Hypothetical Flat-Tire Question

[Carl Fogel]

Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live where
goathead stickers are unknown.

Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.

Two bicyclists will pedal this path at 20 mph for 5 hours.

Let's say that the front and rear tires of each bicycle will sweep a track an inch wide.

One bicyclist--let's call him Carl--is an utterly careless Mr. Magoo who weaves randomly along the
path, but never dodging any hazards.

The other bicyclist--let's call him Hawkeye--is an alert rider with excellent vision who also weaves
randomly along the same path, but enjoys reflexes good enough to swerve around any small hazard that
he sees in time.

Let's stipulate that "in time" is roughly 30 feet ahead of Hawkeye, about 1 second at 20 mph.

To be realistic, let's also stipulate that Hawkeye fails to pay attention for 1 second every minute
because he blinks, sneezes, coughs, stares at birds, gazes at his speedometer, or simply daydreams
about hypothetical flat-tire questions.

I have here a bag of imaginary goat-head stickers, each one guaranteed to puncture any tire that
rolls over it. They are so small that they cannot be seen further than 30 feet away. They stick up
and happily pierce the sides of tires, just like real goatheads, so don't worry about the contact
patch being narrower than the one-inch path swept by the tires.

Here's a picture of a real goathead:

http://home.comcast.net/~carlfogel/download/Goathead.jpg

It's sitting on a dime about 0.705" wide.

How many of these stickers should I sprinkle randomly to give careless Carl a flat tire? And how
many more should I sprinkle to deflate Hawkeye's alert ego?

Hint: does the total number of stickers required have anything to do with whether the distance is
one mile, a century ride, or to the moon and back?

Extra-credit: given ten thousand stickers, how many stickers do both riders avoid through sheer
blind luck? That is, how much dodging is actually useful?

Feel free to email me before posting if you have any uneasy suspicions that this is a
loaded question.

[email protected]

 

[Werehatrack]

On 17 Jan 2004 18:55:42 -0800, [email protected] (Carl Fogel) may
have said:

>How many of these stickers should I sprinkle randomly to give careless Carl a flat tire? And how
>many more should I sprinkle to deflate Hawkeye's alert ego?

Okay, then for any given point on the track where there is a single goathead, assuming mere
random placement of both the tire and the goathead, the chance of each tire intersecting the
goathead is 1 in
100. The chance of a bike hitting a specific goathead is roughly double that, because it is given
that the rider weaves around a lot, which means that the two tires follow essentially
independent paths. I will treat the problem as being 200 events of 1% probability with the
rider's subsequent record irrelevant after a hit.

This does not mean that with 100 goatheads scattered randomly over the length of the course, a 1%
probability per tire, and two tires, that the probability of a hit is 200%. That is not the case.
The non-alert rider's probability with 100 goatheads randomly scattered over that long trail is
86.6% of having hit one.

Regardless of the number of goatheads sprinkled up to a level *less than* total path saturation,
there is a measurable probability of either or both riders successfully avoiding all of them.

OTOH, if 102 are laid out cheek-by-jowl across the path in a .98" spacing line, the lead cyclist
can't fail to hit one, although that's probably pushing it on real-world guarantee of a puncture. No
provision was stated for the alert rider to be able to stop or portage past a goathead, but if
that's a possibility, then there is no absolute-certainty puncture scenario with either random
goathead placement or a solid line.

OTOOH, if neither cyclist is carrying a spare tube and/or patch kit, the probability is high that a
tire will have been flatted before the first goathead can be positioned. (This is due to the
operation of the Laws Of Perversity Of The Universe.)

>Hint: does the total number of stickers required have anything to do with whether the distance is
>one mile, a century ride, or to the moon and back?

It is dependent only upon the total number of goatheads, and at high densities (i.e. more than 1% of
the trail width produces a puncture) their density as a percentage of the width of the trail.

