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NY Times math problem

K

KBH

Okay the circular pond has North and East coordinates at the radius point
Oh...that KBH Code will run in Delphi Console Mode.
Click to expand...


The following is a file for KBH ScratchPlot. Copy to a text file and save
the file as "Name.plt" with the quotation marks and then the file will run
in ScratchPlot. And of course the file should contain text...not the html
being picked up here.

I didn't convert the entire coordinate file to a ScratchPlot file...I just
entered 15 points for the rabbit as lines. Then I laid out 15 points for the
agent from the radius point with the Fan and Azimuth features. Then I added
15 points to the rabbit line so that the points of the agent could be
matched up to the points of the rabbit. For instance both the agent and the
rabbit were at 220 degrees clockwise azimuth but at different times. I could
have added a full circle but that would make the file very large...

663&137@F+bkN0E0
41&759@S+bkN20E20
0
352&448@P+bkN10E10
353&447@P+rdN9.9564E9.9992
385&442@L+rdN8.9295E9.8114
416&429@L+rdN7.9231E9.3937
445&412@L+rdN6.9807E8.8454
473&392@L+rdN6.0868E8.2205
500&372@L+rdN5.2194E7.5591
527&351@L+rdN4.3545E6.8944
555&331@L+rdN3.4713E6.2543
583&312@L+rdN2.5578E5.6584
595&306@L+rdN2.1834E5.4343
600&302@L+rdN1.9944E5.3254
606&299@L+rdN1.8042E5.2184
612&296@L+rdN1.6130E5.1134
618&292@L+rdN1.4208E5.0102
621&291@L+rdN1.3437E4.9694
352&448@P+bkN10E10
352&448@P+bkN10E10
70&579@P+bkN19.0631E14.2262
352&448@+bkN10E10
41&453@P+bkN19.9985E10.1745
352&448@+bkN10E10
152&686@P+bkN16.4279E17.6604
352&448@+bkN10E10
271&748@P+bkN12.5882E19.6593
352&448@+bkN10E10
406&754@P+bkN8.2635E19.8481
352&448@+bkN10E10
530&702@P+bkN4.2642E18.1915
352&448@+bkN10E10
621&603@P+bkN1.3397E15.0000
352&448@+bkN10E10
661&475@P+bkN0.0381E10.8716
352&448@+bkN10E10
644&341@P+bkN0.6031E6.5798
352&448@+bkN10E10
621&292@P+bkN1.3397E5.0000
352&448@+bkN10E10
606&269@P+bkN1.8085E4.2642
352&448@+bkN10E10
590&248@P+bkN2.3396E3.5721
352&448@+bkN10E10
571&228@P+bkN2.9289E2.9289
352&448@+bkN10E10
551&209@P+bkN3.5721E2.3396
352&448@+bkN10E10
543&202@P+bkN3.8434E2.1199
352&448@+bkN10E10
352&448@P+bkN10E10
41&453@P+prN19.9985E10.1745
352&448@+prN10E10
41&453@P+bkN19.9985E10.1745
352&448@+bkN10E10
385&442@P+bkN8.9295E9.8114
416&429@P+bkN7.9231E9.3937
445&412@P+bkN6.9807E8.8454
473&392@P+bkN6.0868E8.2205
500&372@P+bkN5.2194E7.5591
527&351@P+bkN4.3545E6.8944
555&331@P+bkN3.4713E6.2543
583&312@P+bkN2.5578E5.6584
595&306@P+bkN2.1834E5.4343
600&302@P+bkN1.9944E5.3254
606&299@P+bkN1.8042E5.2184
612&296@P+bkN1.6130E5.1134
618&292@P+bkN1.4208E5.0102
621&291@P+bkN1.3437E4.9694
 
K

KBH

CORRECTED CODE !

No...the plot didn't look right. And so in the previous code the coordinates
of the agent were based on a radius of 1 but should have been based on a
radius of 10.

And with this corrected code the Rabbit does not get out of the pond with
this systematic method ! But I'm sure the plot will make some nice
curves...which is what I was looking for.

Here is the corrected KBH code:

Var
ads, nag, eag, rba, rds, sec, rbn, rbe, nd, ed, dgr, dst: double;
flg: integer;


Procedure D;
Begin
If (flg = 1) Then
Begin
{Procedure E}
rba:= rba + Pi;
Exit;
End;
rba:= rba + (Pi * 2);
{The structure of procedures D, E, & F are from the KBH Survey Progam for
the HP35S}
End;

Procedure F;
Begin
If (flg = 1) Then Exit;
rba:= rba + Pi;
End;

begin
{KBH Code}
flg:= 0;
sec:= 1 / 3600;
rbn:= 10;
rbe:= 10;
rds:= 0;
dgr:= 1;

While (dgr < 360) Do
Begin
ads:= (dgr * Pi) / 180;
nag:= 10 + (Cos(ads) * 10);
eag:= 10 + (Sin(ads) * 10);
nd:= rbn - nag;
If (nd = 0) Then nd:= 0.000000000001;
ed:= rbe - eag;
If (ed >= 0) Then flg:= 1;
rba:= ArcTan(ed / nd);
If (rba < 0.000000000001) Then D Else F;
rbn:= rbn + (Cos(rba) * 0.043633231);
rbe:= rbe + (Sin(rba) * 0.043633231);
dst:= Sqrt(Sqr(10 - rbn) + Sqr(10 - rbe));
WriteLn(rbn:15:4, rbe:15:4, dgr:15:4, dst:15:4);
dgr:= dgr + 1;
End;

ReadLn;
end.

