All engineers are mathematicians, who enjoy puzzles.
Probability theory is a stock part of the curriculum, so...

A theater has 15 seats in the first row. Patrons enter
in single file, taking each consecutive seat, as they
arrive. In this particular case, there are 8 boys, 7 girls.

If a boy and girl occupy adjacent seats, we have a candidate
match. If boy sees girl on each side, that's two matches.
Clearly, any possible arrangement will contain 1...14
candidate matches.

What's the average number of matches, for this row?

I found this in a book of math problems, it's a nice little
exercise. However, my motivation here is mainly due
to the fact that the author's reasoning, in his solution,
is severely flawed. I think.

Take a crack at it. In a few days, I'll post the book
solution, with my critique. Then you tell me where I'm wrong.

Or, if I'm right, yet somehow the book's solution is
numerically correct, we have the challenge of explaining
that miracle. Which is more interesting than the problem itself -

--
Rich

On Saturday, July 3, 2021 at 8:33:18 PM UTC-4, RichD wrote:
> All engineers are mathematicians, who enjoy puzzles.
> Probability theory is a stock part of the curriculum, so...
>
> A theater has 15 seats in the first row. Patrons enter
> in single file, taking each consecutive seat, as they
> arrive. In this particular case, there are 8 boys, 7 girls.
>
> If a boy and girl occupy adjacent seats, we have a candidate
> match. If boy sees girl on each side, that's two matches.
> Clearly, any possible arrangement will contain 1...14
> candidate matches.
>
> What's the average number of matches, for this row?
>
> I found this in a book of math problems, it's a nice little
> exercise. However, my motivation here is mainly due
> to the fact that the author's reasoning, in his solution,
> is severely flawed. I think.
>
> Take a crack at it. In a few days, I'll post the book
> solution, with my critique. Then you tell me where I'm wrong.
>
> Or, if I'm right, yet somehow the book's solution is
> numerically correct, we have the challenge of explaining
> that miracle. Which is more interesting than the problem itself -

If the boys and girls are of dating age it is pretty certain there will be 14 candidate matches. They enter single file but no one said they entered randomly.

--

Rick C.

- Get 1,000 miles of free Supercharging
- Tesla referral code - https://ts.la/richard11209

In article <b4b77816-5154-41be-9958-4f7cbd1ee1a0n@googlegroups.com>,
r_delaney2001@yahoo.com says...
>
> A theater has 15 seats in the first row. Patrons enter
> in single file, taking each consecutive seat, as they
> arrive. In this particular case, there are 8 boys, 7 girls.
>
> If a boy and girl occupy adjacent seats, we have a candidate
> match. If boy sees girl on each side, that's two matches.
> Clearly, any possible arrangement will contain 1...14
> candidate matches.
>
> What's the average number of matches, for this row?
>
>
>

Anytime there is talk of averages I think of it as if your head is in
liquid nitrogen and your feet is in a fire. The average temperature may
be 72 deg F. but one end burns up and the other end is frozen solid.
You are dead from both ends.

On Sat, 3 Jul 2021 17:33:15 -0700 (PDT), RichD
<r_delaney2001@yahoo.com> wrote:

>All engineers are mathematicians, who enjoy puzzles.

I used to enjoy puzzles and chess. But I have to think all day at
work, so I don't want to do that for free any more.

Real puzzles are better than made-up ones.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On Sunday, July 4, 2021 at 2:01:51 PM UTC+10, jla...@highlandsniptechnology.com wrote:
> On Sat, 3 Jul 2021 17:33:15 -0700 (PDT), RichD
> <r_dela...@yahoo.com> wrote:
>
> >All engineers are mathematicians, who enjoy puzzles.
> I used to enjoy puzzles and chess. But I have to think all day at
> work, so I don't want to do that for free any more.

Not an all-that-plausible claim. John Larkin likes to claim that what he does at work involves thinking, but what he posts here suggests he lost the capacity to think some years ago.

He doesn't do it all any more, and uses the lack of payment as an excuse for not even trying.

> Real puzzles are better than made-up ones.

Somebody will pay you for appearing to try to solve them - and often just for going through the motions - before you palm them off with a solution that somebody else thought up earlier, for a rather different problem.

