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Groups > sci.physics > #894823 > unrolled thread

Any Math Heads Out There?

Started byFarley Flud <ff@linux.rocks>
First post2025-12-24 13:30 +0000
Last post2026-01-07 18:17 -0600
Articles 16 — 7 participants

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Contents

  Any Math Heads Out There? Farley Flud <ff@linux.rocks> - 2025-12-24 13:30 +0000
    Re: Any Math Heads Out There? Farley Flud <ff@linux.rocks> - 2025-12-24 18:36 +0000
    Re: Any Math Heads Out There? Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2025-12-28 04:27 +0100
    Re: Any Math Heads Out There? Physfitfreak <physfitfreak@gmail.com> - 2026-01-03 20:06 -0600
      Re: Any Math Heads Out There? pothead <pothead@snakebite.com> - 2026-01-04 02:17 +0000
        Re: Any Math Heads Out There? CrudeSausage <crude@sausa.ge> - 2026-01-03 21:36 -0500
          [OT] Netiquette (was: Any Math Heads Out There?) Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-04 15:04 +0100
            Re: [OT] Netiquette "Joel W. Crump" <joelcrump@gmail.com> - 2026-01-04 13:58 -0500
      Re: Any Math Heads Out There? Farley Flud <ff@linux.rocks> - 2026-01-04 11:20 +0000
        Re: Any Math Heads Out There? Farley Flud <ff@linux.rocks> - 2026-01-04 11:31 +0000
          Re: Any Math Heads Out There? Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-04 15:14 +0100
          Re: Any Math Heads Out There? Physfitfreak <physfitfreak@gmail.com> - 2026-01-04 19:21 -0600
            Re: Any Math Heads Out There? Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2026-01-05 18:09 +0100
    Re: Any Math Heads Out There? Physfitfreak <physfitfreak@gmail.com> - 2026-01-06 20:08 -0600
      Re: Any Math Heads Out There? John Hasler <john@sugarbit.com> - 2026-01-06 22:16 -0600
        Re: Any Math Heads Out There? Physfitfreak <physfitfreak@gmail.com> - 2026-01-07 18:17 -0600

#894823 — Any Math Heads Out There?

FromFarley Flud <ff@linux.rocks>
Date2025-12-24 13:30 +0000
SubjectAny Math Heads Out There?
Message-ID<pan$dcbe2$183521f7$63cceda3$1d68c149@linux.rocks>
Any math heads out there?

The following two expressions are equal.  That is, substituting any value
of t in the expressions will produce the same value.

Yet, I cannot reduce one to the other using algebra and the trig identities.
It must be possible but I can't find a way to do it.

Can you?

Eternal fame awaits the person who can solve this.


Expression 1:

⎛                      _______________                                                      ⎞
⎜      _              ╱          2                ⎛           4              2    ⎞         ⎟
⎜8 ⋅ ╲╱3 ⋅ cos(t) ⋅ ╲╱ 3 ⋅ cos(t)  + 1 ⋅ sin(t) + ⎝27 ⋅ cos(t)  + 18 ⋅ cos(t)  + 3⎠ ⋅ sin(t)⎟
⎜───────────────────────────────────────────────────────────────────────────────────────────⎟
⎜                                         4              2                                  ⎟
⎝                              36 ⋅ cos(t)  + 24 ⋅ cos(t)  + 4                              ⎠


Expression 2:

⎛   _______________                                                         ⎞
⎜  ╱          2       ⎛          2    ⎞                  _                  ⎟
⎜╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝9 ⋅ cos(t)  + 3⎠ ⋅ sin(t) + 8 ⋅ ╲╱3 ⋅ cos(t) ⋅ sin(t)⎟
⎜───────────────────────────────────────────────────────────────────────────⎟
⎜                     _______________                                       ⎟
⎜                    ╱          2       ⎛           2    ⎞                  ⎟
⎝                  ╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝12 ⋅ cos(t)  + 4⎠                  ⎠






-- 
Gentoo: the only road to GNU/Linux freedom and perfection.