You didn't state whether the initial conditions include the knowledge of the possible presence of
goatheads on the part of the careful rider, and you didn't state whether the goatheads had a
specific probability of not being recognized if either rider was looking straight at them, or of
being hidden by leaves, etc.

>Extra-credit: given ten thousand stickers, how many stickers do both riders avoid through sheer
>blind luck? That is, how much dodging is actually useful?

I could haul out the stats book and come up with several justifiable answers to this, but my
instincts tell me that they won't get six feet down the trail if either of them borrowed my bike for
this ride.

>Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
>question.

Getting loaded before answering may improve the question. It's Saturday, after all, at least in part
of the world.

--
My email address is antispammed; pull WEEDS if replying via e-mail.
Yes, I have a killfile. If I don't respond to something,
it's also possible that I'm busy.
Words processed in a facility that contains nuts.

 

[Per ElmsäTer]

Carl Fogel wrote:
> Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live where
> goathead stickers are unknown.
>
> Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>
> Two bicyclists will pedal this path at 20 mph for 5 hours.
>

Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
they reached the destination at the end of the 100 mile road without any flats at all.

--
Perre

You have to be smarter than a robot to reply.

 

[Rick Onanian]

On 17 Jan 2004 18:55:42 -0800, [email protected] (Carl Fogel)
wrote:
Carl's Conundrum:
>entertain snowbound riders lucky enough to Let's say that the front and rear tires of each bicycle
>will sweep a track an inch wide.

>The other bicyclist--let's call him Hawkeye--is an alert rider with excellent vision who also
>weaves randomly along the same path, but enjoys reflexes good enough to swerve around any small
>hazard that he sees in time.

>How many of these stickers should I sprinkle randomly to give careless Carl a flat tire? And how
>many more should I sprinkle to deflate Hawkeye's alert ego?

I haven't got the knowledge to tell you how many you need to sprinkle, but I can tell you that it's
the same for Carl as it is for Hawkeye. They will be invisible under the snow; and if they're on top
of the snow, these are the possibilities:
- They'll sink in enough not to be seen
- between color and size, they'll be invisible on top of the snow
- they won't affect either rider because they will penetrate the snow easier than the tire

Of course, none of that matters, as our Skippy eating friend Perre points out, since you've already
told us that they go 100 miles.

>[email protected]
--
Rick Onanian

 

[Tim Lines]

Per Elmsäter wrote:
> Carl Fogel wrote:
>
>>Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live where
>>goathead stickers are unknown.
>>
>>Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>>
>>Two bicyclists will pedal this path at 20 mph for 5 hours.
>>
>
>
> Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
> they reached the destination at the end of the 100 mile road without any flats at all.
>
Oh. I see how YOU are!

 

[Per ElmsäTer]

Tim Lines wrote:
> Per Elmsäter wrote:
>> Carl Fogel wrote:
>>
>>> Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
>>> where goathead stickers are unknown.
>>>
>>> Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>>>
>>> Two bicyclists will pedal this path at 20 mph for 5 hours.
>>>
>>
>>
>> Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
>> they reached the destination at the end of the 100 mile road without any flats at all.
>>
> Oh. I see how YOU are!

huh?
--
Perre

You have to be smarter than a robot to reply.

 

[Tim Lines]

Per Elmsäter wrote:
> Tim Lines wrote:
>
>>Per Elmsäter wrote:
>>
>>>Carl Fogel wrote:
>>>
>>>
>>>>Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
>>>>where goathead stickers are unknown.
>>>>
>>>>Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>>>>
>>>>Two bicyclists will pedal this path at 20 mph for 5 hours.
>>>>
>>>
>>>
>>>Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
>>>they reached the destination at the end of the 100 mile road without any flats at all.
>>>
>>
>>Oh. I see how YOU are!
>
>
> huh?

You're the kind who sees the right answer without knowing WHY it's the right answer. Of course they
got no flats, but WHY?

