## Poll

 I love math! 21 votes (46.66%) Math is great. 14 votes (31.11%) My religion is mathology. 6 votes (13.33%) Women didn't speak to me until I was 30. 2 votes (4.44%) Total eclipse reminder -- 04/08/2024 12 votes (26.66%) I steal cutlery from restaurants. 3 votes (6.66%) I should just say what's on my mind. 6 votes (13.33%) Who makes up these awful names for pandas? 5 votes (11.11%) I like to touch my face. 12 votes (26.66%) Pork chops and apple sauce. 10 votes (22.22%)

45 members have voted

chevy Joined: Apr 15, 2011
• Posts: 138
October 25th, 2023 at 4:42:20 PM permalink
Quote: charliepatrick

I think the trick here is to consider A at (0,2) but B at (5,-3) rather than (5,3). Where the line between A and B meets the X asis (2,0) is where the river is. Since the horizontal and vertical lengths are each 5, the length of the pipe is 5 SQRT(2).
As I'm away the Y solution, for three points, usually requires the three pipes to be at 120 degrees to each other, thus you would construct two lines at 120o connecting to A and B such that, a third line 120o to those, was vertically down to the river.

the 120 degree solution does match what I got finding intersection (x,y) that minimizes pipe distance numerically.

Gialmere Joined: Nov 26, 2018
• Posts: 2801
October 25th, 2023 at 4:46:48 PM permalink
Quote: unJon

Solving the two simultaneous equations:

Hits/ABs = 0.274

(Hits + 3) / (ABs + 4) = 0.289

Gives hits of 33.683

With rounding you get the right answer with 34 Hits and 124 At Bats before the current game.

Correct!!

Well done.
------------------------------------------

Doctor: What did you dream about last night?

Patient: Baseball.

Doctor: Don�t you dream about anything else?

Patient: What, and miss my turn at bat?
Have you tried 22 tonight? I said 22.
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 25th, 2023 at 7:03:26 PM permalink
To follow up on the water pipe problem -- it's tough because there are two things to solve, the x and y coordinates of where to split off from the main pipe to the river to the two houses.

Here is the diagram again. If we say point B is at (0,0) and point D is at (5,0), then I find the point where the three points meet to be at (1.633971768, 1.056623499).

This results in total pipe length of 6.830127019. Can anyone do better?

I admit the calculus got too messy and I used the "solver" feature in Excel to get those coordinates. If you don't have that add-on, then I highly recommend you do.

I'd be very interested to see a closed-form solution, if anyone is up to the challenge.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
chevy Joined: Apr 15, 2011
• Posts: 138
October 25th, 2023 at 9:37:39 PM permalink
Quote: Wizard

To follow up on the water pipe problem -- it's tough because there are two things to solve, the x and y coordinates of where to split off from the main pipe to the river to the two houses.

Here is the diagram again. If we say point B is at (0,0) and point D is at (5,0), then I find the point where the three points meet to be at (1.633971768, 1.056623499).

This results in total pipe length of 6.830127019. Can anyone do better?

I admit the calculus got too messy and I used the "solver" feature in Excel to get those coordinates. If you don't have that add-on, then I highly recommend you do.

I'd be very interested to see a closed-form solution, if anyone is up to the challenge.

I agree with your (x,y). I had numerically solved on a 0.01x0.01 size grid getting (1.63, 1.06). So yours is more precise. And it does give the 120 degree spacing @charliepatrick mentions.

Didn't work out closed form yet....I too gave up on the calculus.....got as far as

x=5*(2-y) / (5-2*y) from partial deriv wrt x set = 0.....it works for your solution, so plenty confident in your numbers
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 25th, 2023 at 9:48:41 PM permalink
Quote: chevy

I agree with your (x,y). I had numerically solved on a 0.01x0.01 size grid getting (1.63, 1.06). So yours is more precise. And it does give the 120 degree spacing @charliepatrick mentions.

Didn't work out closed form yet....I too gave up on the calculus.....got as far as

x=5*(2-y) / (5-2*y) from partial deriv wrt x set = 0.....it works for your solution, so plenty confident in your numbers

Thank you for the confirmation. That is interesting about the 120 angles.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
chevy Joined: Apr 15, 2011
• Posts: 138
October 25th, 2023 at 10:47:38 PM permalink
Quote: Wizard

Quote: chevy

I agree with your (x,y). I had numerically solved on a 0.01x0.01 size grid getting (1.63, 1.06). So yours is more precise. And it does give the 120 degree spacing @charliepatrick mentions.

Didn't work out closed form yet....I too gave up on the calculus.....got as far as

x=5*(2-y) / (5-2*y) from partial deriv wrt x set = 0.....it works for your solution, so plenty confident in your numbers

Thank you for the confirmation. That is interesting about the 120 angles.

