A Fistful of Brain Teasers

Eloise posted:
rjstaines posted:
TOBYJUG posted:

..Almost everyone is familiar with Meatloafs ' I'd do anything for love ( but I won't do that). famous hit from Bat out of Hell.

Does any one know what THAT was, THAT which wouldn't be done ?

The last time I talked to Meatloaf**, he wasn't giving any response to this question

** I'm blatently and unashamedly name dropping here as you have already summised, but it's true... I met Meatloaf back in 1984 at a Heathrow hotel - he'd just dropped his motorcycle trying to park it and wasn't feeling very communicative.  

In other words, he didn't explain in 1984 what the lyrics to a song he wouldn't perform until 1993 meant?  Strange that. :-)

He has explained (since) ... including using blackboard to illustrate on VH-1 (TV).

Innocent Bystander posted:

Not a song I particularly like, but isn’t the whole point that it is up to the listener to imagine?

No; its up to the listener to listen to the lyrics of the song ... each verse has two things he (Meatloaf) would do for love, and one thing he wouldn't...

In other words in the first verse:

And I would do anything for love
I'd run right into hell and back
I would do anything for love
I'd never lie to you and that's a fact
But I'll never forget the way you feel right now
Oh no, no way
And I would do anything for love
Oh I would do anything for love
I would do anything for love, but I won't do that
No, I won't do that

He would "run right into hell and back" and would "never lie to you" but he wouldn't "forget the way you feel right now".

Jim Steinmann thought it would confuse the listeners apparently, but Meatloaf felt it was clear.  I guess Mr Meatloaf was wrong!

Seeing the words (I never had enough interest in the song to follow it fully when sung, let alone look up the lyrics), it is still ambiguous, in that as well as “won’t do that” reinforcing the statement that he’ll never “forget the way you feel right now”, it also could be reinforcing that he would never lie to her (him?).

Of course, if he is lying...

Eloise posted:
rjstaines posted:

The last time I talked to Meatloaf**, he wasn't giving any response to this question

** I'm blatently and unashamedly name dropping here as you have already summised, but it's true... I met Meatloaf back in 1984 at a Heathrow hotel - he'd just dropped his motorcycle trying to park it and wasn't feeling very communicative.  

In other words, he didn't explain in 1984 what the lyrics to a song he wouldn't perform until 1993 meant?  Strange that. :-)

He has explained (since) ... including using blackboard to illustrate on VH-1 (TV).

Hey, at my age what's ten years between friends ?   1974, 1984, 1994... they are all back in the mists of time; we were still on NAT01 and the NAP135 ruled the roost. 

Innocent Bystander posted:
Don Atkinson posted:

Another one, this time from the 7th century..........

Long since, the holy power that made all things

So made me that my master’s dangerous foes

I scatter. Bearing weapons in my jaws,

I soon decide fierce combats; yet flee

Before the lashings of a little child

It’s beaten me, though probably blindingly obviious - weapons in jaws, suggests an animal with fearsome fangs, but fleeing from a child..? So I think not a living thing, but an abstract concept from religeous, possibly Christian teaching, but not satan.

I'm still erring towards an animal. 7th century suggests the weapons will either be sharp or blunt instruments, rather than anything modern in style. My first pass was a bee, it scatters foes of the hive, will fly away from a kid waving their arms, but I suspect even in the 7th century they didn't think the sting was in the mouth, the queen as master of the hive is a stretch, and bees don't decide fierce combats in any way I can stretch to.

Dog isn't a bad shout. A dog's teeth can be seen as weapons. A guard dog can still scatter dangerous foes, and even more so in the 7th century. The fierce combats could be dogfights, hunting or even warhoundsr. And yet most dogs will flee from a kid hitting them. It's a bit of a stretch, but the best I can do.

While my brain hurts thinking about the ladder one I came back to this, 1st thoughts on that soon.

 

I also wondered about dog, but dogs that are not pets  wouldn’t  flee from a child lashing out.

i’ve considered a weapon like a ballista, but though it might be damaged by a child’s flailing arms, it scarcely flees them. 