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#894825

FromFarley Flud <ff@linux.rocks>
Date2025-12-24 18:36 +0000
Message-ID<pan$51cf2$2afcb46d$fdcf1476$ea5b67e1@linux.rocks>
In reply to#894823
On Wed, 24 Dec 2025 13:30:01 +0000, Farley Flud wrote:

> Any math heads out there?
> 
> 
> Eternal fame awaits the person who can solve this.
> 

You may be wondering how I arrived at these particular expressions
and what they have to do with GNU/Linux.

Fret not.  I shall reveal.

I was exploring general tangent plane vectors along a general
sphere geodesic using the open-source GNU/Linux Maxima:

<https://maxima.sourceforge.io/>

I had derived the covariant derivative of the general tangent
vector using both the intrinsic (Christoffel symbols) and extrinsic
(projection onto unit normal) methods.  They should be equal
but the expressions were different.  Yet they are indeed equal.
One must be reducible to the other but I can't figure out how
to do it.

If you can perform this reduction you will gain everlasting
fame.

So get to it.



-- 
Gentoo: the only road to GNU/Linux freedom and perfection.

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#894827

FromThomas 'PointedEars' Lahn <PointedEars@web.de>
Date2025-12-28 04:27 +0100
Message-ID<10iq82i$1pob0$1@gwaiyur.mb-net.net>
In reply to#894823
[pointless Followup-To comp.os.linux.advocacy ignored; F'up2 sci.math set]

Farley Flud wrote:
> Any math heads out there?
> 
> The following two expressions are equal.  That is, substituting any value
> of t in the expressions will produce the same value.
> 
> Yet, I cannot reduce one to the other using algebra and the trig identities.
> It must be possible but I can't find a way to do it.
> 
> Can you?

Next time, use standard (ASCII) notation and not Unicode *art*.  There is a
place and time for ASCII and Unicode art; posting mathematical problems on
Usenet is NOT it.

Notice that at least Thunderbird converts some ASCII character sequences to
Unicode, like x^2 (x followed by circumflex followed by 2).

Before I attempt to simplify this: Are the expressions below those that you
meant?

> Eternal fame awaits the person who can solve this.

8-)

> Expression 1: [rewritten using standard notation as I see it]
> 
> {8 sqrt(3) cos(t) sqrt[3 cos^2(t) + 1] sin(t)
>   + [27 cos^4(t) + 18 cos^2(t) + 3] sin(t)}/
> {36 cos^4(t) + 24 cos^2(t) + 4}
> 
> Expression 2: [ditto]
> [...]
>
> {sqrt[3 cos^2(t) + 1] [9 cos^2(t) + 3] sin(t) + 8 sqrt(3) cos(t) sin(t)}/
> {sqrt[3 cos^2(t) + 1] [12 cos^2(t) + 4]}

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894881

FromPhysfitfreak <physfitfreak@gmail.com>
Date2026-01-03 20:06 -0600
Message-ID<10jchue$1mjsm$1@dont-email.me>
In reply to#894823
On 12/24/25 7:30 AM, Farley Flud wrote:
> Any math heads out there?
> 
> The following two expressions are equal.  That is, substituting any value
> of t in the expressions will produce the same value.
> 
> Yet, I cannot reduce one to the other using algebra and the trig identities.
> It must be possible but I can't find a way to do it.
> 
> Can you?
> 
> Eternal fame awaits the person who can solve this.
> 
> 
> Expression 1:
> 
> ⎛                      _______________                                                      ⎞
> ⎜      _              ╱          2                ⎛           4              2    ⎞         ⎟
> ⎜8 ⋅ ╲╱3 ⋅ cos(t) ⋅ ╲╱ 3 ⋅ cos(t)  + 1 ⋅ sin(t) + ⎝27 ⋅ cos(t)  + 18 ⋅ cos(t)  + 3⎠ ⋅ sin(t)⎟
> ⎜───────────────────────────────────────────────────────────────────────────────────────────⎟
> ⎜                                         4              2                                  ⎟
> ⎝                              36 ⋅ cos(t)  + 24 ⋅ cos(t)  + 4                              ⎠
> 
> 
> Expression 2:
> 
> ⎛   _______________                                                         ⎞
> ⎜  ╱          2       ⎛          2    ⎞                  _                  ⎟
> ⎜╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝9 ⋅ cos(t)  + 3⎠ ⋅ sin(t) + 8 ⋅ ╲╱3 ⋅ cos(t) ⋅ sin(t)⎟
> ⎜───────────────────────────────────────────────────────────────────────────⎟
> ⎜                     _______________                                       ⎟
> ⎜                    ╱          2       ⎛           2    ⎞                  ⎟
> ⎝                  ╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝12 ⋅ cos(t)  + 4⎠                  ⎠
> 
> 
> 
> 
> 
> 