Still can't get calculus to work, but using @charliepatrick's 120 degree theorem (maybe he can chime in on why)....

then just use geometry....calling intersection F

AF is 30 degrees above horizontal.........(2-y)/x = tan(30) = 1/sqrt(3)
CF is 30 degrees above horizontal.........(3-y)/(5-x) = tan (30) = 1/sqrt(3)

Solving these two equations for x,y....making several algebra mistakes in the process, I get

setting the two equal yields x=5(y-2)/(2y-5)......mention it as it matches the local min in x dir from calc)
plugging in to AF equation gives

y=(5/2) * (1 - 1/sqrt(3)) = 1.05662432703

Plugging back in gives

x=(5/2) * (1-sqrt(3) / 5) = 1.63397459622
Gialmere Joined: Nov 26, 2018
• Posts: 2801
October 26th, 2023 at 7:01:06 AM permalink One sunny day four young boys began following the train tracks out of town in search of a dead body. They had begun crossing a high railroad bridge spanning a gorge when, to their horror, they heard a train coming from behind them. Two of the boys were up ahead and easily made it to the other side. But, the other two were only 3/7 of the way across the bridge. As their friends screamed for them to hurry, they began running for their lives to the far side, only just reaching it and jumping off the tracks before the train roared by.

Later, when they had time to think about it, they realized they had been less than halfway across the bridge. They wondered if they could have run back toward the train and still gotten off the tracks in time.

What's the minimum distance away the train would of had to have been in order for the two boys to have escaped in either direction?
Have you tried 22 tonight? I said 22.
chevy Joined: Apr 15, 2011
• Posts: 138
Thanks for this post from: October 26th, 2023 at 7:24:53 AM permalink

If D = length of bridge
They run at speed (4/7)*D / T
Running other way a distance (3/7)*D takes (3/4)*T

If just barely making it to safety either way is required, then the train is at the start of the bridge at t=(3/4)*T and end of bridge at time t=T. It travels the length of the bridge in (1/4)*T.

So if train starting at distance of 4D (from the end of bridge or 3D from start of bridge) they should just barely jump to safety
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 26th, 2023 at 7:38:23 AM permalink
Quote: chevy

Still can't get calculus to work, but using @charliepatrick's 120 degree theorem (maybe he can chime in on why)....

I too would be interested to see a proof or a citation of the 120-degree theorem.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
Wizard Joined: Oct 14, 2009
• Posts: 25961
Thanks for this post from: October 26th, 2023 at 7:47:34 AM permalink
Quote: Gialmere

What's the minimum distance away the train would of had to have been in order for the two boys to have escaped in either direction?

I get the train is 3/4 the length of the bridge away from the entrance to it.

If the question is asking the distance from the boys, that would then be 3/4 + 3/7 = 33/28 the length of the bridge.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
chevy Joined: Apr 15, 2011
• Posts: 138
October 26th, 2023 at 8:17:32 AM permalink
Quote: Wizard

Quote: chevy

Still can't get calculus to work, but using @charliepatrick's 120 degree theorem (maybe he can chime in on why)....

I too would be interested to see a proof or a citation of the 120-degree theorem.

Some googling suggests It seems to be a variant of what is called the Fermat Point. (I will now think of it as the Fermat-CharliePatrick point)

https://en.wikipedia.org/wiki/Fermat_point#:~:text=In%20Euclidean%20geometry%2C%20the%20Fermat,median%20of%20the%20three%20vertices.
charliepatrick Joined: Jun 17, 2011
• Posts: 2933
October 26th, 2023 at 10:05:34 AM permalink
The easiest way to think of it is to consider a flat table with three holes drilled. Then a ring with three pieces of string attached, such that a string goes through each hole, each string having an equal weight hung from it.
For the ring to be in equilibrium the net force must be zero. Therefore thinking along one of the lines, the forces from either side must be equal. Since the weights are equal, the angles must be. Use this for all three lines shows they must be at 120 (assuming the interior angles of the "triangle" are less than 120.) (Remember it's all about the potential energy, so this means the weights being as low as possible, which means the total length of string above the table.)
I did do a mathematical solution ages ago, probably based on the fact that any deviation from 120 will increase the length of one string more than the other one decreases, but if you can visualise the above it is possibly clearer to see.
chevy Joined: Apr 15, 2011
• Posts: 138
October 26th, 2023 at 10:39:59 AM permalink
Quote: charliepatrick

The easiest way to think of it is to consider a flat table with three holes drilled. Then a ring with three pieces of string attached, such that a string goes through each hole, each string having an equal weight hung from it.
For the ring to be in equilibrium the net force must be zero. Therefore thinking along one of the lines, the forces from either side must be equal. Since the weights are equal, the angles must be. Use this for all three lines shows they must be at 120 (assuming the interior angles of the "triangle" are less than 120.) (Remember it's all about the potential energy, so this means the weights being as low as possible, which means the total length of string above the table.)
I did do a mathematical solution ages ago, probably based on the fact that any deviation from 120 will increase the length of one string more than the other one decreases, but if you can visualise the above it is possibly clearer to see.