And that KBH Code will run in Delphi Console mode...

Then plot the coordinates in KBH Scratch Plot...
 
K

KBH

CORRECTED CODE !
No...the plot didn't look right. And so in the previous code the
coordinates of the agent were based on a radius of 1 but should have been
based on a radius of 10.

And with this corrected code the Rabbit does not get out of the pond with
this systematic method ! But I'm sure the plot will make some nice
curves...which is what I was looking for.

Here is the corrected KBH code:

Var
ads, nag, eag, rba, rds, sec, rbn, rbe, nd, ed, dgr, dst: double;
flg: integer;


Procedure D;
Begin
If (flg = 1) Then
Begin
{Procedure E}
rba:= rba + Pi;
Exit;
End;
rba:= rba + (Pi * 2);
{The structure of procedures D, E, & F are from the KBH Survey Progam for
the HP35S}
End;

Procedure F;
Begin
If (flg = 1) Then Exit;
rba:= rba + Pi;
End;

begin
{KBH Code}
flg:= 0;
sec:= 1 / 3600;
rbn:= 10;
rbe:= 10;
rds:= 0;
dgr:= 1;

While (dgr < 360) Do
Begin
ads:= (dgr * Pi) / 180;
nag:= 10 + (Cos(ads) * 10);
eag:= 10 + (Sin(ads) * 10);
nd:= rbn - nag;
If (nd = 0) Then nd:= 0.000000000001;
ed:= rbe - eag;
If (ed >= 0) Then flg:= 1;
rba:= ArcTan(ed / nd);
If (rba < 0.000000000001) Then D Else F;
rbn:= rbn + (Cos(rba) * 0.043633231);
rbe:= rbe + (Sin(rba) * 0.043633231);
dst:= Sqrt(Sqr(10 - rbn) + Sqr(10 - rbe));
WriteLn(rbn:15:4, rbe:15:4, dgr:15:4, dst:15:4);
dgr:= dgr + 1;
End;

ReadLn;
end.

And that KBH Code will run in Delphi Console mode...
Click to expand...

And the output of the KBH code is rabbit North coordinate, rabbit East
coordinate, clockwise azimuth of agent on circle, and rabbit distance from
circle radius of 10.
 
B

Baron

M Rath Inscribed thus:
mrdarrett wrote:

Missing a semicolon somewhere?


M Rath wrote:

No the compiler is not accepting "ads:= Pi / 2" as a test value to
develop formulas in a loop. Now the compiler might be saying "What's
the point of calculating the same result over and over in a loop ?"
but it did not accept "ads:= Pi/2" with the loop commented out. But in
both cases it accepts the numerical value...
Click to expand...

Just a guess, has Pi been defined before use ?
 
D

Dik T. Winter

Is the "agent" wading/swimming, or just walking around the perimeter
of the pond? What's each of their respective speeds? How did the rabbit
get to the middle of the pond in the first place? Was somebody using him
for Muxkie bait? ;-)
Click to expand...

The rabbit had been caught by poachers in an helicopter, but he knew to
escape.
 
D

Dik T. Winter

riverman wrote: ....

When the rabbit moved back towards the centre of the pool, the agents
would move back to positions oppoisite each other.

It appears that the rabbit cannot escape if there are two agents.
Click to expand...

Of course the rabbit can escape if the agents use a stupid strategy.
 
S

Sylvia Else

Dik said:
Of course the rabbit can escape if the agents use a stupid strategy.
Click to expand...

You mean like wandering off to get some donuts and coffee? I think it's
usual to assume in these puzzles that the participants will always use
the optimum strategy.

Sylvia.
 
R

Rob Johnson

Observation: I once chased a jackrabbit. I was on
a motorcycle, he was not. The rabbit got away.

Given his preternatural speed, the agent should fox
the rabbit into quitting the safety of the pond,
then give chase--he'll have the bunny in a few
strides.
Click to expand...

Who says that the land speed of the rabbit is the same as its water
speed? My assumption is that once it makes it to shore, the rabbit
has no difficulty in escaping. However, the problem only requires
that the rabbit makes it to shore without being captured.

Rob Johnson <[email protected]>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
 
J

James Arthur

Rob said:
Who says that the land speed of the rabbit is the same as its water
speed? My assumption is that once it makes it to shore, the rabbit
has no difficulty in escaping.
Click to expand...

I didn't assume that because the problem specifies each party's
speed. Otherwise I might've observed that people swim a lot
faster than rabbits, and the agent might have an easier time
with a running dive & a swim.

However, the problem only requires
that the rabbit makes it to shore without being captured.
Click to expand...

Yep. Two points.

Cheers,
James Arthur
 
E

ehsjr

Dave said:
Please give the greatest agent speed for which the rabbit can escape
using this strategy.