They're equal. Place the two expressions on either sides of an equal 
sign. Factor out and get rid of sin(t). Then do the following substitution:

a = square root of (3cost^2 +1)

Both sides will quickly reduce to the same result in terms of a.

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#894882

Frompothead <pothead@snakebite.com>
Date2026-01-04 02:17 +0000
Message-ID<10jcik0$1mo8o$1@pothead.dont-email.me>
In reply to#894881
On 2026-01-04, Physfitfreak <physfitfreak@gmail.com> wrote:
> On 12/24/25 7:30 AM, Farley Flud wrote:
>> Any math heads out there?
>> 
>> The following two expressions are equal.  That is, substituting any value
>> of t in the expressions will produce the same value.
>> 
>> Yet, I cannot reduce one to the other using algebra and the trig identities.
>> It must be possible but I can't find a way to do it.
>> 
>> Can you?
>> 
>> Eternal fame awaits the person who can solve this.
>> 
>> 
>> Expression 1:
>> 
>> ⎛                      _______________                                                      ⎞
>> ⎜      _              ╱          2                ⎛           4              2    ⎞         ⎟
>> ⎜8 ⋅ ╲╱3 ⋅ cos(t) ⋅ ╲╱ 3 ⋅ cos(t)  + 1 ⋅ sin(t) + ⎝27 ⋅ cos(t)  + 18 ⋅ cos(t)  + 3⎠ ⋅ sin(t)⎟
>> ⎜───────────────────────────────────────────────────────────────────────────────────────────⎟
>> ⎜                                         4              2                                  ⎟
>> ⎝                              36 ⋅ cos(t)  + 24 ⋅ cos(t)  + 4                              ⎠
>> 
>> 
>> Expression 2:
>> 
>> ⎛   _______________                                                         ⎞
>> ⎜  ╱          2       ⎛          2    ⎞                  _                  ⎟
>> ⎜╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝9 ⋅ cos(t)  + 3⎠ ⋅ sin(t) + 8 ⋅ ╲╱3 ⋅ cos(t) ⋅ sin(t)⎟
>> ⎜───────────────────────────────────────────────────────────────────────────⎟
>> ⎜                     _______________                                       ⎟
>> ⎜                    ╱          2       ⎛           2    ⎞                  ⎟
>> ⎝                  ╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝12 ⋅ cos(t)  + 4⎠                  ⎠
>> 
>> 
>> 
>> 
>> 
>> 
>
>
> They're equal. Place the two expressions on either sides of an equal 
> sign. Factor out and get rid of sin(t). Then do the following substitution:
>
> a = square root of (3cost^2 +1)
>
> Both sides will quickly reduce to the same result in terms of a.

Correct.

Factor.
Simplify the fraction
Factor once more.
Simplify the fraction again.

Both expressions are equal.

Easy peasy.




-- 
pothead

“Liberals seem to assume that, if you don’t believe in their particular 
political solutions, then you don’t really care about
the people that they claim to want to help.”