“Lashings”, of course, could be lashings by tongue rather than physical: though it usually refers to acerbic or sarcastic comments -  was that the case in the 7th century and could it be referring to the sound of a screaming child?   (That doesn’t prompt anything for me, unless dragons were reputed to be scared of screaming children?)

  The original recording of " Massachusetts" in 1967 on the 'Horizontal" album by all of the Bee Gees took 2:19. Minutes.

 Later from a live 1997 version without Andy Gibbs the boys took 2:27. Minutes to perform this song.

If Barry was to today perform this song on his own, how many minutes and seconds would it take him ?

TOBYJUG posted:

  The original recording of " Massachusetts" in 1967 on the 'Horizontal" album by all of the Bee Gees took 2:19. Minutes.

 Later from a live 1997 version without Andy Gibbs the boys took 2:27. Minutes to perform this song.

If Barry was to today perform this song on his own, how many minutes and seconds would it take him ?

Did Andy ever perform as part of the Bee Gees ? As opposed to “not” performing at a specific live event ?

Eoink posted:
Innocent Bystander posted:
Don Atkinson posted:

Another one, this time from the 7th century..........

Long since, the holy power that made all things

So made me that my master’s dangerous foes

I scatter. Bearing weapons in my jaws,

I soon decide fierce combats; yet flee

Before the lashings of a little child

It’s beaten me, though probably blindingly obviious - weapons in jaws, suggests an animal with fearsome fangs, but fleeing from a child..? So I think not a living thing, but an abstract concept from religeous, possibly Christian teaching, but not satan.

I'm still erring towards an animal. 7th century suggests the weapons will either be sharp or blunt instruments, rather than anything modern in style. My first pass was a bee, it scatters foes of the hive, will fly away from a kid waving their arms, but I suspect even in the 7th century they didn't think the sting was in the mouth, the queen as master of the hive is a stretch, and bees don't decide fierce combats in any way I can stretch to.

Dog isn't a bad shout. A dog's teeth can be seen as weapons. A guard dog can still scatter dangerous foes, and even more so in the 7th century. The fierce combats could be dogfights, hunting or even warhoundsr. And yet most dogs will flee from a kid hitting them. It's a bit of a stretch, but the best I can do.

While my brain hurts thinking about the ladder one I came back to this, 1st thoughts on that soon.

 

These riddles are never precise in the sense of 2+2=4

A Dog it is !

Well done

The ladder and the cube - nice problem; unexpectedly difficult.

I considered the general case of a cube with side a and a ladder of length 2L. The outer top edge of the cube then divides the ladder into sections of length L+d and L-d. Note that there are two symmetric solutions depending on whether the ladder is placed at a shallow or steep angle.

The first step is to find d. To do this, use Pythagoras on the two small right angled triangles in the diagram. You also need to use the fact that the slope of the ladder is constant: a/x = y/a (where x is the distance between the bottom edge of the cube and the bottom of the ladder, and y is the distance between the inner top edge of the cube and the top of the ladder). After much faffing, I get:

d^2 = (L^2 + a^2) + (or -) sqrt(a^4 + 4L^2a^2)

Then use Pythagoras on the small right angled triangle at the bottom of the diagram:

x^2 = (L-d)^2 - a^2

to obtain x. I prefer to leave the solution as a two-step process rather than write a single formula for x in terms of L.

I checked with L = sqrt(2) and a = 1, and (taking the - in the + or -)  this gives d = 0 so x = 1, as expected (i.e. the ladder is bisected by the top outer edge of the cube to form two right-angled isosceles triangles).

Hi ITBUFFSTER,

Yes, this is a problem that is a bit more difficult to solve than might first appear.

Your solution is interesting and not one that I have seen before. Nonetheless, it should work a treat.