--
Bill Sloman, Sydney

On 4/7/21 1:27 pm, Ralph Mowery wrote:
> In article <b4b77816-5154-41be-9958-4f7cbd1ee1a0n@googlegroups.com>,
> r_delaney2001@yahoo.com says...
>>
>> A theater has 15 seats in the first row. Patrons enter
>> in single file, taking each consecutive seat, as they
>> arrive. In this particular case, there are 8 boys, 7 girls.
>>
>> If a boy and girl occupy adjacent seats, we have a candidate
>> match. If boy sees girl on each side, that's two matches.
>> Clearly, any possible arrangement will contain 1...14
>> candidate matches.
>>
>> What's the average number of matches, for this row?
>>
>>
>>
>
> Anytime there is talk of averages I think of it as if your head is in
> liquid nitrogen and your feet is in a fire. The average temperature may
> be 72 deg F. but one end burns up and the other end is frozen solid.
> You are dead from both ends.
>

Yes - I was going to comment that Rich probably wanted to ask what is
the statistically median number of matches.

CH

On 7/3/2021 5:33 PM, RichD wrote:
> All engineers are mathematicians, who enjoy puzzles.

There's two conclusions, there, without any supporting evidence. :>

I much prefer "puzzles" that highlight flaws in our thinking
(problem solving) process as those can be used to IMPROVE it.

For example:

A snail is at the bottom of a 30 ft deep well. Each morning, it
climbs 3 ft up the wall towards the daylight streaming in from above.
Each evening, it slips back down, 2 ft. How many days 'til the snail
gets out of the well?

The answer is obviously "the morning of the 28th day"; yet casual
"solvers" will inevitably say "30". Is this because they are lazy
thinkers? Or, some innate need for the brain to simplify the
problem (incorrectly!) and, having done so, to resist reevaluating
the assumptions that went into that simplification?

The "missing dollar" hotel problem falls into the same pattern.

In a related, but different, approach:

There are three (identical) bottles and three (identical) chopsticks
on the table in front of you. Arrange the chopsticks on the bottles
so that no chopstick touches the table top.

The most common (universal?) solution has folks arranging the bottle
as the vertices of an equilateral triangle and then, carefully,
"playfully" (?) laying the chopsticks across the bottle tops to
form the sides of said triangle.

Next, one of the bottles is removed and the puzzle reposed:

There are TWO (identical) bottles and three (identical) chopsticks
on the table in front of you. Arrange the chopsticks on the bottles
so that no chopstick touches the table top.

And, finally, removing yet another bottle:

There is ONE bottle and three (identical) chopsticks on the table in
front of you. Arrange the chopsticks on the bottles so that no chopstick
touches the table top.

Each of these successive problems is easily solvable. And, solving any
of them solves each of the preceding ones, as well.

So... why wasn't the solution to the LAST problem used to solve each
of the previous ones? Why did the typical user "settle" for one
solution instead of exploring the solution space for other solutions?

By extension, how does this sort of behavior factor into your solutions
of other *engineering* problems? Or, other problems encountered in
daily living?

Is it lazy thinking? Or, some aspect of (some) brain(s) that considers
a solution as good as any other?

And, if you *do* continue to explore the solution space, when does this
become counterproductive -- when you spend more effort than a "better"
solution is likely to justify?

On Sunday, July 4, 2021 at 5:22:24 PM UTC+10, Clifford Heath wrote:
> On 4/7/21 1:27 pm, Ralph Mowery wrote:
> > In article <b4b77816-5154-41be...@googlegroups.com>,
> > r_dela...@yahoo.com says...
> >>
> >> A theater has 15 seats in the first row. Patrons enter
> >> in single file, taking each consecutive seat, as they
> >> arrive. In this particular case, there are 8 boys, 7 girls.
> >>
> >> If a boy and girl occupy adjacent seats, we have a candidate
> >> match. If boy sees girl on each side, that's two matches.
> >> Clearly, any possible arrangement will contain 1...14
> >> candidate matches.
> >>
> >> What's the average number of matches, for this row?
> >
> > Anytime there is talk of averages I think of it as if your head is in
> > liquid nitrogen and your feet is in a fire. The average temperature may
> > be 72 deg F. but one end burns up and the other end is frozen solid.
> > You are dead from both ends.
> >
> Yes - I was going to comment that Rich probably wanted to ask what is
> the statistically median number of matches.

It's a messy problem. There are factorial 15 possible arrangements. Two of them - all the girls go in first or all the boys go in first - produce only one candidate match each, except that there are factorial eight times factorial seven of such possible arrangements, producing 56 different matches

There's only one - boys and girls alternating - that produces the maximum number of matches - 14 - but there are factorial eight ways of arranging the boys and factorial seven ways of arranging the girls, and you don't really care whether the girl is to the right or to the left of the boy she is matched with

Presumably you have to sort through all the other cases - two boys sitting next to one another at either end of the row, two boys sitting next to one another anywhere else in the row - and so forth, to get an answer.