― Thomas Sowell

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#894883

FromCrudeSausage <crude@sausa.ge>
Date2026-01-03 21:36 -0500
Message-ID<6959d23d$6$27$882e4bbb@reader.netnews.com>
In reply to#894882
On 2026-01-03 21:17, pothead wrote:
> On 2026-01-04, Physfitfreak <physfitfreak@gmail.com> wrote:
>> On 12/24/25 7:30 AM, Farley Flud wrote:
>>> Any math heads out there?
>>>
>>> The following two expressions are equal.  That is, substituting any value
>>> of t in the expressions will produce the same value.
>>>
>>> Yet, I cannot reduce one to the other using algebra and the trig identities.
>>> It must be possible but I can't find a way to do it.
>>>
>>> Can you?
>>>
>>> Eternal fame awaits the person who can solve this.
>>>
>>>
>>> Expression 1:
>>>
>>> ⎛                      _______________                                                      ⎞
>>> ⎜      _              ╱          2                ⎛           4              2    ⎞         ⎟
>>> ⎜8 ⋅ ╲╱3 ⋅ cos(t) ⋅ ╲╱ 3 ⋅ cos(t)  + 1 ⋅ sin(t) + ⎝27 ⋅ cos(t)  + 18 ⋅ cos(t)  + 3⎠ ⋅ sin(t)⎟
>>> ⎜───────────────────────────────────────────────────────────────────────────────────────────⎟
>>> ⎜                                         4              2                                  ⎟
>>> ⎝                              36 ⋅ cos(t)  + 24 ⋅ cos(t)  + 4                              ⎠
>>>
>>>
>>> Expression 2:
>>>
>>> ⎛   _______________                                                         ⎞
>>> ⎜  ╱          2       ⎛          2    ⎞                  _                  ⎟
>>> ⎜╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝9 ⋅ cos(t)  + 3⎠ ⋅ sin(t) + 8 ⋅ ╲╱3 ⋅ cos(t) ⋅ sin(t)⎟
>>> ⎜───────────────────────────────────────────────────────────────────────────⎟
>>> ⎜                     _______________                                       ⎟
>>> ⎜                    ╱          2       ⎛           2    ⎞                  ⎟
>>> ⎝                  ╲╱ 3 ⋅ cos(t)  + 1 ⋅ ⎝12 ⋅ cos(t)  + 4⎠                  ⎠
>>>
>>>
>>>
>>>
>>>
>>>
>>
>>
>> They're equal. Place the two expressions on either sides of an equal
>> sign. Factor out and get rid of sin(t). Then do the following substitution:
>>
>> a = square root of (3cost^2 +1)
>>
>> Both sides will quickly reduce to the same result in terms of a.
> 
> Correct.
> 
> Factor.
> Simplify the fraction
> Factor once more.
> Simplify the fraction again.
> 
> Both expressions are equal.
> 
> Easy peasy.

Man, how I wish I could still do math.

-- 
CrudeSausage
John 14:6
Pop_OS!

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#894887 — [OT] Netiquette (was: Any Math Heads Out There?)

FromThomas 'PointedEars' Lahn <PointedEars@web.de>
Date2026-01-04 15:04 +0100
Subject[OT] Netiquette (was: Any Math Heads Out There?)
Message-ID<10jds17$lbkm$1@gwaiyur.mb-net.net>
In reply to#894883
CrudeSausage <crude@sausa.ge> amok-crossposted in comp.os.linux.advocacy and
sci.physics:
> [full quote] 
>
> Man, how I wish I could still do math.

I wish that one would need something like a driver's license before one
would be allowed to post to Usenet, and that that could be revoked or
suspended if one failed to use it properly on a regular basis -- like
violating foreign namespaces with "funny" "addresses", bothering other
newsgroups with off-topic amok-crossposts, and full-quoting.

F'up2 poster

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894889 — Re: [OT] Netiquette

From"Joel W. Crump" <joelcrump@gmail.com>
Date2026-01-04 13:58 -0500
SubjectRe: [OT] Netiquette
Message-ID<wLy6R.338836$e72b.286418@fx44.iad>
In reply to#894887
On 1/4/26 9:04 AM, Thomas 'PointedEars' Lahn wrote:
> CrudeSausage <crude@sausa.ge> amok-crossposted in comp.os.linux.advocacy and
> sci.physics:
>>
>> Man, how I wish I could still do math.
> 
> I wish that one would need something like a driver's license before one
> would be allowed to post to Usenet, and that that could be revoked or
> suspended if one failed to use it properly on a regular basis -- like
> violating foreign namespaces with "funny" "addresses", bothering other
> newsgroups with off-topic amok-crossposts, and full-quoting.
> 
> F'up2 poster


Oh we're *so* surprised, some geekwad from sci.physics crossposts a 
reply to bitch about crossposting, adding to the issue as he talks about 
having a license to post, full of his esoteric rules he would dictate to 
everyone.  You should really lose your virginity.