Dave
Click to expand...

C + x. Somewhere in the neighborhood of C the agent's perception
of the rabbit's position/speed/direction will be distorted.

Ed
 
R

Rob Johnson

http://groups.google.com/group/rec.puzzles/msg/050273ed70e4a9f6
Click to expand...

The rabbit moves at a rate of 1, the agent at a rate of r. The edge
of the lake is the unit circle centered at (0,0).

As has been discussed, the rabbit can keep the agent on the opposite
side of the center of the lake until the rabbit gets to a distance
of 1/r from the center of the lake. So we assume that the rabbit
starts at (-1/r,0) and the agent starts at (1,0). Then the agent
moves counterclockwise and the rabbit head straight down until it
reaches the unit circle after time t (if it headed further left, it
would meet the agent sooner, if it headed further right, it would
decrease its distance to the center of the lake). The positions at
time t are

rabbit: ( -1/r , -t )

agent: ( cos(rt) , sin(rt) )

To solve for the extreme case, we need to solve the equations

cos(rt) + 1/r = 0

sin(rt) + t = 0

Eliminating the sin and cos using Pythagoras' Theorem, we get

rt = sqrt(r^2 - 1)

Thus, we need to solve

r cos(sqrt(r^2 - 1)) + 1 = 0 [1]

while making sure that

sin(sqrt(r^2 - 1)) < 0 [2]

The first two solutions of [1] are

2.261826334114651437542573924426
4.603338848751700352556582029103

For the first solution, inequality [2] fails to hold, so the
solution we are interested in is the second, for which [2] is
satisfied.

Rob Johnson <[email protected]>
take out the trash before replying
to view any ASCII art, display article in a monospaced font
 
K

KBH

CORRECTED CODE !
And the output of the KBH code is rabbit North coordinate, rabbit East
coordinate, clockwise azimuth of agent on circle, and rabbit distance from
circle radius of 10.
Click to expand...

Okay...with the systematic method used here...the Rabbit does get out at
North and East coordinates of 10.5006 , 19.9976 where the radius point has
North and East coordinates of 10 , 10. And the rabbit gets out when the
agent is at a clockwise azimuth on the circle of 834 degrees !

Well...I have only plotted the first 360 degrees so far but here is the KBH
Scratch Plot file:

663&137@F+wtN0E0
41&759@S+wtN20E20
0
352&448@P+bkN10E10
353&447@L+rdN9.9564E9.9992
365&446@L+rdN9.5661E9.9588
378&443@L+rdN9.1452E9.8458
390&437@L+rdN8.7477E9.6669
402&430@L+rdN8.3826E9.4288
412&421@L+rdN8.0572E9.1387
421&410@L+rdN7.7774E8.8044
428&399@L+rdN7.5477E8.4338
433&386@L+rdN7.3716E8.0349
437&373@L+rdN7.2515E7.6158
439&360@L+rdN7.1886E7.1844
439&346@L+rdN7.1834E6.7484
437&333@L+rdN7.2357E6.3155
434&320@L+rdN7.3441E5.8932
429&307@L+rdN7.5067E5.4886
422&295@L+rdN7.7209E5.1089
414&285@L+rdN7.9834E4.7607
405&275@L+rdN8.2899E4.4506
394&267@L+rdN8.6357E4.1851
382&260@L+rdN9.0151E3.9703
369&255@L+rdN9.4215E3.8124
356&252@L+rdN9.8469E3.7173
343&251@L+rdN10.2819E3.6911
329&253@L+rdN10.7150E3.7398
316&257@L+rdN11.1311E3.8687
305&263@L+rdN11.5107E4.0819
295&273@L+rdN11.8285E4.3792
289&284@L+rdN12.00538E4.7515
285&297@L+rdN12.1525E5.1743
286&311@L+rdN12.1075E5.6063
292&323@L+rdN11.9231E5.9996
301&333@L+rdN11.6270E6.318
312&340@L+rdN11.2554E6.5445
325&344@L+rdN10.8406E6.6769
339&346@L+rdN10.4072E6.7212
325&347@L+rdN10.8414E6.7579
312&350@L+rdN11.2621E6.8715
352&448@P+bkN10E10
41&453@P+blN19.9985E10.1745
352&448@+blN10E10
45&502@P+blN19.8481E11.7365
352&448@+blN10E10
59&554@P+blN19.3969E13.4202
352&448@+blN10E10
82&603@P+blN18.6603E15.0000
352&448@+blN10E10
113&647@P+blN17.6604E16.4279
352&448@+blN10E10
152&686@P+blN16.4279E17.6604
352&448@+blN10E10
196&717@P+blN15.0000E18.6603
352&448@+blN10E10
245&740@P+blN13.4202E19.3969
352&448@+blN10E10
297&754@P+blN11.7365E19.8481
352&448@+blN10E10
352&759@P+blN10.0000E20.0000
352&448@+blN10E10
406&754@P+blN8.2635E19.8481
352&448@+blN10E10
458&740@P+blN6.5798E19.3969
352&448@+blN10E10
507&717@P+blN5.0000E18.6603
352&448@+blN10E10
551&686@P+blN3.5721E17.6604
352&448@+blN10E10
590&647@P+blN2.3396E16.4279
352&448@+blN10E10
621&603@P+blN1.3397E15.0000
352&448@+blN10E10
644&554@P+blN0.6031E13.4202
352&448@+blN10E10
658&502@P+blN0.1519E11.7365
352&448@+blN10E10
663&448@P+blN0.0000E10.0000
352&448@+blN10E10
658&393@P+blN0.1519E8.2635
352&448@+blN10E10
644&341@P+blN0.6031E6.5798
352&448@+blN10E10
621&292@P+blN1.3397E5.0000
352&448@+blN10E10
590&248@P+blN2.3396E3.5721
352&448@+blN10E10
551&209@P+blN3.5721E2.3396
352&448@+blN10E10
507&178@P+blN5.0000E1.3397
352&448@+blN10E10
458&155@P+blN6.5798E0.6031
352&448@+blN10E10
406&141@P+blN8.2635E0.1519
352&448@+blN10E10
352&137@P+blN10.0000E0.0000
352&448@+blN10E10
297&141@P+blN11.7365E0.1519
352&448@+blN10E10
245&155@P+blN13.4202E0.6031
352&448@+blN10E10
196&178@P+blN15.0000E1.3397
352&448@+blN10E10
152&209@P+blN16.4279E2.3396
352&448@+blN10E10
113&248@P+blN17.6604E3.5721
352&448@+blN10E10
82&292@P+blN18.6603E5.0000
352&448@+blN10E10
59&341@P+blN19.3969E6.5798
352&448@+blN10E10
45&393@P+blN19.8481E8.2635
352&448@+blN10E10
51&448@Agent->+blN19.65E10
352&463@Rabbit Begins Here+rdN10E10.5