Mathematicians can see way of looking a this kind of problem that let them get to an answer easily, but often the answer is too long to scribble in the margin.

--
Bill Sloman, Sydney

On Sun, 4 Jul 2021 00:49:23 -0700, Don Y <blockedofcourse@foo.invalid>
wrote:

>On 7/3/2021 5:33 PM, RichD wrote:
>> All engineers are mathematicians, who enjoy puzzles.
>
>There's two conclusions, there, without any supporting evidence. :>

Kids can get an EE degree without acquiring much in the way of math
skills, or learning much about electricity.

A lot of kids "learn" to code, which requires no math at all.
"Software Engineering" is kind of an oxymoron. One neighbor is a way
overpaid Google s.e. and we just had a conversation about light bulbs,
watts and KWH and such. It was hilarious.

They are very nice people, actually, from Bulgaria or something. We
exchange baked goods. We share a lemon tree. But he doesn't know which
end of a light bulb screws into the socket.

We're near a Google bus stop so we have several OPGSE's on the block.
They do goofy things.

>
>I much prefer "puzzles" that highlight flaws in our thinking
>(problem solving) process as those can be used to IMPROVE it.
>
>For example:
>
> A snail is at the bottom of a 30 ft deep well. Each morning, it
> climbs 3 ft up the wall towards the daylight streaming in from above.
> Each evening, it slips back down, 2 ft. How many days 'til the snail
> gets out of the well?
>
>The answer is obviously "the morning of the 28th day"; yet casual
>"solvers" will inevitably say "30". Is this because they are lazy
>thinkers?

It's because snails have good days, and bad days.

--

John Larkin Highland Technology, Inc

The best designs are necessarily accidental.

On 7/3/21 10:01 PM, jlarkin@highlandsniptechnology.com wrote:
> Real puzzles are better than made-up ones.

I've found that hypothetical puzzles, or other people's puzzles, can be
a worthwhile academic exercise as the underlying method to solve them
often becomes helpful later in life (personal and / or work).

--
Grant. . . .
unix || die

On Sun, 4 Jul 2021 01:03:54 -0700 (PDT), Anthony William Sloman
<bill.sloman@ieee.org> wrote:

>On Sunday, July 4, 2021 at 5:22:24 PM UTC+10, Clifford Heath wrote:
>> On 4/7/21 1:27 pm, Ralph Mowery wrote:
>> > In article <b4b77816-5154-41be...@googlegroups.com>,
>> > r_dela...@yahoo.com says...
>> >>
>> >> A theater has 15 seats in the first row. Patrons enter
>> >> in single file, taking each consecutive seat, as they
>> >> arrive. In this particular case, there are 8 boys, 7 girls.
>> >>
>> >> If a boy and girl occupy adjacent seats, we have a candidate
>> >> match. If boy sees girl on each side, that's two matches.
>> >> Clearly, any possible arrangement will contain 1...14
>> >> candidate matches.
>> >>
>> >> What's the average number of matches, for this row?
>> >
>> > Anytime there is talk of averages I think of it as if your head is in
>> > liquid nitrogen and your feet is in a fire. The average temperature may
>> > be 72 deg F. but one end burns up and the other end is frozen solid.
>> > You are dead from both ends.
>> >
>> Yes - I was going to comment that Rich probably wanted to ask what is
>> the statistically median number of matches.
>
>It's a messy problem. There are factorial 15 possible arrangements.

No, way more than that. What about if one or more of the boys identify
as girls? What if some girls identify as boys? After all, you Lefties
constantly remind us that gender is nothing more than a personal
choice.

On Sat, 3 Jul 2021 23:31:03 -0700 (PDT), Anthony William Sloman
<bill.sloman@ieee.org> wrote:

>On Sunday, July 4, 2021 at 2:01:51 PM UTC+10, jla...@highlandsniptechnology.com wrote:
>> On Sat, 3 Jul 2021 17:33:15 -0700 (PDT), RichD
>> <r_dela...@yahoo.com> wrote:
>>
>> >All engineers are mathematicians, who enjoy puzzles.
>> I used to enjoy puzzles and chess. But I have to think all day at
>> work, so I don't want to do that for free any more.
>
>Not an all-that-plausible claim. John Larkin likes to claim that what he does at work involves thinking, but what he posts here suggests he lost the capacity to think some years ago.
>
>He doesn't do it all any more, and uses the lack of payment as an excuse for not even trying.