-- 
Joel W. Crump

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#894885

FromFarley Flud <ff@linux.rocks>
Date2026-01-04 11:20 +0000
Message-ID<pan$6537d$17e3e9f0$284d33c9$114762b@linux.rocks>
In reply to#894881
On Sat, 3 Jan 2026 20:06:06 -0600, Physfitfreak wrote:

> 
> They're equal. Place the two expressions on either sides of an equal 
> sign. Factor out and get rid of sin(t). Then do the following substitution:
> 
> a = square root of (3cost^2 +1)
> 

Yes.  Now try the following.  Both expressions are equal.  Reduce one
to the other:

1)

 ⎛                   _______________⎞
 ⎜  _         2     ╱              2⎟
 ⎜╲╱3 ⋅ sin(t)  ⋅ ╲╱ 4 - 3 ⋅ sin(t) ⎟
-⎜──────────────────────────────────⎟
 ⎜             4             2      ⎟
 ⎝   9 ⋅ cos(t)  + 6 ⋅ cos(t)  + 1  ⎠



2)

   _         2     _
 ╲╱3 ⋅ cos(t)  - ╲╱3
────────────────────
                 ⎛3⎞
                 ⎜─⎟
                 ⎝2⎠
⎛          2    ⎞   
⎝3 ⋅ cos(t)  + 1⎠



Again, these are another component of the 2 vectors produced by
Maxima.  The vectors are equal but Maxima has simplified each
differently.  But Maxima cannot reduce one to the other, at least
not with the default procedures.





-- 
Gentoo: the only road to GNU/Linux freedom and perfection.

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#894886

FromFarley Flud <ff@linux.rocks>
Date2026-01-04 11:31 +0000
Message-ID<pan$9e274$80a3ac2$fc4d8a35$77171c9a@linux.rocks>
In reply to#894885
On Sun, 04 Jan 2026 11:20:03 +0000, Farley Flud wrote:

> 
> 2)
> 
>    _         2     _
>  ╲╱3 ⋅ cos(t)  - ╲╱3
> ────────────────────
>                  ⎛3⎞
>                  ⎜─⎟
>                  ⎝2⎠
> ⎛          2    ⎞   
> ⎝3 ⋅ cos(t)  + 1⎠
> 
> 
> 

For some reason, #2 did not print properly.  The numerator should
be:

sqrt(3) * cos(t)^2 - sqrt(3)





-- 
Gentoo: the only road to GNU/Linux freedom and perfection.

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#894888

FromThomas 'PointedEars' Lahn <PointedEars@web.de>
Date2026-01-04 15:14 +0100
Message-ID<10jdsjj$lcdc$1@gwaiyur.mb-net.net>
In reply to#894886
Farley Flud amok-crossposted to comp.os.linux.advocacy and sci.physics again:
> For some reason, #2 did not print properly.  The numerator should
> be:
> 
> sqrt(3) * cos(t)^2 - sqrt(3)

Nobody *here* cares about that.  FOAD.

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894892

FromPhysfitfreak <physfitfreak@gmail.com>
Date2026-01-04 19:21 -0600
Message-ID<10jf3m7$2mpfs$1@solani.org>
In reply to#894886
On 1/4/26 5:31 AM, Farley Flud wrote:
> On Sun, 04 Jan 2026 11:20:03 +0000, Farley Flud wrote:
> 
>>
>> 2)
>>
>>     _         2     _
>>   ╲╱3 ⋅ cos(t)  - ╲╱3
>> ────────────────────
>>                   ⎛3⎞
>>                   ⎜─⎟
>>                   ⎝2⎠
>> ⎛          2    ⎞
>> ⎝3 ⋅ cos(t)  + 1⎠
>>
>>
>>
> 
> For some reason, #2 did not print properly.  The numerator should
> be:
> 
> sqrt(3) * cos(t)^2 - sqrt(3)
> 
> 
> 
> 
> 



On my screen it is properly printed.