Copy to a text file, save the text file as "Name.plt", and the file will run
in KBH Scratch Plot...
 
K

KBH

CORRECTED CODE !
Okay...with the systematic method used here...the Rabbit does get out at
North and East coordinates of 10.5006 , 19.9976 where the radius point has
North and East coordinates of 10 , 10. And the rabbit gets out when the
agent is at a clockwise azimuth on the circle of 834 degrees !

Well...I have only plotted the first 360 degrees so far but here is the
KBH Scratch Plot file:

663&137@F+wtN0E0
41&759@S+wtN20E20
0
352&448@P+bkN10E10
353&447@L+rdN9.9564E9.9992
365&446@L+rdN9.5661E9.9588
378&443@L+rdN9.1452E9.8458
390&437@L+rdN8.7477E9.6669
402&430@L+rdN8.3826E9.4288
412&421@L+rdN8.0572E9.1387
421&410@L+rdN7.7774E8.8044
428&399@L+rdN7.5477E8.4338
433&386@L+rdN7.3716E8.0349
437&373@L+rdN7.2515E7.6158
439&360@L+rdN7.1886E7.1844
439&346@L+rdN7.1834E6.7484
437&333@L+rdN7.2357E6.3155
434&320@L+rdN7.3441E5.8932
429&307@L+rdN7.5067E5.4886
422&295@L+rdN7.7209E5.1089
414&285@L+rdN7.9834E4.7607
405&275@L+rdN8.2899E4.4506
394&267@L+rdN8.6357E4.1851
382&260@L+rdN9.0151E3.9703
369&255@L+rdN9.4215E3.8124
356&252@L+rdN9.8469E3.7173
343&251@L+rdN10.2819E3.6911
329&253@L+rdN10.7150E3.7398
316&257@L+rdN11.1311E3.8687
305&263@L+rdN11.5107E4.0819
295&273@L+rdN11.8285E4.3792
289&284@L+rdN12.00538E4.7515
285&297@L+rdN12.1525E5.1743
286&311@L+rdN12.1075E5.6063
292&323@L+rdN11.9231E5.9996
301&333@L+rdN11.6270E6.318
312&340@L+rdN11.2554E6.5445
325&344@L+rdN10.8406E6.6769
339&346@L+rdN10.4072E6.7212
325&347@L+rdN10.8414E6.7579
312&350@L+rdN11.2621E6.8715
352&448@P+bkN10E10
41&453@P+blN19.9985E10.1745
352&448@+blN10E10
45&502@P+blN19.8481E11.7365
352&448@+blN10E10
59&554@P+blN19.3969E13.4202
352&448@+blN10E10
82&603@P+blN18.6603E15.0000
352&448@+blN10E10
113&647@P+blN17.6604E16.4279
352&448@+blN10E10
152&686@P+blN16.4279E17.6604
352&448@+blN10E10
196&717@P+blN15.0000E18.6603
352&448@+blN10E10
245&740@P+blN13.4202E19.3969
352&448@+blN10E10
297&754@P+blN11.7365E19.8481
352&448@+blN10E10
352&759@P+blN10.0000E20.0000
352&448@+blN10E10
406&754@P+blN8.2635E19.8481
352&448@+blN10E10
458&740@P+blN6.5798E19.3969
352&448@+blN10E10
507&717@P+blN5.0000E18.6603
352&448@+blN10E10
551&686@P+blN3.5721E17.6604
352&448@+blN10E10
590&647@P+blN2.3396E16.4279
352&448@+blN10E10
621&603@P+blN1.3397E15.0000
352&448@+blN10E10
644&554@P+blN0.6031E13.4202
352&448@+blN10E10
658&502@P+blN0.1519E11.7365
352&448@+blN10E10
663&448@P+blN0.0000E10.0000
352&448@+blN10E10
658&393@P+blN0.1519E8.2635
352&448@+blN10E10
644&341@P+blN0.6031E6.5798
352&448@+blN10E10
621&292@P+blN1.3397E5.0000
352&448@+blN10E10
590&248@P+blN2.3396E3.5721
352&448@+blN10E10
551&209@P+blN3.5721E2.3396
352&448@+blN10E10
507&178@P+blN5.0000E1.3397
352&448@+blN10E10
458&155@P+blN6.5798E0.6031
352&448@+blN10E10
406&141@P+blN8.2635E0.1519
352&448@+blN10E10
352&137@P+blN10.0000E0.0000
352&448@+blN10E10
297&141@P+blN11.7365E0.1519
352&448@+blN10E10
245&155@P+blN13.4202E0.6031
352&448@+blN10E10
196&178@P+blN15.0000E1.3397
352&448@+blN10E10
152&209@P+blN16.4279E2.3396
352&448@+blN10E10
113&248@P+blN17.6604E3.5721
352&448@+blN10E10
82&292@P+blN18.6603E5.0000
352&448@+blN10E10
59&341@P+blN19.3969E6.5798
352&448@+blN10E10
45&393@P+blN19.8481E8.2635
352&448@+blN10E10
51&448@Agent->+blN19.65E10
352&463@Rabbit Begins Here+rdN10E10.5
Click to expand...