I note that you have used “d” to differentiate the length of ladder above the chest, and the length below the chest. The original question uses “d” to define the required solution for the distance of the foot of the ladder from the chest – you have used “x” for this dimension. No problem, I just mention it for clarity, especially for others who might be following your solution. In a similar vein, you have used “2L” for the length of the ladder as opposed to “L” as per the teaser text. Again, no problem, but we need to be aware.

I tested my own solutions (I have three solutions) by drawing graphs in Excel. They all co-incide. And the solutions to specific lengths of ladder, eg 8m, all co-incide exactly. In this instance d (or in your terms “x”) = 0.144557m. For a 5m ladder, my “d” would be 0.2605m (your “x”)

I have had a little difficulty with your formula for “d” (or d2 ). When I substitute L=4 (Ladder length 2L = 8m) I get unrealistic answers. Could you check your formula for d2 and (in particular) clarify the power of the element “4L”. I have presumed the power is “4” when a=1 (as per the teaser).

I have no doubt your solution works, but tonight I do seem to be suffering from some form of algebra-blindness !

Cheers, Don

Hi Don

Yes, no harm in clarifying notation. I used a ladder length of 2L to save writing L/2 all the time.

The formula for d^2 works if you use a minus sign between the two terms (I originally wrote + or - because it was the solution of a quadratic equation). It then reads:

d^2 = (L^2 + a^2) - sqrt(a^4 + 4*a^2*L^2)

In the square root sign, the first term is a raised to the power of 4, and the second term is 4 x L squared x a squared.

For any length of ladder there are two solutions (as long as the ladder isn't too short). These correspond to the ladder making an angle of A degrees or (90 - A) degrees with the ground.

My solution generates x values (distance of the foot of the ladder from the edge of the chest) that *decrease* as the ladder gets longer - in other words, the angle the ladder makes with the ground increases.

I agree that when the ladder is 8m long, x (your d) is 0.14457m but I get x = 0.111882m when the ladder is 10m long (less than the former case, as expected).

I'll be interested to see your solutions when you're ready to share them. It would be nice if there was a really elegant way to crack this.

 

 

itbuffster posted:

Hi Don

Yes, no harm in clarifying notation. I used a ladder length of 2L to save writing L/2 all the time.

The formula for d^2 works if you use a minus sign between the two terms (I originally wrote + or - because it was the solution of a quadratic equation). It then reads:

d^2 = (L^2 + a^2) - sqrt(a^4 + 4*a^2*L^2)

In the square root sign, the first term is a raised to the power of 4, and the second term is 4 x L squared x a squared.

For any length of ladder there are two solutions (as long as the ladder isn't too short). These correspond to the ladder making an angle of A degrees or (90 - A) degrees with the ground.

My solution generates x values (distance of the foot of the ladder from the edge of the chest) that *decrease* as the ladder gets longer - in other words, the angle the ladder makes with the ground increases.

I agree that when the ladder is 8m long, x (your d) is 0.14457m but I get x = 0.111882m when the ladder is 10m long (less than the former case, as expected).

I'll be interested to see your solutions when you're ready to share them. It would be nice if there was a really elegant way to crack this.

 

 

Ah ! in the square root sign,  the second term is 4 * L² * a²........................got it !

So for L =4 (ie an 8m ladder) and a = 1 (the side of the chest),  this  second term is = 64 and (your) "d" = 2.989606

I had previously read your solution as sqrt [a4 + ((4L)2a)2]  

ie sqrt [1 + 164]    when a=1

My distance "d" 0.2605m was for a ladder 5m long

For a ladder 10m long, the distance is 0.1119m ie the same as your 10m ladder.

Yes, there are always two solutions, one for the "upright" ladder and one for the "flat" ladder. In fact, the two solutions simply provide the horizontal distance from the chest to the toe of the ladder and the vertical distance from the top of the chest to the head of the ladder.

I'll post my solution later, but remember, the original question set the chest (cube) as side "unity", which makes the arithmetic and solution a bit more simple.

Cheers Don

I agree with all that. Here are the results from my Excel spreadsheet:

20.3622
2.50.260518
30.205013
40.144557
50.111882
60.091322
70.077169
80.066825
90.058932
100.052708

 

The first column is my L (so double it to get the length of the ladder). The second column is my x (your d).