Very nice....Thanks
MichaelBluejay Joined: Sep 17, 2010
• Posts: 1589
October 26th, 2023 at 11:36:08 AM permalink
Why did the obtuse angle jump in the pool?
Presidential Election polls and odds: https://2605.me/p
unJon Joined: Jul 1, 2018
• Posts: 4344
Thanks for this post from:  October 26th, 2023 at 11:43:54 AM permalink
Quote: MichaelBluejay

Why did the obtuse angle jump in the pool? The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
Gialmere Joined: Nov 26, 2018
• Posts: 2801
Thanks for this post from: October 26th, 2023 at 7:26:53 PM permalink
Quote: chevy

If D = length of bridge
They run at speed (4/7)*D / T
Running other way a distance (3/7)*D takes (3/4)*T

If just barely making it to safety either way is required, then the train is at the start of the bridge at t=(3/4)*T and end of bridge at time t=T. It travels the length of the bridge in (1/4)*T.

So if train starting at distance of 4D (from the end of bridge or 3D from start of bridge) they should just barely jump to safety

Quote: Wizard

I get the train is 3/4 the length of the bridge away from the entrance to it.

If the question is asking the distance from the boys, that would then be 3/4 + 3/7 = 33/28 the length of the bridge.

Correct!!

Good show.
------------------------------------

Once I saw this guy on a bridge about to jump. I said, "Don't do it!" He said, "Nobody loves me."
I said, "God loves you. Do you believe in God?"

He said, "Yes." I said, "Are you a Christian or a Jew?" He said, "A Christian." I said, "Me, too! Protestant or Catholic?" He said, "Protestant." I said, "Me, too! What franchise?" He said, "Baptist." I said, "Me, too! Northern Baptist or Southern Baptist?" He said, "Northern Baptist." I said, "Me, too! Northern Conservative Baptist or Northern Liberal Baptist?"

He said, "Northern Conservative Baptist." I said, "Me, too! Northern Conservative Baptist Great Lakes Region, or Northern Conservative Baptist Eastern Region?" He said, "Northern Conservative Baptist Great Lakes Region." I said, "Me, too!"

Northern Conservative�Baptist Great Lakes Region Council of 1879, or Northern Conservative Baptist Great Lakes Region Council of 1912?" He said, "Northern Conservative Baptist Great Lakes Region Council of 1912." I said, "Die, heretic!" And I pushed him over.

Have you tried 22 tonight? I said 22.
Gialmere Joined: Nov 26, 2018
• Posts: 2801
Thanks for this post from: October 27th, 2023 at 7:05:43 AM permalink The numerical expression above is actually the mathematical representation of a limerick. Translate the numbers and symbols into the English words they commonly represent, and the expression will scan perfectly into the rhythm of a limerick. There are several possibilities for translating so try being fussy to make it work.

Can you translate it?
Have you tried 22 tonight? I said 22.
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 27th, 2023 at 7:18:10 AM permalink

I'm not saying you need to be a Baptist to get into heaven ... Buy why take any chances?
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 27th, 2023 at 7:25:44 AM permalink
Quote: chevy

Some googling suggests It seems to be a variant of what is called the Fermat Point. (I will now think of it as the Fermat-CharliePatrick point)

The Wikipedia page does indeed say 120-degree angles are formed between the center point and the three corners of a triangle.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
ThatDonGuy Joined: Jun 22, 2011
• Posts: 5947
Thanks for this post from:  October 27th, 2023 at 7:50:06 AM permalink
Quote: Gialmere The numerical expression above is actually the mathematical representation of a limerick. Translate the numbers and symbols into the English words they commonly represent, and the expression will scan perfectly into the rhythm of a limerick. There are several possibilities for translating so try being fussy to make it work.

Can you translate it?

It's in Lytton's Problematical Recreations:

A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Is nine squared and not a bit more

GM Joined: Jun 16, 2021
• Posts: 48
Thanks for this post from: October 27th, 2023 at 4:18:35 PM permalink
The train question reminds me of a recent puzzle on CarTalk.
Last edited by: GM on Oct 27, 2023
Gialmere Joined: Nov 26, 2018
• Posts: 2801
October 27th, 2023 at 6:51:39 PM permalink
Quote: ThatDonGuy

Quote: Gialmere The numerical expression above is actually the mathematical representation of a limerick. Translate the numbers and symbols into the English words they commonly represent, and the expression will scan perfectly into the rhythm of a limerick. There are several possibilities for translating so try being fussy to make it work.

Can you translate it?

It's in Lytton's Problematical Recreations:

A dozen, a gross, and a score
Plus three times the square root of four
Divided by seven
Plus five times eleven
Is nine squared and not a bit more

Correct!!

Very good.
-----------------------------------------

A mathematician confided
That the M"obius band is one-sided
And you'll get quite a laugh
If you cut one in half
'Cause it stays in one piece when divided.
----------

'Tis a favorite project of mine
A new value of pi to assign.
I would fix it at 3
For it's simpler, you see,
Than 3 point 1 4 1 5 9
----------

Integral z-squared dz
from 1 to the cube root of 3
times the cosine
of three pi over 9
equals log of the cube root of 'e'.