Copy to a text file, save the text file as "Name.plt" with quotation marks,
and the file will run in KBH Scratch Plot...
 
K

KBH

CORRECTED CODE !
Copy to a text file, save the text file as "Name.plt" with quotation
marks, and the file will run in KBH Scratch Plot...
Click to expand...

Looking at the plot...it seems that the rabbit path should continue to
spiral westward rather than turn back northward and eastward at the sharp
point.

But I don't yet see why the coordinate output of the KBH Code would be
correct at 340 degrees but incorrect at 350 degrees. Why I get this worked
out it will probably require a new subject posting...
 
K

KBH

Okay the circular pond has North and East coordinates at the radius
The following is a file for KBH ScratchPlot. Copy to a text file and save
the file as "Name.plt" with the quotation marks and then the file will run
in ScratchPlot. And of course the file should contain text...not the html
being picked up here.

I didn't convert the entire coordinate file to a ScratchPlot file...I just
entered 15 points for the rabbit as lines. Then I laid out 15 points for
the agent from the radius point with the Fan and Azimuth features. Then I
added 15 points to the rabbit line so that the points of the agent could
be matched up to the points of the rabbit. For instance both the agent and
the rabbit were at 220 degrees clockwise azimuth but at different times. I
could have added a full circle but that would make the file very large...

663&137@F+bkN0E0
41&759@S+bkN20E20
0
352&448@P+bkN10E10
353&447@P+rdN9.9564E9.9992
385&442@L+rdN8.9295E9.8114
416&429@L+rdN7.9231E9.3937
445&412@L+rdN6.9807E8.8454
473&392@L+rdN6.0868E8.2205
500&372@L+rdN5.2194E7.5591
527&351@L+rdN4.3545E6.8944
555&331@L+rdN3.4713E6.2543
583&312@L+rdN2.5578E5.6584
595&306@L+rdN2.1834E5.4343
600&302@L+rdN1.9944E5.3254
606&299@L+rdN1.8042E5.2184
612&296@L+rdN1.6130E5.1134
618&292@L+rdN1.4208E5.0102
621&291@L+rdN1.3437E4.9694
352&448@P+bkN10E10
352&448@P+bkN10E10
70&579@P+bkN19.0631E14.2262
352&448@+bkN10E10
41&453@P+bkN19.9985E10.1745
352&448@+bkN10E10
152&686@P+bkN16.4279E17.6604
352&448@+bkN10E10
271&748@P+bkN12.5882E19.6593
352&448@+bkN10E10
406&754@P+bkN8.2635E19.8481
352&448@+bkN10E10
530&702@P+bkN4.2642E18.1915
352&448@+bkN10E10
621&603@P+bkN1.3397E15.0000
352&448@+bkN10E10
661&475@P+bkN0.0381E10.8716
352&448@+bkN10E10
644&341@P+bkN0.6031E6.5798
352&448@+bkN10E10
621&292@P+bkN1.3397E5.0000
352&448@+bkN10E10
606&269@P+bkN1.8085E4.2642
352&448@+bkN10E10
590&248@P+bkN2.3396E3.5721
352&448@+bkN10E10
571&228@P+bkN2.9289E2.9289
352&448@+bkN10E10
551&209@P+bkN3.5721E2.3396
352&448@+bkN10E10
543&202@P+bkN3.8434E2.1199
352&448@+bkN10E10
352&448@P+bkN10E10
41&453@P+prN19.9985E10.1745
352&448@+prN10E10
41&453@P+bkN19.9985E10.1745
352&448@+bkN10E10
385&442@P+bkN8.9295E9.8114
416&429@P+bkN7.9231E9.3937
445&412@P+bkN6.9807E8.8454
473&392@P+bkN6.0868E8.2205
500&372@P+bkN5.2194E7.5591
527&351@P+bkN4.3545E6.8944
555&331@P+bkN3.4713E6.2543
583&312@P+bkN2.5578E5.6584
595&306@P+bkN2.1834E5.4343
600&302@P+bkN1.9944E5.3254
606&299@P+bkN1.8042E5.2184
612&296@P+bkN1.6130E5.1134
618&292@P+bkN1.4208E5.0102
621&291@P+bkN1.3437E4.9694
Click to expand...