The answer? Since Carl and Hawkeye are regular readers of rbt, they know that the only proper and
acceptable way of dealing with road debris of any kind is to stop, pick the bike up, and carry it
past the the dangerous area. So that's what they did. They RAN the final 99.9 miles with their bikes
slung across there backs, thereby completing their century while protecting their precious tires
from the onslaught of potentially thousands of sharp thingamajigs.

Obvious, isn't it?

 

[Marty]

[email protected] (Carl Fogel) wrote in message news:<[email protected]>...
>

>
> Here's a picture of a real goathead:
>
> http://home.comcast.net/~carlfogel/download/Goathead.jpg
>
> It's sitting on a dime about 0.705" wide.
>

What kind of a person says "about" 0.705" wide?

I suppose if you describe it to the nearest thousandths of an inch then that's "near enough".

> Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
> question.
>
> [email protected]

 

[Marty]

"Per Elmsäter" <[email protected]> wrote in message news:<[email protected]>...
> Carl Fogel wrote:
> > Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
> > where goathead stickers are unknown.
> >
> > Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
> >
> > Two bicyclists will pedal this path at 20 mph for 5 hours.
> >
>
> Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
> they reached the destination at the end of the 100 mile road without any flats at all.

But he doesn't specifcally say that those two bicyclists are Carl and Hawkeye. They could be two
other cyclist riding along the same bicycle path. And if some of the thorns are picked up in the
tyres of the othe bicyclists, does that upset the computations.

Marty

 

[Carl Fogel]

[email protected] (Marty) wrote in message news:<[email protected]>...
> [email protected] (Carl Fogel) wrote in message
> news:<[email protected]>...
> >
>
> >
> > Here's a picture of a real goathead:
> >
> > http://home.comcast.net/~carlfogel/download/Goathead.jpg
> >
> > It's sitting on a dime about 0.705" wide.
> >
>
> What kind of a person says "about" 0.705" wide?
>
> I suppose if you describe it to the nearest thousandths of an inch then that's "near enough".
>
>
> > Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
> > question.
> >
> > [email protected]

Dear Marty,

Touché!

Actually, a dime is just a hair under 52/73rds of an inch wide.

Alas, my dial calipers don't have fine enough ends to fit in between the itty-bitty ridges, so I was
forced to use Kentucky windage and honorably added the qualifying "about" lest I be accused of
excessive precision.

(Or was joking.)

So how many stickers do ya reckon, plus or minus half a dozen?

Carl Fogel

 

[Per ElmsäTer]

Marty wrote:
> "Per Elmsäter" <[email protected]> wrote in message
> news:<[email protected]>...
>> Carl Fogel wrote:
>>> Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
>>> where goathead stickers are unknown.
>>>
>>> Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>>>
>>> Two bicyclists will pedal this path at 20 mph for 5 hours.
>>>
>>
>> Since you just told us they would be pedaling for five hours at 20 mph you have also let on that
>> they reached the destination at the end of the 100 mile road without any flats at all.
>
> But he doesn't specifcally say that those two bicyclists are Carl and Hawkeye. They could be two
> other cyclist riding along the same bicycle path. And if some of the thorns are picked up in the
> tyres of the othe bicyclists, does that upset the computations.
>

Well that would explain how the road was cleared from debris and thorns. Smart thinking.

--
Perre

You have to be smarter than a robot to reply.

 

[Per ElmsäTer]