(And it's correct, too)
Have you tried 22 tonight? I said 22.
DogHand Joined: Sep 24, 2011
• Posts: 1314
Thanks for this post from: October 28th, 2023 at 12:09:56 AM permalink
Quote: Gialmere

<snip>

Very good.
-----------------------------------------

A mathematician confided
That the M"obius band is one-sided
And you'll get quite a laugh
If you cut one in half
'Cause it stays in one piece when divided.
----------

'Tis a favorite project of mine
A new value of pi to assign.
I would fix it at 3
For it's simpler, you see,
Than 3 point 1 4 1 5 9
----------

Integral z-squared dz
from 1 to the cube root of 3
times the cosine
of three pi over 9
equals log of the cube root of 'e'.

(And it's correct, too)

Gialmere,

I remember that first limerick somewhat differently:

A mathematician confided,
That a Möbius strip is one-sided,
You'll get quite a laugh,
If you cut it in half,
'Cause it's twice as long when divided.

A burlesque dancer, a pip,
Named Virginia, could peel in a zip,
And died of constriction,
Attempting a Möbius strip.

Another favorite:

Equations, when spoken aloud,
To children in class sound profound,
But you'll get quite a stare,
When you say "Pi r square",
For they all know that Mom's pies are round.

Dog Hand
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 28th, 2023 at 7:39:15 AM permalink
Quote: chevy

Still can't get calculus to work, but using @charliepatrick's 120 degree theorem (maybe he can chime in on why)....

then just use geometry....calling intersection F

AF is 30 degrees above horizontal.........(2-y)/x = tan(30) = 1/sqrt(3)
CF is 30 degrees above horizontal.........(3-y)/(5-x) = tan (30) = 1/sqrt(3)

Solving these two equations for x,y....making several algebra mistakes in the process, I get

setting the two equal yields x=5(y-2)/(2y-5)......mention it as it matches the local min in x dir from calc)
plugging in to AF equation gives

y=(5/2) * (1 - 1/sqrt(3)) = 1.05662432703

Plugging back in gives

x=(5/2) * (1-sqrt(3) / 5) = 1.63397459622

I independently had the same idea.

A 30-60-90 triangle has sides proportional to 1, sqrt(3) and 2.

The equations of the two pipes would be:

From left house: y = -x/sqrt(3) + 2
From right house: y = +x/(sqrt(3) + (9-5*sqrt(3))/3

Solving for x gives us x = (5-sqrt(3))/2 =~ 1.633975
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
ThatDonGuy Joined: Jun 22, 2011
• Posts: 5947
October 28th, 2023 at 11:40:42 AM permalink
Okay, back to the "easy" math puzzles...

Express 1 / (x^8 - 1) as the sum of five fractions of the form (a / (x + b)) and/or (ax + b) / (x^2 + cx + d), where all a, b, c, d are real numbers.

x^4 + 1 = (x^2 + sqrt(2) x + 1) (x^2 - sqrt(2) x + 1)

charliepatrick Joined: Jun 17, 2011
• Posts: 2933
October 28th, 2023 at 1:54:14 PM permalink
Quote: Wizard

To follow up on the water pipe problem...This results in total pipe length of 6.830127019. Can anyone do better?...

I agree with this - it's 5/2 (1 + SQRT(3) ). Construct lines down from A and B so they meet at 120 degrees, and extend these to the river. Construct other lines as shown.
BH=3, so BD=6 (D is beyond E), so DH=3*SQRT(3) (about 5.196).
Note AJ=AK so BJ+AJ=BK. This is therefore 6 * 5 / (3 SQRT(3) = 10/SQRT(3).
So let AL=x and so JF=2-x. Similarly BM=1+x giving BJ=2*(1+x).
Using BK=2+4x, this gives 4x=10/SQRT(3)-2 or x=5/(2 SQRT(3)) - 1/2.

The total required is 2 + 2x +2x+2 - x = 4+3x = 4 + 3(above) = 4-3/2+5*SQRT(3)/2 = 5/2(1+SQRT(3))
This is about 6.830127 and as BK was 5.773503 gives the y-coordinate as about 1.056624.
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 28th, 2023 at 2:44:50 PM permalink
Limericks have been split off to the Limirick Thread. Enjoy!
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
Wizard Joined: Oct 14, 2009
• Posts: 25961
October 29th, 2023 at 2:22:30 PM permalink
If I may get back to the pipe puzzle, we already agreed on the location of the Fermat point. That said, it's easy Pythagorean application to get a sum of the pipes of 6.830127. Any agreements or disagreements?
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
charliepatrick Joined: Jun 17, 2011
• Posts: 2933
Thanks for this post from: October 29th, 2023 at 2:28:12 PM permalink
^ Yes I used the Pythagorean theorem and that where the triangle is 90-60-30, the sides are 1:SQRT(3):2. Also there were similar triangles, so the lengths of other sides were proportional to the known sides.