All the above is voided...the agent coordinates should have been based on a
circle radius of 10 instead of a circle radius of 1 .
 
K

KBH

The KBH code below is edited to reset the value of "flg" to zero at the end
of the loop.

And according to this systematic method...the rabbit gets out at North and
East coordinates of 8.1332 , 0.1612 with the agent at a clockwise azimuth of
535 degrees on the circle edge.

The KBH Scratch Plot file below is correct until the sharp point...but is
edited below to stop before the sharp point.
 
K

KBH

KBH Code:

{The rabbit at the center of a 10 radius circular pond and the chasing agent
is at the edge of the pond.}

Var
ads, nag, eag, rba, rbn, rbe, nd, ed, dgr, dst: double;
flg: integer;


Procedure D;
Begin
If (flg = 1) Then
Begin
{Procedure E}
rba:= rba + Pi;
Exit;
End;
rba:= rba + (Pi * 2);
{The structure of procedures D, E, & F are from the KBH Survey Progam for
the HP35S}
End;

Procedure F;
Begin
If (flg = 1) Then Exit;
rba:= rba + Pi;
End;

begin
{KBH Code}
flg:= 0;
rbn:= 10;
rbe:= 10;
dgr:= 1;

While (dgr < 361) Do {Change value for desired output range}
Begin
ads:= (dgr * Pi) / 180;
nag:= 10 + (Cos(ads) * 10);
eag:= 10 + (Sin(ads) * 10);
nd:= rbn - nag;
If (nd = 0) Then nd:= 0.000000000001;
ed:= rbe - eag;
If (ed >= 0) Then flg:= 1;
rba:= ArcTan(ed / nd);
If (rba < 0.000000000001) Then D Else F;
rbn:= rbn + (Cos(rba) * 0.043633231);
rbe:= rbe + (Sin(rba) * 0.043633231);
dst:= Sqrt(Sqr(10 - rbn) + Sqr(10 - rbe));
rba:= (rba / Pi) * 180;
WriteLn(rbn:15:4, rbe:15:4, dgr:15:4, dst:15:4);
dgr:= dgr + 1;
flg:= 0;
End;

ReadLn;
end.

And the above KBH code runs in Delphi Console Mode...

Now the rabbit gets out at North and East coordinates of 8.1332 , 0.1612
while the agent is at an azimuth of 535 degrees on the edge of the circular
pond and on his second trip around the pond. Of course this is a systematic
method only...

And here is the KBH Scratch Plot file for the rabbit path:

663&137@F+wtN0E0
41&759@S+wtN20E20
0
352&448@P+bkN10E10
353&447@L+rdN9.9564E9.9992
365&446@L+rdN9.5661E9.9588
378&443@L+rdN9.1452E9.8458
390&437@L+rdN8.7477E9.6669
402&430@L+rdN8.3826E9.4288
412&421@L+rdN8.0572E9.1387
421&410@L+rdN7.7774E8.8044
428&399@L+rdN7.5477E8.4338
433&386@L+rdN7.3716E8.0349
437&373@L+rdN7.2515E7.6158
439&360@L+rdN7.1886E7.1844
439&346@L+rdN7.1834E6.7484
437&333@L+rdN7.2357E6.3155
434&320@L+rdN7.3441E5.8932
429&307@L+rdN7.5067E5.4886
422&295@L+rdN7.7209E5.1089
414&285@L+rdN7.9834E4.7607
405&275@L+rdN8.2899E4.4506
394&267@L+rdN8.6357E4.1851
382&260@L+rdN9.0151E3.9703
369&255@L+rdN9.4215E3.8124
356&252@L+rdN9.8469E3.7173
343&251@L+rdN10.2819E3.6911
329&253@L+rdN10.7150E3.7398
316&257@L+rdN11.1311E3.8687
305&263@L+rdN11.5107E4.0819
295&273@L+rdN11.8285E4.3792
289&284@L+rdN12.00538E4.7515
285&297@L+rdN12.1525E5.1743
286&311@L+rdN12.1075E5.6063
292&323@L+rdN11.9231E5.9996
301&333@L+rdN11.6270E6.318
312&340@L+rdN11.2554E6.5445
325&344@L+rdN10.8406E6.6769
339&346@L+rdN10.4072E6.7212
352&344@L+rdN9.9729E6.6856
365&341@L+rdN9.5502E6.5793
378&336@L+rdN9.1482E6.4106
390&329@L+rdN8.7736E6.1875
400&321@L+rdN8.4315E5.9171
410&311@L+rdN8.1258E5.6061
418&300@L+rdN7.8594E5.2609
425&288@L+rdN7.6345E4.8873
431&276@L+rdN7.4526E4.4909
435&263@L+rdN7.3148E4.0771
438&250@L+rdN7.2216E3.6511
439&237@L+rdN7.1730E3.2176
440&223@L+rdN7.1687E2.7815
438&209@L+rdN7.2079E2.3471
436&196@L+rdN7.2895E1.9186
432&183@L+rdN7.4120E1.5
427&171@L+rdN7.5736E1.0949
421&158@L+rdN7.7720E.7064
414&147@L+rdN8.0048E0.3376
410&142@L+rdN8.1332E0.1612
352&448@P+bkN10E10
41&453@P+blN19.9985E10.1745
352&448@+blN10E10
45&502@P+blN19.8481E11.7365
352&448@+blN10E10
59&554@P+blN19.3969E13.4202
352&448@+blN10E10
82&603@P+blN18.6603E15.0000
352&448@+blN10E10
113&647@P+blN17.6604E16.4279
352&448@+blN10E10
152&686@P+blN16.4279E17.6604
352&448@+blN10E10
196&717@P+blN15.0000E18.6603
352&448@+blN10E10
245&740@P+blN13.4202E19.3969
352&448@+blN10E10
297&754@P+blN11.7365E19.8481
352&448@+blN10E10
352&759@P+blN10.0000E20.0000
352&448@+blN10E10
406&754@P+blN8.2635E19.8481
352&448@+blN10E10
458&740@P+blN6.5798E19.3969
352&448@+blN10E10
507&717@P+blN5.0000E18.6603
352&448@+blN10E10
551&686@P+blN3.5721E17.6604
352&448@+blN10E10
590&647@P+blN2.3396E16.4279
352&448@+blN10E10
621&603@P+blN1.3397E15.0000
352&448@+blN10E10
644&554@P+blN0.6031E13.4202
352&448@+blN10E10
658&502@P+blN0.1519E11.7365
352&448@+blN10E10
663&448@P+blN0.0000E10.0000
352&448@+blN10E10
658&393@P+blN0.1519E8.2635
352&448@+blN10E10
644&341@P+blN0.6031E6.5798
352&448@+blN10E10
621&292@P+blN1.3397E5.0000
352&448@+blN10E10
590&248@P+blN2.3396E3.5721
352&448@+blN10E10
551&209@P+blN3.5721E2.3396
352&448@+blN10E10
507&178@P+blN5.0000E1.3397
352&448@+blN10E10
458&155@P+blN6.5798E0.6031
352&448@+blN10E10
406&141@P+blN8.2635E0.1519
352&448@+blN10E10
352&137@P+blN10.0000E0.0000
352&448@+blN10E10
297&141@P+blN11.7365E0.1519
352&448@+blN10E10
245&155@P+blN13.4202E0.6031
352&448@+blN10E10
196&178@P+blN15.0000E1.3397
352&448@+blN10E10
152&209@P+blN16.4279E2.3396
352&448@+blN10E10
113&248@P+blN17.6604E3.5721
352&448@+blN10E10
82&292@P+blN18.6603E5.0000
352&448@+blN10E10
59&341@P+blN19.3969E6.5798
352&448@+blN10E10
45&393@P+blN19.8481E8.2635
352&448@+blN10E10
51&448@Agent->+blN19.65E10
352&463@Rabbit Begins Here+rdN10E10.5
352&448@P+bkN10E10
661&475@P+blN0.0381E10.8716
599&475@L+blN2.0381E10.8716
597&475@Agent Stops Here on 2nd Loop+blN2.1E10.88

Copy to a text file, save the text file as "Name.plt" with quotation marks,
and the file will run in KBH Scratch Plot...

And the rabbit path is a small-cap handwritten-style "e"...
 
D

Dave

KBH Code:

{The rabbit at the center of a 10 radius circular pond and the chasing agent
is at the edge of the pond.}

Var
ads, nag, eag, rba, rbn, rbe, nd, ed, dgr, dst: double;
flg: integer;

Procedure D;
Begin
If (flg = 1) Then
 Begin
 {Procedure E}
 rba:= rba + Pi;
 Exit;
 End;
rba:= rba + (Pi * 2);
{The structure of procedures D, E, & F are from the KBH Survey Progam for
the HP35S}
End;

Procedure F;
Begin
If (flg = 1) Then Exit;
rba:= rba + Pi;
End;

begin
{KBH Code}
flg:= 0;
rbn:= 10;
rbe:= 10;
dgr:= 1;

While (dgr < 361) Do  {Change value for desired output range}
 Begin
 ads:= (dgr * Pi) / 180;
 nag:= 10 + (Cos(ads) * 10);
 eag:= 10 + (Sin(ads) * 10);
 nd:= rbn - nag;
 If (nd = 0) Then nd:= 0.000000000001;
 ed:= rbe - eag;
 If (ed >= 0) Then flg:= 1;
 rba:= ArcTan(ed / nd);
 If (rba < 0.000000000001) Then D Else F;
 rbn:= rbn + (Cos(rba) * 0.043633231);
 rbe:= rbe + (Sin(rba) * 0.043633231);
 dst:= Sqrt(Sqr(10 - rbn) + Sqr(10 - rbe));
 rba:= (rba / Pi) * 180;
 WriteLn(rbn:15:4, rbe:15:4, dgr:15:4, dst:15:4);
 dgr:= dgr + 1;
 flg:= 0;
 End;

ReadLn;
end.

