person closes every 2 doors. The third person counts every three locker and does the opposite to the locker. (Open/close it.) The fourth person counts every fourth …
9. Math

There are 10,000 lockers. One person opens all the lockers. The second person closes every 2 doors. The third person counts every three locker and does the opposite to the locker. (Open/close it.) The fourth person counts every fourth …
10. Algrebra

A new high school has just been opened, with an enrollment of 1000 students. The school has 1000 lockers, numbered 1 to 1000, to accommodate them. On the first day of school all 1000 lockers are closed. The first student enters the …

More Similar Questions