What was interesting is I think you get the same result if both A and B were 2.5 away from the river, so I'm guessing this is where the 5/2 comes from.

edit Further thoughts, an easier approach. If you consider two squares, size SQRT(3), you get the lengths would be 2+2+(SQRT(3)-1) = 3 + SQRT(3). So if it had been unit squares it would have been SQRT(3)+1. In our puzzle it is 5/2 sides, hence 5/2 (SQRT(3)+1). Last edited by: charliepatrick on Oct 29, 2023
Gialmere Joined: Nov 26, 2018
• Posts: 2801
November 1st, 2023 at 7:00:03 AM permalink In a non sequitur, today is both Hump Day and the middle of the Dia de los Muertos festivities. Here are three puzzles... A nasty number is a positive integer having at least four different factors such that the difference between one pair of factors equals the sum of another pair of factors. For instance,

6 is nasty because 6 � 1 = 2 + 3
24 is nasty because 12 � 2 = 4 + 6

30 is nasty because 15 � 2 = 10 + 3.

Find the other six nasty numbers that are less than 180. What is the closest planet to Earth on average? Up in a forest fire lookout tower in California�s Sierra Nevada, two watchers need supplies. Three mules, led by a ranch hand, can carry supplies up to them.

The mules are named Geranium, Delphinium, and Ammonium. Geranium can carry 200 pounds, Delphinium can carry 190 pounds, and Ammonium can carry 180 pounds.

What supplies should each mule carry so that everything fits?

Drinking water 28 lb
Ice 35 lb
Kitty litter 42 lb
Camera Equipment 44 lb
Backup Telescope 48 lb

Batteries 61 lb
Detective Novels 63 lb
Window Glass Cleaner 77 lb
Ping Pong Table 84 lb
Mountain Dew 88 lb Have you tried 22 tonight? I said 22.
unJon Joined: Jul 1, 2018
• Posts: 4344
Thanks for this post from: November 1st, 2023 at 7:28:19 AM permalink
Mercury. I�ve always liked this question. Venus and Mars would seem to be correct until you realize they are very far away for half the time or so that they are across the Sun from Earth.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
charliepatrick Joined: Jun 17, 2011
• Posts: 2933
Thanks for this post from: November 1st, 2023 at 7:44:07 AM permalink
The trick seems to be to realise the total of the numbers means there's no slack. Hence the odd numbers have to be divided 2-2-0 or 4-0-0. Since all four add up to more than 200, it's 2-2-0. Thus just work through all the permutations of two pairs.
Just work through using 61 with each of the other odd numbers in turn...
61 63 (124) 28 48 = 200
35 77 (112) 88 = 200 (no good as already got a 200)
61 35 (96) 84 = 180
63 77 (140) now impossible
61 77 (138) 42 = 180
35 63 (98) 42 48 = 190
leaves 28 84 88 = 200.
charliepatrick Joined: Jun 17, 2011
• Posts: 2933
Thanks for this post from: November 1st, 2023 at 7:51:51 AM permalink
Assuming all the planets have circular motion it's easier to think of the Earth and Sun not moving and look at how the other planets go round the sun (rather like being on a clock at the "12" and watching the tip of an hour hand go round). In this case consider the effects of having longer or shorter hour hands.

At three o'clock and nine o'click the distance to the "12" is least for the shortest hour hand. At 12 o'clock and 6 o'clock (combined) it's the same. At times imbetween, the total distance of (say) 1 o'clock and 5 o'clock is least with the smallest hands.
Thus the average is smallest for the smallest hour hand.
unJon Joined: Jul 1, 2018
• Posts: 4344
Thanks for this post from: November 1st, 2023 at 7:54:18 AM permalink
I thought this would be trickier:

Drinking Water + Ping Pong Table + Mountain Dew = 200 lbs for Geranium.

Ice + Camera Equipment + Backup Telescope + Detective Novels = 190 lbs for Delphinium.

Kitty Litter + Batteries + Window Glass Cleaner = 180 lbs for Ammonium.
The race is not always to the swift, nor the battle to the strong; but that is the way to bet.
charliepatrick Joined: Jun 17, 2011
• Posts: 2933
November 1st, 2023 at 8:01:27 AM permalink
Nasty numbers - just checking you meant for every pair of factors of the number, their difference will always have another (?) two factors that add up to that difference. Otherwise any multiple of 6 would have the property.
Gialmere Joined: Nov 26, 2018
• Posts: 2801
Thanks for this post from: November 1st, 2023 at 8:32:28 AM permalink
It's any two pair. So 18 is a multiple of 6 but not nasty since none of the three factor pairs add up to the difference nor subtract to the sum of another pair.