Tim Lines wrote:
> Per Elmsäter wrote:
>> Tim Lines wrote:
>>
>>> Per Elmsäter wrote:
>>>
>>>> Carl Fogel wrote:
>>>>
>>>>
>>>>> Here's a hypothetical flat-tire question to entertain snowbound riders lucky enough to live
>>>>> where goathead stickers are unknown.
>>>>>
>>>>> Imagine a bicycle path 100 miles long and 8 feet 4 inches wide.
>>>>>
>>>>> Two bicyclists will pedal this path at 20 mph for 5 hours.
>>>>>
>>>>
>>>>
>>>> Since you just told us they would be pedaling for five hours at 20 mph you have also let on
>>>> that they reached the destination at the end of the 100 mile road without any flats at all.
>>>>
>>>
>>> Oh. I see how YOU are!
>>
>>
>> huh?
>
> You're the kind who sees the right answer without knowing WHY it's the right answer. Of course
> they got no flats, but WHY?
>
> The answer? Since Carl and Hawkeye are regular readers of rbt, they know that the only proper and
> acceptable way of dealing with road debris of any kind is to stop, pick the bike up, and carry it
> past the the dangerous area. So that's what they did. They RAN the final
> 99.9 miles with their bikes slung across there backs, thereby completing their century while
> protecting their precious tires from the onslaught of potentially thousands of sharp
> thingamajigs.
>
> Obvious, isn't it?

You should be careful when you start generalizing without having something on your feet. As a matter
of fact you should be very careful even then. Of course I know WHY it's the right answer and if you
learn to read you'll figure out what I said too Once you've understood what Carl wrote you'll
realze that they didn't run or walk a single step since he clearly states that they *pedaled* for 5
hours going 20 mph.

--
Perre

You have to be smarter than a robot to reply.

 

[Tim Lines]

Please accept my apology for failing to treat earlier response with the seriousness that it was
apparently intended. My assumption ("I see how YOU are!") was that you take things lightly. A
person's tone is hard to read via USEnet, I clearly missed yours. I should know better than to have
made that assumption. So I'm sorry if you took that as an attack of any kind.

 

[Per ElmsäTer]

Tim Lines wrote:
> Please accept my apology for failing to treat earlier response with the seriousness that it was
> apparently intended. My assumption ("I see how YOU are!") was that you take things lightly. A
> person's tone is hard to read via USEnet, I clearly missed yours. I should know better than to
> have made that assumption. So I'm sorry if you took that as an attack of any kind.

No problems Tim. I wasn't offended, just surprised at your assumptions. However you are quite
correct in that I take this thread rather lightly as in humorous. Although my solution to the
problem was 100% serious and of course the only correct answer offered so far.

--
Perre

You have to be smarter than a robot to reply.

 

[David Damerell]

Per Elmsäter <[email protected]> wrote:
>Marty wrote:
>>But he doesn't specifcally say that those two bicyclists are Carl and Hawkeye. They could be two
>>other cyclist riding along the same bicycle path. And if some of the thorns are picked up in the
>>tyres of the othe bicyclists, does that upset the computations.
>Well that would explain how the road was cleared from debris and thorns. Smart thinking.

Obviously the answer is to always ride behind steamrollers in thorny conditions.
--
David Damerell <[email protected]> flcl?

 

[Charles Ramsey]

[email protected] (Carl Fogel) wrote in message news:<[email protected]>...
> [email protected] (Marty) wrote in message news:<[email protected]>...
> > [email protected] (Carl Fogel) wrote in message
> > news:<[email protected]>...
> > >
>
> > >
> > > Here's a picture of a real goathead:
> > >
> > > http://home.comcast.net/~carlfogel/download/Goathead.jpg
> > >
> > > It's sitting on a dime about 0.705" wide.
> > >
> >
> > What kind of a person says "about" 0.705" wide?
> >
> > I suppose if you describe it to the nearest thousandths of an inch then that's "near enough".
> >
> >
> > > Feel free to email me before posting if you have any uneasy suspicions that this is a loaded
> > > question.
> > >
> > > [email protected]
>
> Dear Marty,
>
> Touché!
>
> Actually, a dime is just a hair under 52/73rds of an inch wide.
>
> Alas, my dial calipers don't have fine enough ends to fit in between the itty-bitty ridges, so I
> was forced to use Kentucky windage and honorably added the qualifying "about" lest I be accused of
> excessive precision.
>
> (Or was joking.)
>
> So how many stickers do ya reckon, plus or minus half a dozen?
>
> Carl Fogel

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