Yes Steve,
That formula plus Excel should produce a result pretty quickly, and earn that initial 1 point.
I have a solution for the 10 pointer, but it is not what one would call elegant, let alone ultra elegant!
I tried a number of substitutions such as x = 1 + d or x = 1 + 1/d but they didn’t help. Here’s where I ended up, minus the several pages of scribbles.
As already stated:
L² = (d + 1)² + (1/d + 1)²
Εxpanding:
L² = d² + 1 + 2d + 1/d² + 1 + 2/d
Rearranging:
L² = (d² + 1/d² + 2) + 2(d + 1/d) = (d + 1/d)² + 2(d + 1/d)
Now we can see that substituting x = d + 1/d gives us a quadratic:
x² + 2x - L² = 0, which has one positive solution: x = -1 + √ (1 + L²)
Replacing x with d + 1/d gives us another quadratic:
d² - (-1 + √ (1 + L²))d + 1 = 0
Solving:
d = (-1 + √ (1 + L²) +/- √ ((-1 + √ (1 + L²))² - 4)) / 2
Tidying:
d = (x +/- √ (x² - 4)) / 2, where: x = -1 + √ (1 + L²) as above
I won’t reveal the answer for L = 10m, but Steve is correct that d lies between 1/8 and 1/9 metres. The top of the ladder is within a few centimetres of 10m above the ground.
Well done Ravvie.
If you all look back in this thread to post No 661 (c.Feb 2020 ie about a year ago) you will find my formula.
I simply wrote some of the components in a different order to avoid putting “-1” at the start of the sub formula and I personally find that (L^2 + 1) simply looks nicer than (1 + L^2) - but both ways are correct.
At the same time (Feb 2020) and just a couple of posts later, I set out ken c’s formula. I personally prefer mine, but many others prefer ken’s.
enjoy (as ken would say !)
Does Ken C’s formula have a derivation?
Yes it looks fancy but invariably I prefer the story or explanation behind an answer, rather than the answer itself.
I have to admit I have completely forgotten what acosh etc are all about!
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Yes, but i’ll save that for another day, unless ken would like to post it.
(you’ll need to brush up on logarithms as well as acosh etc) 
Laddered Nights - Part 2
The engineers didn’t work out “d” before sunrise, mainly because they knew even less about acosh than Ravvie. So they decided to return the following night. HQ found out about their plan and insisted that they took the Health and Safety Compliance Officer (HSCO) along.
The HSCO quickly spotted that the wooden ladder actually met an old British Standard, so was an exact number of feet in length, with rungs exactly one foot apart. This allowed it to be lengthened or shortened to any exact length in feet.
The HSCO ruled that for safety (i) the ladder must exactly reach the roofline at the top of the wall and (ii) the ladder slope must not be less than 45 degrees.
The engineers forgot to bring a tape measure. The HSCO spotted that the wall was built to the standard three courses per foot in height. An engineer counted N brick courses and found that it was a multiple of three.
The engineer that had taken Don’s advice to brush up on all aspects of supported ladders thought about it for a while. He then announced: “Don’s right, we’re gonna need a bigger crate!”
They could only find crates like the one the previous night. The HSCO noted that they were actually three feet in dimension. He looked at the square in Don’s diagram and ruled that multiple crates could be used as long as any such crate structure was an exact square when viewed from the side. The crate structure only needed to be one crate deep to support the ladder.
Q1 (1 point) What was the engineer’s solution? (N brick courses, number of crates, L and d)
Q2 (3 points) Find another solution where the slope of the ladder differs from Q1. (hint: it may be rather longer!)
Q3 (10 points) Give a general formula or method for determining solutions.
Q4 (Predicted A* grade at A-level) Prove your answer to Q3 gives all solutions.
I had my money on @SteveD getting there first with Q1 at least. I had rather liked Steve’s intuitive approach of “I wonder if…” to Don’s original ladder puzzle, so this variation is intended to fit with that approach whilst being plausibly close to the original specifications.
I have to confess to being somewhat unsure as to the essence of Ravvie’s version of the Ladder teaser. I found the wording difficult to follow - but that might be a deliberate part of any teaser ie grasping the meaning is the teaser itself !!
The lack of response from others, suggests I might not be alone in failing to grasp the concept of the described situation.
Nonetheless, I have been sitting on an answer for some time now, involving 16 crates and a 35’ ladder. But I think it’s time to move on, unless somebody else is also sitting on possible solutions ?
I had found the original ladder to puzzle be a bit challenging in that it needed higher level maths to solve, such as how to develop then reduce a quartic equation, or alternatively knowledge of the acosh function.
I developed the variation of the puzzle to allow an intuitive solution, along the lines that Steve was pursuing. Yes, I suppose the key aspects of the back story need to be spotted from all the noise, but after that it only requires school level maths.
A 35 foot ladder supported by a 4x4 stack of crates is a correct answer.
apologies Ravvie,
I’ve had some family issues to deal with recently so haven’t paid as much attention to the forum as I normally do.
Back to normal now hopefully!
Cheers, Steve
Mr Fish loves fruit and vegetables. He can buy four apples and two pears for £1.54.
He can also buy two pears and four oranges for £1.70.
How much would he expect to pay if he bought an apple, an orange and a pear ?
(Assume the price of individual items is the same regardless of quantity or combination.)
Hopefully this relatively simple teaser will restore confidence that at least some teasers are actually solvable by the proverbial man on the Clapham omnibus 
I’m just drafting out a neat solution to the Apples & Oranges teaser above.
I’ll post it later this evening.
First however, I need to simplify my explanation, to avoid the Techno-babble overload situation illustrated by litemotiv in the Best Jokes thread
Sorry, I was on school camp with 70 girls and didn’t get to this… but I survived the camp!
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Well done Mike !!
If you managed to survive school-camp with 70 girls, all of these teasers must seem like a doddle !!
Mind you, I think you need your head examining for agreeing to go on school camp with 70 girls in the first place, but that’s a different story
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Here’s a famous one…
Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a high-end NAIM system; behind the others, nothing. You pick a door, say No.1, and the host, who knows what’s behind the doors, opens another door, say No.3, which has nothing. He then says to you:
“Do you want to switch your choice to No.2?”
Is it to your advantage to stick with your original choice [No.1] or switch your choice to No.2?
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Cluffy, nice teaser, and as you say, well known.
I’ll leave others to come up with the answer, but as you know, even when the correct answer is stated, there will be dissenters who will argue otherwise
It’s always worth asking what your chances of winning would be if you could pre-select two boxes
Nice teaser,
Cheers
Don
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Hi Cluffy,
Perhaps we could persuade you to publish the solution to your “Three-Door” Brain Teaser ? (there have been about 100 Views of the thread since you posted, so I guess a few people are interested in your solution )
Ditto Ravvie to your “Ladder up the Wall” teaser ? (those 100 hundred views could also be looking for the solution to your teaser )
Cheers
Don
I think:
- the dot is at the bottom again
- you should take the other door
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