I haven't seen any evidence of YOU doing any work, lately, Bill. And
since you seem to spend every waking minute tapping away on that
keyboard of yours posting a lot of nonsense, I'll wager you don't even
lift a finger to help your wife around the house either.
--

"You must therefore confess that by 'individual' you mean no other person
than the bourgeois; than the middle-class owner of property. This person
must indeed be swept out of the way, and made impossible."

- Marx & Engels, The Communist Manifesto

On 04/07/21 20:02, Cursitor Doom wrote:
> I haven't seen any evidence of YOU doing any work, lately, Bill.

Bill is retired.

We haven't seen any evidence of YOU EVER doing ANY work,
other than posting support for Putin's TV station and
inane pseudo-anonymised zerohedge garbage.

On Sunday, July 4, 2021 at 4:03:58 AM UTC-4, bill....@ieee.org wrote:
> On Sunday, July 4, 2021 at 5:22:24 PM UTC+10, Clifford Heath wrote:
> > On 4/7/21 1:27 pm, Ralph Mowery wrote:
> > > In article <b4b77816-5154-41be...@googlegroups.com>,
> > > r_dela...@yahoo.com says...
> > >>
> > >> A theater has 15 seats in the first row. Patrons enter
> > >> in single file, taking each consecutive seat, as they
> > >> arrive. In this particular case, there are 8 boys, 7 girls.
> > >>
> > >> If a boy and girl occupy adjacent seats, we have a candidate
> > >> match. If boy sees girl on each side, that's two matches.
> > >> Clearly, any possible arrangement will contain 1...14
> > >> candidate matches.
> > >>
> > >> What's the average number of matches, for this row?
> > >
> > > Anytime there is talk of averages I think of it as if your head is in
> > > liquid nitrogen and your feet is in a fire. The average temperature may
> > > be 72 deg F. but one end burns up and the other end is frozen solid.
> > > You are dead from both ends.
> > >
> > Yes - I was going to comment that Rich probably wanted to ask what is
> > the statistically median number of matches.
> It's a messy problem. There are factorial 15 possible arrangements. Two of them - all the girls go in first or all the boys go in first - produce only one candidate match each, except that there are factorial eight times factorial seven of such possible arrangements, producing 56 different matches
>
> There's only one - boys and girls alternating - that produces the maximum number of matches - 14 - but there are factorial eight ways of arranging the boys and factorial seven ways of arranging the girls, and you don't really care whether the girl is to the right or to the left of the boy she is matched with
>
> Presumably you have to sort through all the other cases - two boys sitting next to one another at either end of the row, two boys sitting next to one another anywhere else in the row - and so forth, to get an answer.
>
> Mathematicians can see way of looking a this kind of problem that let them get to an answer easily, but often the answer is too long to scribble in the margin.

Mainly they use much more ordered thinking than you're capable of. The 15 individuals have no identity, so the factorial permutation business is off the table. Boys might as well be blue marbles, and girls are pink marbles. Replace the stupid "match" nonsense with different color marbles adjacent. Then start creating some simplifying notation like D(n) = arrangements with n mixed color adjacencies. Then you need to let T be the total number of arrangements of the 15 marbles possible. By definition of randomness, each arrangement is as likely as any other and therefore has probability of occurrence of 1/T. The probability of occurrence of n different color adjacencies is therefore D(n)/T. The average, or expected number of adjacencies, M, is therefore: M= 1 x D(1)/T + 2 x D(2)/T + .....+14 x D(14)/T + 15 x D(15)/T, and yes, yes- we see D(1), D(15)=0 but don't really care.
T should be T=15!/(8! x 7!)
And using other observations about sums, a simplified answer can be derived..

>
> --
> Bill Sloman, Sydney

On Monday, July 5, 2021 at 5:02:34 AM UTC+10, Cursitor Doom wrote:
> On Sat, 3 Jul 2021 23:31:03 -0700 (PDT), Anthony William Sloman
> <bill....@ieee.org> wrote:
>
> >On Sunday, July 4, 2021 at 2:01:51 PM UTC+10, jla...@highlandsniptechnology.com wrote:
> >> On Sat, 3 Jul 2021 17:33:15 -0700 (PDT), RichD
> >> <r_dela...@yahoo.com> wrote:
> >>
> >> >All engineers are mathematicians, who enjoy puzzles.
> >> I used to enjoy puzzles and chess. But I have to think all day at
> >> work, so I don't want to do that for free any more.
> >
> >Not an all-that-plausible claim. John Larkin likes to claim that what he does at work involves thinking, but what he posts here suggests he lost the capacity to think some years ago.
> >
> >He doesn't do it all any more, and uses the lack of payment as an excuse for not even trying.
>
> I haven't seen any evidence of YOU doing any work, lately, Bill.

