Forumophilia - 18 Years Online!
ramblin'man
baralis wrote:
Baralis, you get the Gold Star, even if you did leave out #9.
Next time you're in Toronto, stop in at this place to collect your, uhh, bonus points.
Good explanation too. The problem deals with FACTORS of a number, which are other numbers that divide into the original evenly (with no remainder). For example, consider the number 80. Factors of 80 include: 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80 itself. All the girls with those numbers will change the state of locker #80.
All positive integers have an even number of factors, except for Perfect Squares. This includes Prime Numbers, which only have two factors: 1, and themselves (e.g. 2, 3, 5, 7, 11, 13, 17, etc).
So lockers with an even number of factors will end up closed. Those with an odd number of factors -- the perfect squares -- will end up open.
RM
RM,
I'm going to suggest that after the models have done their stuff they will have left the following lockers open:
1, 4, 16, 25, 36, 49, 64, 81, 100
The reason being that each locker needs to have an odd number of visitors in order for it to stay open. For the general case, locker N is visited by:
girl 1 and girl N (always) [for locker 1 this is the same girl]
if the locker is visited by girl X then it must also be visited by girl Y (where Y=N/X).
Therefore the only way that a locker can have an odd number of visits is if at some point X=Y (i.e. it is a square number).
That's my logic anyway! Phew - I don't normally come on here to exercise the largest organ in the body (just the second largest
)
I'm going to suggest that after the models have done their stuff they will have left the following lockers open:
1, 4, 16, 25, 36, 49, 64, 81, 100
The reason being that each locker needs to have an odd number of visitors in order for it to stay open. For the general case, locker N is visited by:
girl 1 and girl N (always) [for locker 1 this is the same girl]
if the locker is visited by girl X then it must also be visited by girl Y (where Y=N/X).
Therefore the only way that a locker can have an odd number of visits is if at some point X=Y (i.e. it is a square number).
That's my logic anyway! Phew - I don't normally come on here to exercise the largest organ in the body (just the second largest
)Baralis, you get the Gold Star, even if you did leave out #9.
Next time you're in Toronto, stop in at this place to collect your, uhh, bonus points.
Good explanation too. The problem deals with FACTORS of a number, which are other numbers that divide into the original evenly (with no remainder). For example, consider the number 80. Factors of 80 include: 1, 2, 4, 5, 8, 10, 16, 20, 40, and 80 itself. All the girls with those numbers will change the state of locker #80.
All positive integers have an even number of factors, except for Perfect Squares. This includes Prime Numbers, which only have two factors: 1, and themselves (e.g. 2, 3, 5, 7, 11, 13, 17, etc).
So lockers with an even number of factors will end up closed. Those with an odd number of factors -- the perfect squares -- will end up open.
RM
ramblin'man
Here's a somewhat easier problem. Anyone out there besides Baralis and melmac? 
=============
In the United States, what is the most amount of coin money I can have without being able to make EXACT change for a dollar ($)? And how many coins of each would be included (multiple possible answers)?
Countin'Man
p.s. For those of you not in the US and/or not familiar with US coin money...
1$ = 100 cents
1$ = 2 Half-Dollars, or 50-cent coins
1$ = 4 Quarters, where a quarter is 25 cents
1$ = 10 Dimes, where a dime is 10 cents
1$ = 20 Nickels, where a nickel is 5 cents
1$ = 100 pennies, where a penny is 1 cent
=============
In the United States, what is the most amount of coin money I can have without being able to make EXACT change for a dollar ($)? And how many coins of each would be included (multiple possible answers)?
Countin'Man
p.s. For those of you not in the US and/or not familiar with US coin money...
1$ = 100 cents
1$ = 2 Half-Dollars, or 50-cent coins
1$ = 4 Quarters, where a quarter is 25 cents
1$ = 10 Dimes, where a dime is 10 cents
1$ = 20 Nickels, where a nickel is 5 cents
1$ = 100 pennies, where a penny is 1 cent
baralis
Poster
spdspk
Sugarman
Very Respected Poster
Sugarman
Very Respected Poster
ramblin'man
Sugarman wrote:
Well if you're going to allow "Division by Zero", then we can prove that any number equals any other number.
I've seen proofs that can do this but do not use division by zero, will research. Who was it that said "Black Holes are where God tried to divide by zero"?
Apologies for the delay, I've been away for a while. Sugarman (and others), thanks for your replies before. $1.19 was of course the correct answer, but you get an A for creativity. Kudos.
C'mon fellas, let's exercise that other head for a bit.
NEXT PROBLEM
Consider a non-digital clock with 2 hands of equal width:
1. The Hour hand moves at a constant slow speed around the clock, e.g. it takes exactly 60 minutes to move between each number, 30 minutes to move halfway between each number, and so on.
2. The Minute hand moves at a constant faster speed around the clock, e.g. it takes exactly 300 seconds to travel between each number.
(the key point here is that both hands are ALWAYS constantly moving, no clicks or jerks).
At precisely 12:00 noon -- 12:00:00.00 -- the Hour and Minute hands perfectly overlap each other width-wise for a teensy-tiny fraction of a moment.
What will be the exact time when next they overlap?
