Sherlock Holmes Puzzle 3

[Sema]

While working on a case Dr Watson accidentally fell down
a 30ft dry wishing well. Sherlock Holmes lowered him
down a rope.
"Can you climb up?" shouted Holmes.
"I'll be out before you know it!" came Watson's reply.
But the climb wasn't as easy as Watson had first
imagined. Each hour he managed to climb 3ft - but
slipped back 2ft.
How long did it take Watson to get out?

I am not sure what I did his correct. What I have done is in the spoiler tag.

Spoiler:

I assumed that at each hour the displacement is 1ft, since he moves up 3 ft and slips 2ft down in hour. So I multiplied Ihr/ft with 30feet to get 30 hrs. But I have doubts about this.

[Tirian]

You are right to be suspicious of your answer. Try solving the problem assuming that the well was only 5 feet deep, carefully noting Watson's depth when he climbs three feet and then when he drops two feet instead of combining them into a single operation.

[Sema]

You are right to be suspicious of your answer. Try solving the problem assuming that the well was only 5 feet deep, carefully noting Watson's depth when he climbs three feet and then when he drops two feet instead of combining them into a single operation.

Spoiler:

Thanks for your hint. Following what you posted, Watson is actually 2 ft above the ground of the well, not 1ft above like I previously thought. I will try and see If I can solve the puzzle right now.

my answer(im still not sure)

Spoiler:

15 hrs

[Gopher of Pern]

You are right to be suspicious of your answer. Try solving the problem assuming that the well was only 5 feet deep, carefully noting Watson's depth when he climbs three feet and then when he drops two feet instead of combining them into a single operation.

Spoiler:

Thanks for your hint. Following what you posted, Watson is actually 2 ft above the ground of the well, not 1ft above like I previously thought. I will try and see If I can solve the puzzle right now.

my answer(im still not sure)

Spoiler:

15 hrs

Not quite.

Spoiler:

Imagine the well is 5ft. After the first hour, Watson reaches 3ft, but then slips 2ft, putting him at 1 ft. After the second hour, he climbs another 3ft, putting him at 4ft, then slips 2ft, to a height of 2ft. After the third hour, he climbs another 3ft, reaching the top of the 5ft well, thus climbing out.

How would this work for a 30ft well?

[unlofl]

where t is hours elapsed, d is depth of well in feet

we assume that during each hour he first climbs 3 feet, and only then slips down 2 feet; rather than doing some other combination of climb/slip/climb/slip which gives him the same total progress/slipping per hour.
also, we assume he is starting 30ft below the top of the well, or is 0ft tall, or whatever; ignore his actual height when starting.

Spoiler:

* at the end of each hour (after climbing and then slipping) he will be at a height of t
* during each hour of climbing he will reach a maximum height of t+2

I was going to come up with general solution for any amount of climbing and slipping in any depth well and do a nice proof by induction, but I just realized its been way too long since I took discrete math. It would be a more fun problem for that if he were to get tired as he progressed, or the rope caused less slipping near the top of the well where it was dry, or something like that.

[Zalzidrax]

Spoiler:

Depending on Watson's height he should be able to flop over the edge of the well after he's climbed up 27 or 28 feet. So if he's close to six feet or taller, he will take about 25 hours to escape, otherwise a bit under 26 hours.

Of course if Holmes isn't scrawny or a jackass, and Watson isn't too chubby, Holmes could probably pull Watson up after he gets up 24 or 25 feet, which would be 22 to 23 hours.

[redrogue]

Imagine a perfectly spherical Watson in simple harmonic motion...

I get 20.5 hours, roughly.

Spoiler:

This problem is normally worded with a frog that can summon power for a 3 foot leap once per hour, during which he slips back 2 feet. However, this version of the problem is not worded to suggest there's instant addition of 3 feet every hour. Thus, the assumption is that when he gets to 27 feet, he can climb 3 feet and be out, in reality, climbing progress and slipping rates will be somewhat steady.

The other assumption is that Watson is zero feet tall.


Assume Watson is 5.5 feet tall, with a 2.5 foot reach, and lets give him a conservative vertical leap 1.5 feet (these numbers based on my height, my ability to grab the top of a door frame, and some Googling... average male vertical leap is 50cm).

Watson wants the best possible start, so he leaps into the air, grabbing the rope at his apex. His hands are now 9.5 feet in the air, and once they reach the ledge, he should be able to grab and mantle out.

His net progress is 1 foot per hour. Since he only has 20.5 feet to go, he should be out in that many hours.

Of course, if we knew his height, reach, and vertical leap, we could find the exact number of hours as such:
(depth of well - (height + reach + leap)) / (climb rate - slip rate)

[Sema]

You are right to be suspicious of your answer. Try solving the problem assuming that the well was only 5 feet deep, carefully noting Watson's depth when he climbs three feet and then when he drops two feet instead of combining them into a single operation.

Spoiler:

Thanks for your hint. Following what you posted, Watson is actually 2 ft above the ground of the well, not 1ft above like I previously thought. I will try and see If I can solve the puzzle right now.

my answer(im still not sure)

Spoiler:

15 hrs

Not quite.