You wouldn't. The thread "Low noise, high bias voltage on picoAmp TIA's input, howto?" isn't the sort of stuff you could follow, even if you found it interesting, which you wouldn't.
Or you could look at

http://site.ieee.org/nsw/files/2021/04/Circuit_March_2021.pdf

which I happen to edit. I've got the July 2021 issue about half done - the new and up-dated members data for the past three months has just come in and I've still got to slot that in.
And I'm currently the treasurer for the NSW branch of the IEEE, which doesn't take up much time either

>And since you seem to spend every waking minute tapping away on that keyboard of yours posting a lot of nonsense,

I spend an hour or so a day at it. Most of it is beyond your understanding, so it may well look like nonsense to you, and not the kind of nonsense to which you are addicted.

> I'll wager you don't even lift a finger to help your wife around the house either.

I'm retired. My wife isn't. We pay cleaners to go over the place once a week, but I look after everything else (including the cooking and the shopping). That doesn't take long either.

You really are remarkably silly, and unpleasant with it.

--
Bill Sloman, Sydney

On Monday, July 5, 2021 at 7:57:12 AM UTC+10, Fred Bloggs wrote:
> On Sunday, July 4, 2021 at 4:03:58 AM UTC-4, bill....@ieee.org wrote:
> > On Sunday, July 4, 2021 at 5:22:24 PM UTC+10, Clifford Heath wrote:
> > > On 4/7/21 1:27 pm, Ralph Mowery wrote:
> > > > In article <b4b77816-5154-41be...@googlegroups.com>,
> > > > r_dela...@yahoo.com says...
> > > >>
> > > >> A theater has 15 seats in the first row. Patrons enter
> > > >> in single file, taking each consecutive seat, as they
> > > >> arrive. In this particular case, there are 8 boys, 7 girls.
> > > >>
> > > >> If a boy and girl occupy adjacent seats, we have a candidate
> > > >> match. If boy sees girl on each side, that's two matches.
> > > >> Clearly, any possible arrangement will contain 1...14
> > > >> candidate matches.
> > > >>
> > > >> What's the average number of matches, for this row?
> > > >
> > > > Anytime there is talk of averages I think of it as if your head is in
> > > > liquid nitrogen and your feet is in a fire. The average temperature may
> > > > be 72 deg F. but one end burns up and the other end is frozen solid..
> > > > You are dead from both ends.
> > > >
> > > Yes - I was going to comment that Rich probably wanted to ask what is
> > > the statistically median number of matches.
> > It's a messy problem. There are factorial 15 possible arrangements. Two of them - all the girls go in first or all the boys go in first - produce only one candidate match each, except that there are factorial eight times factorial seven of such possible arrangements, producing 56 different matches
> >
> > There's only one - boys and girls alternating - that produces the maximum number of matches - 14 - but there are factorial eight ways of arranging the boys and factorial seven ways of arranging the girls, and you don't really care whether the girl is to the right or to the left of the boy she is matched with
> >
> > Presumably you have to sort through all the other cases - two boys sitting next to one another at either end of the row, two boys sitting next to one another anywhere else in the row - and so forth, to get an answer.
> >
> > Mathematicians can see way of looking a this kind of problem that let them get to an answer easily, but often the answer is too long to scribble in the margin.
> Mainly they use much more ordered thinking than you're capable of. The 15 individuals have no identity,

Of course they do.

> so the factorial permutation business is off the table. Boys might as well be blue marbles, and girls are pink marbles. Replace the stupid "match" nonsense with different color marbles adjacent.

Change the problem so it's easier to tackle.

> Then start creating some simplifying notation like D(n) = arrangements with n mixed color adjacencies. Then you need to let T be the total number of arrangements of the 15 marbles possible. By definition of randomness, each arrangement is as likely as any other and therefore has probability of occurrence of 1/T. The probability of occurrence of n different color adjacencies is therefore D(n)/T. The average, or expected number of adjacencies, M, is therefore: M= 1 x D(1)/T + 2 x D(2)/T + .....+14 x D(14)/T + 15 x D(15)/T, and yes, yes- we see D(1), D(15)=0 but don't really care.

The boys and girls would.

> T should be T=15!/(8! x 7!)
> And using other observations about sums, a simplified answer can be derived.

But it won't fit in the margin.