Uploaded any of those videos of yours recently? :)

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#894893

FromThomas 'PointedEars' Lahn <PointedEars@web.de>
Date2026-01-05 18:09 +0100
Message-ID<10jgr80$sr16$1@gwaiyur.mb-net.net>
In reply to#894892
Physfitfreak crossposted to comp.os.linux.advocacy and sci.physics without
Followup-To:
> On 1/4/26 5:31 AM, Farley Flud wrote:
>> On Sun, 04 Jan 2026 11:20:03 +0000, Farley Flud wrote:
>>> 2)
>>>
>>>     _         2     _
>>>   ╲╱3 ⋅ cos(t)  - ╲╱3
>>> ────────────────────
>>>                   ⎛3⎞
>>>                   ⎜─⎟
>>>                   ⎝2⎠
>>> ⎛          2    ⎞
>>> ⎝3 ⋅ cos(t)  + 1⎠
>>>
>>>
>>>
>>
>> For some reason, #2 did not print properly.  The numerator should
>> be:
>>
>> sqrt(3) * cos(t)^2 - sqrt(3)
> 
> On my screen it is properly printed.

To me the term looks like

  [√(3) cos(t)^2 - √(3)]/[3 cos^(t)^2 + 1]^(3/2),

but only if I use DejaVu Sans Mono 13pt (not, for example, with Consolas
12pt, another monospace font).

So this is more evidence that it is not a good idea (that should even
convince the staunchest supporter) to *draw* equations (using Unicode) in
Usenet.  [As you can see, I can find nothing wrong with *writing* Unicode
characters when using ASCII would be clumsy instead.]

F'up2 sci.math

-- 
PointedEars

Twitter: @PointedEars2
Please do not cc me. / Bitte keine Kopien per E-Mail.

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#894901

FromPhysfitfreak <physfitfreak@gmail.com>
Date2026-01-06 20:08 -0600
Message-ID<10jkf6e$aheh$1@dont-email.me>
In reply to#894823
On 12/24/25 7:30 AM, Farley Flud wrote:
> Any math heads out there?
> 
> 
> 


This one befits a math head better:

An Iranian Ayatollah invites a penniless Iranian scientist to give him a 
visit. When the scientist visits him, he goes like:

Ayatollah: We gave this question to every engineer we could and none of 
them could answer it. Fat, thin, short, tall, rich, or super rich, 
didn't matter. None of those donkeys could answer it. Then someone said 
why not asking a scientist, so now you're here.

Scientist: What is the question?

Ayatollah: I have gathered fund to distribute a large number of spies 
throughout occupied Palestine (Israel.) All of them are non-Iranian, and 
have different ethnicity. We have divided up the entire land into as 
many zones as we plan to assign spies to. The spies will only contact 
each other if they're stationed in neighboring zones. But we do not want 
any spy have a neighboring spy of the same ethnicity.

Scientist: !.. Why not?

Ayatollah: Because if of the same ethnicity, their stupid wives will be 
talking each other's brains on the phone all day and Mossad can easily 
get suspicious of the two households.

Scientist: Oh!..

Ayatollah: So here's the problem we have. We prefer to use as few 
different ethnicity among the spies as possible. So what is the minimum 
number of different ethnicity we can limit our process to, to distribute 
them in a way to prevent any neighboring spies of same ethnicity?

And he continues:

Ayatollah: So are we done here?

Scientist: Of course not. What's in it for me? I'm not one of your 5 
million Basij slave to do it free for you.

Ayatollah: Hmm.. how about ... much in advance and another ... much 
after you come up with the answer?

The scientist takes out his smartphone and uses the astronomical number 
to convert the amount into U.S. dollar, and then:

Scientist: Holy fucking shit.. I'm in it!

So he gets the first payment and goes home, works on the problem for a 
day, and goes back and gives the answer to the Ayatollah and gets the 
final payment.

What was his answer?

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#894902

FromJohn Hasler <john@sugarbit.com>
Date2026-01-06 22:16 -0600
Message-ID<87344i5a7m.fsf@sugarbit.com>
In reply to#894901
Four, of course.
-- 
John Hasler 
john@sugarbit.com
Dancing Horse Hill
Elmwood, WI USA

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#894906

FromPhysfitfreak <physfitfreak@gmail.com>
Date2026-01-07 18:17 -0600
Message-ID<10jmt3h$2rqic$1@solani.org>
In reply to#894902
On 1/6/26 10:16 PM, John Hasler wrote:
> Four, of course.


:-)

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