Also, I see the question should read "less than or equal to 180".
Last edited by: Gialmere on Nov 1, 2023
Have you tried 22 tonight? I said 22.
Wizard Joined: Oct 14, 2009
• Posts: 25961
Thanks for this post from: November 1st, 2023 at 11:45:19 AM permalink
Mercury
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
Gialmere Joined: Nov 26, 2018
• Posts: 2801
November 1st, 2023 at 8:32:38 PM permalink
Quote: unJon

Mercury. I�ve always liked this question. Venus and Mars would seem to be correct until you realize they are very far away for half the time or so that they are across the Sun from Earth.

Quote: charliepatrick

The trick seems to be to realise the total of the numbers means there's no slack. Hence the odd numbers have to be divided 2-2-0 or 4-0-0. Since all four add up to more than 200, it's 2-2-0. Thus just work through all the permutations of two pairs.
Just work through using 61 with each of the other odd numbers in turn...
61 63 (124) 28 48 = 200
35 77 (112) 88 = 200 (no good as already got a 200)
61 35 (96) 84 = 180
63 77 (140) now impossible
61 77 (138) 42 = 180
35 63 (98) 42 48 = 190
leaves 28 84 88 = 200.

Quote: charliepatrick

Assuming all the planets have circular motion it's easier to think of the Earth and Sun not moving and look at how the other planets go round the sun (rather like being on a clock at the "12" and watching the tip of an hour hand go round). In this case consider the effects of having longer or shorter hour hands.

At three o'clock and nine o'click the distance to the "12" is least for the shortest hour hand. At 12 o'clock and 6 o'clock (combined) it's the same. At times imbetween, the total distance of (say) 1 o'clock and 5 o'clock is least with the smallest hands.
Thus the average is smallest for the smallest hour hand.

Quote: unJon

I thought this would be trickier:

Drinking Water + Ping Pong Table + Mountain Dew = 200 lbs for Geranium.

Ice + Camera Equipment + Backup Telescope + Detective Novels = 190 lbs for Delphinium.

Kitty Litter + Batteries + Window Glass Cleaner = 180 lbs for Ammonium.

Quote: Wizard

Mercury

Correct!!

Well done.

Admittedly this one was for solvers with time on their hands...

54 is nasty because 18 � 3 = 6 + 9
60 is nasty because 20 � 3 = 12 + 5
96 is nasty because 24 � 4 = 12 + 8
120 is nasty because 30 � 4 = 20 + 6
150 is nasty because 30 � 5 = 10 + 15
180 is nasty because 45 � 4 = 36 + 5

I have no idea why "nasty" is used to describe these numbers.

Yes Mercury. In fact, because of its central location and Kepler's 3rd law, Mercury is (when averaged over 4.5 billion years) the closest planet to every planet in the solar system.

This was another "just fiddle with the numbers" type of puzzle although, I confess, it's made me spend time wondering about the best way to rig a ping pong table onto a mule.

------------------------------------------

No joke today. Instead here's a link to an article at "Physics Today" discussing calculations and computer data on Mercury's proximity to other planets.

Venus is not Earth�s closest neighbor

The funny part is the comments at the end. If you think a gambling forum is filled with piss and vinegar, you should see astronomers going at it.
--------------------------------------------

Still in play but got lost amongst the limericks...

Quote: ThatDonGuy

Okay, back to the "easy" math puzzles...

Express 1 / (x^8 - 1) as the sum of five fractions of the form (a / (x + b)) and/or (ax + b) / (x^2 + cx + d), where all a, b, c, d are real numbers.

x^4 + 1 = (x^2 + sqrt(2) x + 1) (x^2 - sqrt(2) x + 1)

Have you tried 22 tonight? I said 22.
ThatDonGuy Joined: Jun 22, 2011
• Posts: 5947
November 8th, 2023 at 4:12:00 PM permalink
Quote: Gialmere

Still in play but got lost amongst the limericks...

Quote: ThatDonGuy

Okay, back to the "easy" math puzzles...

Express 1 / (x^8 - 1) as the sum of five fractions of the form (a / (x + b)) and/or (ax + b) / (x^2 + cx + d), where all a, b, c, d are real numbers.

x^4 + 1 = (x^2 + sqrt(2) x + 1) (x^2 - sqrt(2) x + 1)

Apparently, nobody is interested, so here is the answer

(x^2 + sqrt(2) x + 1)(x^2 - sqrt(2) x + 1)(x^2 + 1)(x + 1)(x - 1)

Let q = sqrt(2):