Looks like I get the same answers as you do.

itbuffster posted:

Cheers Don

I agree with all that. Here are the results from my Excel spreadsheet:

20.3622
2.50.260518
30.205013
40.144557
50.111882
60.091322
70.077169
80.066825
90.058932
100.052708

 

The first column is my L (so double it to get the length of the ladder). The second column is my x (your d).

Looks like I get the same answers as you do.

Well done.

I do like your general formula which incorporates "a", ie the dimension of the cubic chest.

The minimum length of ladder that fits the bill is 2√2 or in your case, such that "L" ie the half length is √2

In this case the toe of the ladder is 1m away from the base of the chest, as you noted previously.

My answers are the same as yours viz  (L; d)

3; 0.6702    4; 0.3622    5; 0.2605    6; 0.2050    7; 0.1694    8; 0.1446    9; 0.1261    10; 0.1119

15; 0.0716    20; 0.0527     etc

The most straightforward formula that I have, (like yours best expressed in two parts) is as follows:-

d = ½[y ± √(y² - 4)]

y = [√(L² + 1)] - 1

Note "d" is the distance required ie from the base of the chest to the toe of the ladder. You need to take the "minus" option if the ladder is "upright"

L is the length of the ladder (not the half length)

 

Slide2

This is the second solution. You will note that the plot is identical to the first solution, despite the different input.

You will notice that the plot values are based on integer values of ladder lengths "L" as they were in solution 1 above ("L" is plotted on the X axis; "d" is plotted on the Y axis)

Slide3

And this is the third solution.

OK, it's not quite a real solution, but it's easy enough to input successive values of "d" until you get the "given" ladder length.

So, although the graph is clearly the same curve as solutions 1 and 2, the actual plot values are based on given values of "d" (on the Y Axis)

There are many puzzles and situations in life with complex solutions, which can only be arrived at in stages. A methodical approach, designed to narrow the search in successive steps as per detective work or a forensic exercise is often successful. For example :-

An oil executive lives on the twenty fourth floor of a tall apartment building in Dubai. Each morning when he goes to work he calls the lift, pushes the ground floor button and is driven by chauffeured limo to his downtown office.

On his return home, he enters the lift, usually pushes the twelfth floor button and walks up the rest of the stairs. At other times he rides straight up to his floor.

Why does he not always take the lift to the twenty fourth floor ?

The ground rules for these puzzles are simple, the solution must :-

       fit all the given facts

      conform to acceptable norms of behaviour

      obey the laws of the physical world as we know them

I'll outline a typical solution later, once you've all had time to consider, an if necessary clarify the puzzle concept.

 

The lift button panel could look like this:

24  25

22  23

20  21

...

12  13

10  11

 8  9

 6  7

 4  5

 2  3

 G  1

 

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Minh Nguyen posted:

The lift button panel could look like this:

24  25

22  23

20  21

...

12  13

10  11

 8  9

 6  7

 4  5

 2  3

 G  1

 

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Why not press the 13?  No more difficult than the 12.

Possibly worse still for this solution depends on whether in Dubai they have a G, or whether as in America the first floor is what in the UK (and Europe, IIRC) is the Ground floor, in which case he might only be able to reach the 12.  Or the joke started out in the US, with no G button. 

Beachcomber posted:
Minh Nguyen posted:

The lift button panel could look like this:

24  25

22  23

20  21

...

12  13

10  11

 8  9

 6  7

 4  5

 2  3

 G  1

 

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Why not press the 13?  No more difficult than the 12.

Possibly worse still for this solution depends on whether in Dubai they have a G, or whether as in America the first floor is what in the UK (and Europe, IIRC) is the Ground floor, in which case he might only be able to reach the 12.  Or the joke started out in the US, with no G button. 

Perhaps I should have defined the lift button panel as follows:

23  24

...

11  12

...