--
Bill Sloman, Sydney

On 04-Jul-21 10:33 am, RichD wrote:
> All engineers are mathematicians, who enjoy puzzles.
> Probability theory is a stock part of the curriculum, so...
>
> A theater has 15 seats in the first row. Patrons enter
> in single file, taking each consecutive seat, as they
> arrive. In this particular case, there are 8 boys, 7 girls.
>
> If a boy and girl occupy adjacent seats, we have a candidate
> match. If boy sees girl on each side, that's two matches.
> Clearly, any possible arrangement will contain 1...14
> candidate matches.
>
> What's the average number of matches, for this row?
>
> I found this in a book of math problems, it's a nice little
> exercise. However, my motivation here is mainly due
> to the fact that the author's reasoning, in his solution,
> is severely flawed. I think.
>
> Take a crack at it. In a few days, I'll post the book
> solution, with my critique. Then you tell me where I'm wrong.
>
> Or, if I'm right, yet somehow the book's solution is
> numerically correct, we have the challenge of explaining
> that miracle. Which is more interesting than the problem itself -
>
> --
> Rich
>

Programming the permutations gives me 7.466666.. which is (7 * 16 / 15).
Those numbers are suggestive.

This matches the result I get by simulation.

Trying to do it algebraicly suggests it will still produce 28 distinct
terms to be calculated, which is presumably not the intent.

Sylvia.

On 05/07/2021 07:07, Sylvia Else wrote:
> On 04-Jul-21 10:33 am, RichD wrote:
>> All engineers are mathematicians, who enjoy puzzles.
>> Probability theory is a stock part of the curriculum, so...
>>
>> A theater has 15 seats in the first row.  Patrons enter
>> in single file, taking each consecutive seat, as they
>> arrive.  In this particular case, there are 8 boys, 7 girls.
>>
>> If a boy and girl occupy adjacent seats, we have a candidate
>> match.  If boy sees girl on each side, that's two matches.
>> Clearly, any possible arrangement will contain 1...14
>> candidate matches.
>>
>> What's the average number of matches, for this row?
>>
>> I found this in a book of math problems, it's a nice little
>> exercise.  However, my motivation here is mainly due
>> to the fact that the author's reasoning, in his solution,
>> is severely flawed.  I think.
>>
>> Take a crack at it.  In a few days, I'll post the book
>> solution, with my critique.  Then you tell me where I'm wrong.
>>
>> Or, if I'm right, yet somehow the book's solution is
>> numerically correct, we have the challenge of explaining
>> that miracle.  Which is more interesting than the problem itself -
>>
>> --
>> Rich
>>
>
> Programming the permutations gives me 7.466666.. which is (7 * 16 / 15).
> Those numbers are suggestive.
>
> This matches the result I get by simulation.
>
> Trying to do it algebraicly suggests it will still produce 28 distinct
> terms to be calculated, which is presumably not the intent.

There is a trick in that all the middle seats are equivalent through
having two nearest neighbours so you can easily compute the probability
of a seat gap being a 0-1 or 1-0 transition. The gaps are all identical!
(each gap between the seats has exactly one seat on either side)

So the closed form solution for N = 2n+1 seats is:

(N+1)(N-1)/(2N)

Or substituting N = 2n+1

(n+1)(2n)/(2n+1)

Where in this case n=7, N=15 (though I would write it as 14*8/15)
(since there are 14 adjacent seats and 8 boys)

That gives the same result as you have observed experimentally.

I did check it by brute force simulation going from N=3 & N=5 where I
started by hand out to N=31 by computer and it still holds. You could
easily generalise it for any ratio of girls and boys.

--
Regards,
Martin Brown

On Sun, 4 Jul 2021 21:18:50 -0700 (PDT), Anthony William Sloman
<bill.sloman@ieee.org> wrote:

>On Monday, July 5, 2021 at 5:02:34 AM UTC+10, Cursitor Doom wrote:
>> On Sat, 3 Jul 2021 23:31:03 -0700 (PDT), Anthony William Sloman
>> <bill....@ieee.org> wrote:
>>
>> >On Sunday, July 4, 2021 at 2:01:51 PM UTC+10, jla...@highlandsniptechnology.com wrote:
>> >> On Sat, 3 Jul 2021 17:33:15 -0700 (PDT), RichD
>> >> <r_dela...@yahoo.com> wrote:
>> >>
>> >> >All engineers are mathematicians, who enjoy puzzles.
>> >> I used to enjoy puzzles and chess. But I have to think all day at
>> >> work, so I don't want to do that for free any more.
>> >
>> >Not an all-that-plausible claim. John Larkin likes to claim that what he does at work involves thinking, but what he posts here suggests he lost the capacity to think some years ago.
>> >
>> >He doesn't do it all any more, and uses the lack of payment as an excuse for not even trying.
>>
>> I haven't seen any evidence of YOU doing any work, lately, Bill.
>
>You wouldn't. The thread "Low noise, high bias voltage on picoAmp TIA's input, howto?" isn't the sort of stuff you could follow, even if you found it interesting, which you wouldn't.
>Or you could look at
>
>http://site.ieee.org/nsw/files/2021/04/Circuit_March_2021.pdf
>
>which I happen to edit. I've got the July 2021 issue about half done - the new and up-dated members data for the past three months has just come in and I've still got to slot that in.
>And I'm currently the treasurer for the NSW branch of the IEEE, which doesn't take up much time either
>
>>And since you seem to spend every waking minute tapping away on that keyboard of yours posting a lot of nonsense,
>
>I spend an hour or so a day at it. Most of it is beyond your understanding, so it may well look like nonsense to you, and not the kind of nonsense to which you are addicted.
>
>> I'll wager you don't even lift a finger to help your wife around the house either.
>
>I'm retired. My wife isn't. We pay cleaners to go over the place once a week, but I look after everything else (including the cooking and the shopping). That doesn't take long either.
>
>You really are remarkably silly, and unpleasant with it.

Whereas YOUR conduct here is unfailingly exemplary; polite and
informative. <sarc>

On Wednesday, July 7, 2021 at 3:23:45 AM UTC+10, Cursitor Doom wrote:
> On Sun, 4 Jul 2021 21:18:50 -0700 (PDT), Anthony William Sloman
> <bill....@ieee.org> wrote:
>
> >On Monday, July 5, 2021 at 5:02:34 AM UTC+10, Cursitor Doom wrote:
> >> On Sat, 3 Jul 2021 23:31:03 -0700 (PDT), Anthony William Sloman
> >> <bill....@ieee.org> wrote:
> >>
> >> >On Sunday, July 4, 2021 at 2:01:51 PM UTC+10, jla...@highlandsniptechnology.com wrote:
> >> >> On Sat, 3 Jul 2021 17:33:15 -0700 (PDT), RichD
> >> >> <r_dela...@yahoo.com> wrote:
> >> >>
> >> >> >All engineers are mathematicians, who enjoy puzzles.
> >> >> I used to enjoy puzzles and chess. But I have to think all day at
> >> >> work, so I don't want to do that for free any more.
> >> >
> >> >Not an all-that-plausible claim. John Larkin likes to claim that what he does at work involves thinking, but what he posts here suggests he lost the capacity to think some years ago.
> >> >
> >> >He doesn't do it all any more, and uses the lack of payment as an excuse for not even trying.
> >>
> >> I haven't seen any evidence of YOU doing any work, lately, Bill.
> >
> >You wouldn't. The thread "Low noise, high bias voltage on picoAmp TIA's input, howto?" isn't the sort of stuff you could follow, even if you found it interesting, which you wouldn't.
> >Or you could look at
> >
> >http://site.ieee.org/nsw/files/2021/04/Circuit_March_2021.pdf
> >
> >which I happen to edit. I've got the July 2021 issue about half done - the new and up-dated members data for the past three months has just come in and I've still got to slot that in.
> >And I'm currently the treasurer for the NSW branch of the IEEE, which doesn't take up much time either
> >
> >>And since you seem to spend every waking minute tapping away on that keyboard of yours posting a lot of nonsense,
> >
> >I spend an hour or so a day at it. Most of it is beyond your understanding, so it may well look like nonsense to you, and not the kind of nonsense to which you are addicted.
> >
> >> I'll wager you don't even lift a finger to help your wife around the house either.
> >
> >I'm retired. My wife isn't. We pay cleaners to go over the place once a week, but I look after everything else (including the cooking and the shopping). That doesn't take long either.
> >
> >You really are remarkably silly, and unpleasant with it.
>
> Whereas YOUR conduct here is unfailingly exemplary; polite and informative. <sarc>

It is informative. You frequently don't like the information I impart because it doesn't line up with your favourite delusions.

I am polite to people who deserve it. I like to think that I conform to definition of a gentleman - never unintentionally rude. In reality everybody makes mistakes and some - like you - make them more frequently than others, and deserve all the rudeness they get.

--
Bill Sloman, Sydney

On July 4, 2021, Grant Taylor wrote:
> I've found that hypothetical puzzles, or other people's puzzles, can be
> a worthwhile academic exercise as the underlying method to solve them
> often becomes helpful later in life

Tennis isn't push ups, but push ups are a useful exercise for a tennis player -

--
Rich

On July 4, 2021, Don Y wrote:

>> All engineers are mathematicians, who enjoy puzzles.
>
> I much prefer "puzzles" that highlight flaws in our thinking
> (problem solving) process as those can be used to IMPROVE it.
> The "missing dollar" hotel problem falls into the same pattern.