(ax + b) (x^2 - q x + 1)(x^4 - 1)
(cx + d) (x^2 + q x + 1)(x^4 - 1)
(ex + f) (x^4 + 1)(x^2 - 1)
g (x^4 + 1)(x^2 + 1)(x - 1)
h (x^4 + 1)(x^2 + 1)(x + 1)

a (x^7 - q x^6 + x^5 - x^3 + q x^2 - x)
b (x^6 - q x^5 + x^4 - x^2 + q x - 1)
c (x^7 + q x^6 + x^5 - x^3 - q x^2 - x)
d (x^6 + q x^5 + x^4 - x^2 - q x - 1)
e (x^7 - x^5 + x^3 - x)
f (x^6 - x^4 + x^2 - 1)
g (x^7 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1)
h (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)

a (x^7 - q x^6 + x^5 - x^3 + q x^2 - x)
b ( x^6 - q x^5 + x^4 - x^2 + q x - 1)
c (x^7 + q x^6 + x^5 - x^3 - q x^2 - x)
d ( x^6 + q x^5 + x^4 - x^2 - q x - 1)
e (x^7 - x^5 + x^3 - x)
f ( x^6 - x^4 + x^2 - 1)
g (x^7 - x^6 + x^5 - x^4 + x^3 - x^2 + x - 1)
h (x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)

 a + c + e + g + h = 0
 -q a + b + q c + d + f - g + h = 0
 a - q b + c + q d - e + g + h = 0
 b + d - f - g + h = 0
 -a - c + e + g + h = 0
 q a - b - q c - d + f - g + h = 0
 -a + q b - c - q d - e + g + h = 0
 -b - d - f - g + h = 1

 b + d - f - g + h = 0
 -b - d - f - g + h = 1
[4-0] 2b + 2d = -1 -> d = -1/2 - b
[4+0] -2f -2g + 2h = 1

 a + c + e + g + h = 0
 -a - c + e + g + h = 0
[7-3] 2a + 2c = 0 -> c = -a
[7+3] 2e + 2g + 2h = 0 -> e = -g - h

 a - q b + c + q d - e + g + h = 0
 -a + q b - c - q d - e + g + h = 0
[5+1] -2e + 2g + 2h = 0 -> e = g + h
By [5+1] and [7+3], e = 0 -> h = -g -> f = 2g
Substitute into [4+0]: -4g - 2g - 2g = 1 -> g = -1/8 -> h = 1/8, f = -1/4

 -q a + b + q c + d + f - g + h = 0
 q a - b - q c - d + f - g + h = 0
[2-6] 2q a - 2 b - 2q c - 2 d = 0

Substitute for e, f, g, h:
 c = -a
 -q a + b + q c + d = 0
 a - q b + c + q d = 0
 b + d = -1/2 -> d = -1/2 - b
{ is the negative of ;  is the negative of ;  is the negative of ;  is the negative of }

Substitute c = -a in :
- q b + q d = 0 -> d = b = -1/2 - b -> b = -1/4 -> d = -1/4
Substitue for b and d (as well as c = -a) in :
a = b / q = -1 / (4q) -> c = 1 / (4q)

The five fractions are:
(-sqrt(2)/8 x - 1/4) / (x^2 + sqrt(2) x + 1)
(sqrt(2)/8 x - 1/4) / (x^2 - sqrt(2) x + 1)
(-1/4) / (x^2 + 1)
(-1/8) / (x + 1)
(1/8) / (x - 1)

Wizard Joined: Oct 14, 2009
• Posts: 25961
November 16th, 2023 at 5:43:14 PM permalink
Getting back to the closest planet puzzle, here are my calculations. All distances are in Astronomical Units (AU).

Planet AU Cosest distance Furthest distance Average distance
Mercury 0.39 0.61 1.39 1.038401104
Venus 0.72 0.28 1.72 1.134493617
Earth 2 1 0 2 1.273231464
Mars 1.52 0.52 2.52 1.689508849
Jupiter 5.2 4.2 6.2 5.24818909
Saturn 9.54 8.54 10.54 9.566223496
Uranus 19.2 18.2 20.2 19.21302304
Neptune 30.06 29.06 31.06 30.06831728

As for the Earth 2 row, I am referring to a hypothetical second earth that has the same orbit, but different speed. Yes, I know that violates one of Kepler's laws, so take it as hypothetical only. Interesting that Mercury would be closer than another Earth.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
ChesterDog Joined: Jul 26, 2010
• Posts: 1369
November 16th, 2023 at 6:19:32 PM permalink
Quote: Wizard

Getting back to the closest planet puzzle, here are my calculations. All distances are in Astronomical Units (AU).

Planet AU Cosest distance Furthest distance Average distance
Mercury 0.39 0.61 1.39 1.038401104
Venus 0.72 0.28 1.72 1.134493617
Earth 2 1 0 2 1.273231464
Mars 1.52 0.52 2.52 1.689508849
Jupiter 5.2 4.2 6.2 5.24818909
Saturn 9.54 8.54 10.54 9.566223496
Uranus 19.2 18.2 20.2 19.21302304
Neptune 30.06 29.06 31.06 30.06831728

As for the Earth 2 row, I am referring to a hypothetical second earth that has the same orbit, but different speed. Yes, I know that violates one of Kepler's laws, so take it as hypothetical only. Interesting that Mercury would be closer than another Earth.