G  -1

The second approach from Minh seems to be right when the executive pushes the button himself. At the other times another taller person is with him in the elevator and pushes the top floor button for him.

“On his return home, he enters the lift, usually pushes the twelfth floor button and walks up the rest of the stairs. At other times he rides straight up to his floor.”

Minh Nguyen posted:

The lift button panel could look like this:

24  25

22  23

20  21

...

12  13

10  11

 8  9

 6  7

 4  5

 2  3

 G  1

 

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Now this was a good start and is basically correct.

But it didn't account for those times when he rides straight up to his 24th floor. But perhaps that was so obvious that it wasn't worth mentioning .

Eoink posted:

I can't help wondering if the 13th floor is relevant here.

ooooooo!!!! spooky !!!!!!! but no, on this occasion the 13th floor is not relevant

btw, many hotels and cruise ships don't have a number 13 floor. For sure floor No 14 is the 13th floor, but it avoids that inevitable anxiety that some people have !

Beachcomber posted:
Minh Nguyen posted:

The lift button panel could look like this:

24  25

22  23

20  21

...

12  13

10  11

 8  9

 6  7

 4  5

 2  3

 G  1

 

The person could be very small, so on the way up, the highest button they can reach is the button for the twelfth floor. On the way down, they can always reach the button for the ground floor.

Why not press the 13?  No more difficult than the 12.

Possibly worse still for this solution depends on whether in Dubai they have a G, or whether as in America the first floor is what in the UK (and Europe, IIRC) is the Ground floor, in which case he might only be able to reach the 12.  Or the joke started out in the US, with no G button. 

Yes, floor numbering does get confusing if you visit the USA then the UK. But even in the UK, some hotels number the guest rooms on the ground floor as 1xx.

Mulberry posted:

The second approach from Minh seems to be right when the executive pushes the button himself. At the other times another taller person is with him in the elevator and pushes the top floor button for him.

“On his return home, he enters the lift, usually pushes the twelfth floor button and walks up the rest of the stairs. At other times he rides straight up to his floor.”

Now that's it in a nutshell. Well done Mulberry !

Minh Nguyen posted:
Mulberry posted:

The second approach from Minh seems to be right when the executive pushes the button himself. At the other times another taller person is with him in the elevator and pushes the top floor button for him.

“On his return home, he enters the lift, usually pushes the twelfth floor button and walks up the rest of the stairs. At other times he rides straight up to his floor.”

I agree with your addition. I wonder whether he is a cross dresser that likes to wear stilettos on the weekends? No that wouldn't make sense because it would be the equivalent of being on tip toes.

In the first layout, he may avoid the thirteen floor due to superstition.

I am trying to erase a disturbing image from my mind..............................

...........this is Dubai........................

Bruce has volunteered to mow the Yorkshire Dales No.1 village cricket ground. He has a tractor-mower that cuts a strip that is 2 metres wide.

The ground is circular with a circumference of 400 metres. He starts from the outside of the ground and goes around in ever-decreasing circles towards the centre.

Estimate the total distance he has travelled on his journey to the centre. You can ignore the inevitable “wiggles” he must make at the end of each circular transit.

What would be the annual percentage rate for the 2018 global population of 7.6bn to reach 10bn by 2038 ?

Assume the annual percentage growth rate is the same for every year for the next 20 years.

3 decimal places will be enough.

 

A starter........................

Population growth - Starter JPEG

"bn" means Billion, nothing to do with the "unknown" which I have called "n"

PS. don't be put off by the large white space beneath the two starter lines. I only used about another five lines and that was just to make every step crystal clear !

1.382% based on Excel's ability to calculate roots.

 

As you shown, of the current level is 7.6bn and the gross rate of growth is n, then to reach 10bn in 20 years the calculation is 7.6bn * n^20 = 10bn.

So resolving, n^20 = 10bn/7.7bn = 10/7.6 = 1.316

 

So n = 20th root of 1.316, Excel tells me that is 1.013816, which is a percentage growth rate of 1.382% to 3dp.

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