Please give your answer down to the nearest HUNDREDTH of a SECOND. This is a non-trivial problem, worthy of such brainiacs as we have here in this forum, and not as easy as it sounds. But its also relatively simple once you figure out the key idea. And yeah, you'll need a calculator.
Good luck.
RM
p.s. a man's alarm clock.
Divide both sides of the equation by the common factor (Y-Y)[/b]
Well if you're going to allow "Division by Zero", then we can prove that any number equals any other number.
I've seen proofs that can do this but do not use division by zero, will research. Who was it that said "Black Holes are where God tried to divide by zero"?
Apologies for the delay, I've been away for a while. Sugarman (and others), thanks for your replies before. $1.19 was of course the correct answer, but you get an A for creativity. Kudos.
C'mon fellas, let's exercise that other head for a bit.
NEXT PROBLEM
Consider a non-digital clock with 2 hands of equal width:
1. The Hour hand moves at a constant slow speed around the clock, e.g. it takes exactly 60 minutes to move between each number, 30 minutes to move halfway between each number, and so on.
2. The Minute hand moves at a constant faster speed around the clock, e.g. it takes exactly 300 seconds to travel between each number.
(the key point here is that both hands are ALWAYS constantly moving, no clicks or jerks).
At precisely 12:00 noon -- 12:00:00.00 -- the Hour and Minute hands perfectly overlap each other width-wise for a teensy-tiny fraction of a moment.
What will be the exact time when next they overlap?
Please give your answer down to the nearest HUNDREDTH of a SECOND. This is a non-trivial problem, worthy of such brainiacs as we have here in this forum, and not as easy as it sounds. But its also relatively simple once you figure out the key idea. And yeah, you'll need a calculator.
Good luck.
RM
p.s. a man's alarm clock.
Sugarman
Very Respected Poster
ramblin'man
Sugarman wrote:
Correct! But I have to ask ... how did you get your answer, you sweet guy you? I ask because there are basically 2 ways to get it, the long way and the sneaky way.
RM
Correct - one of the rules of algebra is that you cannot divide by zero. The result is undefined
The clock will show the time as 01:05:27.27
The exact (military) time will be 13:05:27.27
The clock will show the time as 01:05:27.27
The exact (military) time will be 13:05:27.27
Correct! But I have to ask ... how did you get your answer, you sweet guy you? I ask because there are basically 2 ways to get it, the long way and the sneaky way.
RM
ramblin'man
Sugarman, here's a little proof that 2 = 1 which doesn't use any division by zero. Instead, it uses a little Calculus, which I'm sure everyone remembers. 
2 = 1
First, let X be a non-zero number. Then
X = 1 + 1 + 1 + ... + 1
(X times, the "1" is listed X times on the right hand side)
Multiply both sides of the equation by X:
X^2 = X + X + X + ... + X
Now take the Derivative with respect to X:
2X = 1 + 1 + 1 + ... + 1
But the Right Hand Side is simply equal to X, so substitute:
2X = X
Divide both sides by X, which is non-zero, to get
2 = 1
Q.E.D.
Anyone see the problem? Anyone? Anyone? Beuhler? Beuhler?
RM
2 = 1
First, let X be a non-zero number. Then
X = 1 + 1 + 1 + ... + 1
(X times, the "1" is listed X times on the right hand side)
Multiply both sides of the equation by X:
X^2 = X + X + X + ... + X
Now take the Derivative with respect to X:
2X = 1 + 1 + 1 + ... + 1
But the Right Hand Side is simply equal to X, so substitute:
2X = X
Divide both sides by X, which is non-zero, to get
2 = 1
Q.E.D.
Anyone see the problem? Anyone? Anyone? Beuhler? Beuhler?
RM
Sugarman
Very Respected Poster
ramblin'man wrote:
Me sneaky...never!
Basically, the minute hand travels at twelve times the speed of the hour hand. So when the hour hand is at 1 o'clock, the minute hand is at noon (obviously). For the minute hand to reach '1' takes 300 seconds - by then the hour hand has progressed 300/12 or 25 seconds. The formula is therefore 300 + 300/12 + 300/12/12 and so on.
Seemed like a logical approach.
Apparently, using calculus you can turn a tennis ball inside-out without cutting it.
Here's another:
According to Darwin's Theory of Evolution,
what came first logically- the chicken or the egg?
Sugarman wrote:
Correct! But I have to ask ... how did you get your answer, you sweet guy you? I ask because there are basically 2 ways to get it, the long way and the sneaky way.
RM
The clock will show the time as 01:05:27.27
The exact (military) time will be 13:05:27.27
The exact (military) time will be 13:05:27.27
Correct! But I have to ask ... how did you get your answer, you sweet guy you? I ask because there are basically 2 ways to get it, the long way and the sneaky way.
RM
Me sneaky...never!
Basically, the minute hand travels at twelve times the speed of the hour hand. So when the hour hand is at 1 o'clock, the minute hand is at noon (obviously). For the minute hand to reach '1' takes 300 seconds - by then the hour hand has progressed 300/12 or 25 seconds. The formula is therefore 300 + 300/12 + 300/12/12 and so on.
Seemed like a logical approach.
Apparently, using calculus you can turn a tennis ball inside-out without cutting it.
Here's another:
According to Darwin's Theory of Evolution,
what came first logically- the chicken or the egg?