I got the same answer for the 2nd earth, which is 4 / pi. And I had the same problem you had about a planet traveling in earth's orbit with a different speed. Also, the planets would have trouble passing each other.
Wizard Joined: Oct 14, 2009
• Posts: 25961
November 16th, 2023 at 7:02:29 PM permalink
Quote: ChesterDog

I got the same answer for the 2nd earth, which is 4 / pi. And I had the same problem you had about a planet traveling in earth's orbit with a different speed. Also, the planets would have trouble passing each other.

Interesting about the 4/pi. That must have been some messy integration.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
ChesterDog Joined: Jul 26, 2010
• Posts: 1369
November 16th, 2023 at 7:40:28 PM permalink
Quote: Wizard

Quote: ChesterDog

I got the same answer for the 2nd earth, which is 4 / pi. And I had the same problem you had about a planet traveling in earth's orbit with a different speed. Also, the planets would have trouble passing each other.

Interesting about the 4/pi. That must have been some messy integration.

I thought it would be a messy integration (integral from x = 0 to x = pi of sqrt( 2 - 2 cos x ) dx / pi ), so I had WolframAlpha do it.
But just now, I see that the half-angle formulas make it easy.

Your other planet integrations must have been messy.
Wizard Joined: Oct 14, 2009
• Posts: 25961
November 20th, 2023 at 9:38:28 AM permalink
Quote: ChesterDog

Your other planet integrations must have been messy.

I was lazy and didn't even try. I used Excel. I put earth at point (1,0) and the other planet at every point around the sun from 0 to 359 degrees, in groups of one degree. It was simple trigonometry to get the coordinates of the other planet and then the Pythagorean formula to find the distance.
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
Wizard Joined: Oct 14, 2009
• Posts: 25961
November 20th, 2023 at 9:40:54 AM permalink
I may have asked this one before, but here goes:

You are allowed a total of one square foot of metal to make a can, including the top and bottom. What is the radius of the can that maximizes the volume? Last edited by: Wizard on Nov 20, 2023
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
TinMan Joined: Nov 17, 2009
• Posts: 441
November 20th, 2023 at 10:10:11 AM permalink
There was a question earlier that referenced the circumference of the earth. This is my favorite similar puzzle (I didn�t create it; from YouTube):

A band is stretched taut over the circumference of the earth. Assume the earths circumference is static. There�s no room between the band and the earth. Treat the earth like a solid spherical object so no issues with oceans/mountains/etc. suddenly, the band increases in size by 1 foot. What does that do to the space between the band and the earth? can you slip a playing card between them? Your hand? Will the difference be indistinguishable from before?
If anyone gives you 10,000 to 1 on anything, you take it. If John Mellencamp ever wins an Oscar, I am going to be a very rich dude.
TigerWu Joined: May 23, 2016
• Posts: 5386
November 20th, 2023 at 10:14:42 AM permalink
Quote: Wizard

I may have asked this one before, but here goes:

You are allowed 1 square foot of metal to make a can, including the top and bottom. What is the radius of the can that maximizes the volume?

Roll up the metal and join the ends, giving a circumference of 12 inches. Pinch the ends together like this. That makes a can that maximizes volume without wasting metal by cutting.

If you're talking about a traditional can like the one in the picture, I have no idea.
Wizard Joined: Oct 14, 2009
• Posts: 25961
November 20th, 2023 at 11:38:02 AM permalink
Quote: TigerWu

Roll up the metal and join the ends, giving a circumference of 12 inches. Pinch the ends together like this. That makes a can that maximizes volume without wasting metal by cutting.

If you're talking about a traditional can like the one in the picture, I have no idea.

To make it clear, you can form the one square foot of metal any way you like. In other words, you can cut it, melt it, and shape it however you wish. The goal is to make a normal shaped can.
Last edited by: Wizard on Nov 20, 2023
�Extraordinary claims require extraordinary evidence.� -- Carl Sagan
chevy Joined: Apr 15, 2011
• Posts: 138
November 20th, 2023 at 1:25:46 PM permalink
Quote: Wizard

I may have asked this one before, but here goes:

You are allowed 1 square foot of metal to make a can, including the top and bottom. What is the radius of the can that maximizes the volume? Or maybe rusty calculus...

Area
2*pi*r^2 + 2*pi*r*h=1
h=1/(2*pi*r) - r

Volume
V=pi * r^2 * h = r/2 - pi*r^3

dV/dr = 1/2 - 3*pi*r^2 = 0

r= 1/sqrt(6*pi)

or some typo permutation of that....

Wizard Joined: Oct 14, 2009
• Posts: 25961
November 20th, 2023 at 2:46:05 PM permalink
Quote: chevy

Or maybe rusty calculus...

Area
2*pi*r^2 + 2*pi*r*h=1
h=1/(2*pi*r) - r

Volume
V=pi * r^2 * h = r/2 - pi*r^3

dV/dr = 1/2 - 3*pi*r^2 = 0

r= 1/sqrt(6*pi)

or some typo permutation of that....