I used to know that one, but alas now rusty.
Remind me -

> There are three (identical) bottles and three (identical) chopsticks
> on the table in front of you. Arrange the chopsticks on the bottles
> so that no chopstick touches the table top.

Given seven cylindrical chalks, 5" long, 1/4" diameter, cut flush
at each end. Arrange them on a table top, for common contact,
i.e. such that each touches every other.

Give yourself an Einstein Award if you solve it.
Or prove it's impossible -

--
Rich

On 7/7/2021 5:19 PM, RichD wrote:
> On July 4, 2021, Don Y wrote:
>
>>> All engineers are mathematicians, who enjoy puzzles.
>>
>> I much prefer "puzzles" that highlight flaws in our thinking
>> (problem solving) process as those can be used to IMPROVE it.
>> The "missing dollar" hotel problem falls into the same pattern.
>
> I used to know that one, but alas now rusty.
> Remind me -

Three "salesmen" walk into a hotel looking for rooms.
Each is charged $10 and given a room. 3 * 10 = $30

Some time later, the maitre d'hotel discovers that the
night desk clerk had failed to give them the "3 for $25"
special. So, he gives $5 to the bellboy and tells him
to run it up to the men.

The enterprising bell boy realizes that he can, instead,
give each of the men $1 and pocket the remaining $2
for himself! (3 * 1 + 2 = 5)

So, each man has paid $9 (the initial 10 - the $1 refund)
which is $27. And, the bellboy has $2 in his pocket
(27 + 2 = 29) -- where did the other $1 go??

The point being that this is defective reasoning. Yet,
tell it to most folks and they'll be drawn into the flawed
argument.

Why??

*Clearly* the hotel has $25 in their till. The bell boy has $2
in his pocket and this exactly matches the $27 that was laid
out by the three travelers (3 * 9). There is no "missing dollar"!

My interest is in trying to understand why folks are so
easily coerced by such obviously defective arguments.
These aren't "challenging" problems that require any
special skillsets!

[Or, the "30 days" for the snail to exit the well. Any
serious thought would make it clear that this is NOT the case!
Or, the sub-optimal solutions to the three bottles problems.
So, why do people allow themselves to be seduced by faulty
reasoning or "lazy" thinking? How does this sort of
thinking permeate other "technical" decision-making?]

On July 4, 2021, Fred Bloggs wrote:
>>> >> A theater has 15 seats in the first row. Patrons enter
>>> >> in single file, taking each consecutive seat, as they
>>> >> arrive. In this particular case, there are 8 boys, 7 girls.
>>> >> What's the average number of matches, for this row?
>
>> It's a messy problem. There are factorial 15 possible arrangements.
>> Mathematicians can see way of looking a this kind of problem that let them get to an answer easily,
>> but often the answer is too long to scribble in the margin.
>
> Then start creating some simplifying notation like D(n) = arrangements with n mixed color
> adjacencies. Then you need to let T be the total number of arrangements of the 15 marbles possible.
> The probability of occurrence of n different color adjacencies is therefore D(n)/T. The average,
> or expected number of adjacencies, M, is therefore: M= 1 x D(1)/T + 2 x D(2)/T + .....+14 x D(14)/T + 15 x D(15)/T
> T should be T=15!/(8! x 7!)

oh brother -

Guys, you ever heard of gamesmanship? Likewise, there's puzzlemanship.

Any worthy puzzle is "aha!" A light bulb has to flash. If it actually involves
computation of factorials, it won't get posted!

I didn't realize it's such a baffler.
All right, try this: select a SINGLE pair. Calculate the expected number of matches
for that. Then...

This is the approach taken by the author, utilizing a theorem he hammers repeatedly:
the expectation of the sum is the sum of the expectations.

Simple, hey? um, not so fast. Because any theorem is only as strong as its
underlying assumptions. pop quiz: in the case of this theorem, what's the
assumed condition? It's clearly (to me) violated by this problem.

Yet, does it lead to the right answer?
"Fascinating." -- Mr. Spock

--
Rich

On July 4, 2021, Sylvia Else wrote:
> Programming the permutations gives me 7.466666.. which is (7 * 16 / 15).
> Those numbers are suggestive.
> Trying to do it algebraicly suggests it will still produce 28 distinct
> terms to be calculated, which is presumably not the intent.

Finally, someone shows some sense!

--
Rich