Spoiler:

Imagine the well is 5ft. After the first hour, Watson reaches 3ft, but then slips 2ft, putting him at 1 ft. After the second hour, he climbs another 3ft, putting him at 4ft, then slips 2ft, to a height of 2ft. After the third hour, he climbs another 3ft, reaching the top of the 5ft well, thus climbing out.

How would this work for a 30ft well?

Spoiler:

Using Tirian's logic, I wrote this code in python. I got 28 hours

Code: Select all

 def solve_for_time(h):
                time = 0
                a = 3
                b = 2
                x = 0
                while x<h:
                        x = x+a-b
                        time = time + 1
                        if h-x == 3:
                                time = time + 1
                                return time
                return time

[Moonbeam]

Spoiler:

Using Tirian's logic, I wrote this code in python. I got 28 hours

Code: Select all

 def solve_for_time(h):
                time = 0
                a = 3
                b = 2
                x = 0
                while x<h:
                        x = x+a-b
                        time = time + 1
                        if h-x == 3:
                                time = time + 1
                                return time
                return time

... it's all very well putting it into a computer to solve (I honestly can't believe you felt the need to do that, but hey - who am I to ask???) - but do you understand the logic behind it??

If not and you need an explanation, then look here:

Spoiler:

C'mon, surely you can work it out !!!

[ircmaxell]

Well, I got 19 hours.

Spoiler:

This is using two assumptions. First, he's 5 feet tall. Second, both Sherlock and Watson can reach approximately 2 feet from their body.

So Watson's head will be at the top of the well when his feet are 25 feet from the bottom. But both himself and Sherlock can reach 2 feet, so he will be rescue-able when he's at 21 feet from the bottom of the well (since both can reach 2 feet,so 4 feet total from the top).

Hour 1: max: 3 feet, final: 1 foot
Hour 2: max 4 feet, final: 2 feet
Hour 3: max 5 feet, final: 3 feet
Hour n: max n+2 feet, final: n feet

Now, since we can model the maximum height from the floor as a function of time, h = t + 2. So since he can be rescued at 21 feet, 21 = h + 2, so t = 19.

If you don't count his being able to use his hands (and needs to step out without climbing at all), he will be able to walk at 30 = t + 2, 28 hours...

[freddyfish]

to introduce a point of complexity (I dont actually know if this changes the answer, I'll work it out later and let you know)
we know that he climbs up three ft in 1 hr and he slips back 2 ft. But clearly he does slip down on the hour every hour. It happens gradually the whole way up. Now whats the answer?
(interestingly, im fairly sure this makes the problem easier)

[Krealr]

... Dr Watson accidentally fell down
a 30ft dry wishing well.

After falling down a 30ft well Watson is dead.

It takes 30 minutes for Watson to realize he is a ghost and float to the top of the well

[tckthomas]

Gosh, so long?

Spoiler:

5 feet well = climb (3 ft) slip (1 ft) climb (4 ft) slip (2 ft) climb (5 ft) 3 hours

6 feet well = climb (3 ft) slip (1 ft) climb (4 ft) slip (2 ft) climb (5 ft) slip (3 ft) climb (6 ft) 4 hours

9 feet well = climb (3 ft) slip (1 ft) climb (4 ft) slip (2 ft) climb (5 ft) slip (3 ft) climb (6 ft) slip (4 ft) climb (7 ft) slip (5 ft) climb (8 ft) slip (6 ft) climb (9 ft) 7 hours

n feet well = n-2 hours

[SomeFloridaKid]

Spoiler:

it takes him 27 hours to get to 27 feet, the 28'th hour he would have been at the top, so twenty eight hours.

may be wrong, but this is how we did it in my bio class (warm up brain teaser)

[math]

Okay, you people overanalyze, but the original poster UNDERanalyzed.

Spoiler:

At each hour, it adds 3, but then subtracts two AFTERWARDS. So in 28 hours, he will have climbed 27 feet and jumped up 3 more, reaching 30 feet before slipping back down.

[HarvesteR]

Gosh, so long?

Spoiler:

n feet well = n-2 hours

My thoughts exactly..

Spoiler:

once Watson reaches 30 feet (at hour 28, just before slipping), he is already at the top, therefore we either assume he's pulled out, or that he can grab onto the ledge where he doesn't slip, thus being able to climb to safety

So, the answer (in a discrete step scenario, where any other parameters are not considered) is:

Spoiler:

28 hours

Cheers

[Xias]

Spoiler:

0 hours.

The wishing well was so full of coins (since humans practically live on wishes) that the fall was actually less than a foot, and he easily stepped out without the need of a rope.

[fimzo]

In case anyone finds it relevant, Sherlock Holmes is very fit and tall, but Watson was shot in either the arm or leg (it varies throughout Doyle's books) during the second Anglo-Afghan war. Watson is of average height and fitness. Assuming Holmes can't do anything to assist Watson's ascent, you would also have to integrate the fact that spending over a day (28 hours, without any additional factors) climbing out of a well would be exhausting, and require some rest. Of course, none of these external variables need apply to the puzzle.

[Vesuvius]

Imagine a perfectly spherical Watson...

Lol!