And the above KBH code runs in Delphi Console Mode...

Now the rabbit gets out at North and East coordinates of 8.1332 , 0.1612
while the agent is at an azimuth of 535 degrees on the edge of the circular
pond and on his second trip around the pond. Of course this is a systematic
method only...

And here is the KBH Scratch Plot file for the rabbit path:

663&137@F+wtN0E0
41&759@S+wtN20E20
0
352&448@P+bkN10E10
353&...@L+rdN9.9564E9.9992
365&...@L+rdN9.5661E9.9588
378&...@L+rdN9.1452E9.8458
390&...@L+rdN8.7477E9.6669
402&...@L+rdN8.3826E9.4288
412&...@L+rdN8.0572E9.1387
421&...@L+rdN7.7774E8.8044
428&...@L+rdN7.5477E8.4338
433&...@L+rdN7.3716E8.0349
437&...@L+rdN7.2515E7.6158
439&...@L+rdN7.1886E7.1844
439&...@L+rdN7.1834E6.7484
437&...@L+rdN7.2357E6.3155
434&...@L+rdN7.3441E5.8932
429&...@L+rdN7.5067E5.4886
422&...@L+rdN7.7209E5.1089
414&...@L+rdN7.9834E4.7607
405&...@L+rdN8.2899E4.4506
394&...@L+rdN8.6357E4.1851
382&...@L+rdN9.0151E3.9703
369&...@L+rdN9.4215E3.8124
356&...@L+rdN9.8469E3.7173
343&...@L+rdN10.2819E3.6911
329&...@L+rdN10.7150E3.7398
316&...@L+rdN11.1311E3.8687
305&...@L+rdN11.5107E4.0819
295&...@L+rdN11.8285E4.3792
289&...@L+rdN12.00538E4.7515
285&...@L+rdN12.1525E5.1743
286&...@L+rdN12.1075E5.6063
292&...@L+rdN11.9231E5.9996
301&...@L+rdN11.6270E6.318
312&...@L+rdN11.2554E6.5445
325&...@L+rdN10.8406E6.6769
339&...@L+rdN10.4072E6.7212
352&...@L+rdN9.9729E6.6856
365&...@L+rdN9.5502E6.5793
378&...@L+rdN9.1482E6.4106
390&...@L+rdN8.7736E6.1875
400&...@L+rdN8.4315E5.9171
410&...@L+rdN8.1258E5.6061
418&...@L+rdN7.8594E5.2609
425&...@L+rdN7.6345E4.8873
431&...@L+rdN7.4526E4.4909
435&...@L+rdN7.3148E4.0771
438&...@L+rdN7.2216E3.6511
439&...@L+rdN7.1730E3.2176
440&...@L+rdN7.1687E2.7815
438&...@L+rdN7.2079E2.3471
436&...@L+rdN7.2895E1.9186
432&...@L+rdN7.4120E1.5
427&...@L+rdN7.5736E1.0949
421&...@L+rdN7.7720E.7064
414&...@L+rdN8.0048E0.3376
410&...@L+rdN8.1332E0.1612
352&448@P+bkN10E10
41&...@P+blN19.9985E10.1745
352&448@+blN10E10
45&...@P+blN19.8481E11.7365
352&448@+blN10E10
59&...@P+blN19.3969E13.4202
352&448@+blN10E10
82&...@P+blN18.6603E15.0000
352&448@+blN10E10
113&...@P+blN17.6604E16.4279
352&448@+blN10E10
152&...@P+blN16.4279E17.6604
352&448@+blN10E10
196&...@P+blN15.0000E18.6603
352&448@+blN10E10
245&...@P+blN13.4202E19.3969
352&448@+blN10E10
297&...@P+blN11.7365E19.8481
352&448@+blN10E10
352&...@P+blN10.0000E20.0000
352&448@+blN10E10
406&...@P+blN8.2635E19.8481
352&448@+blN10E10
458&...@P+blN6.5798E19.3969
352&448@+blN10E10
507&...@P+blN5.0000E18.6603
352&448@+blN10E10
551&...@P+blN3.5721E17.6604
352&448@+blN10E10
590&...@P+blN2.3396E16.4279
352&448@+blN10E10
621&...@P+blN1.3397E15.0000
352&448@+blN10E10
644&...@P+blN0.6031E13.4202
352&448@+blN10E10
658&...@P+blN0.1519E11.7365
352&448@+blN10E10
663&...@P+blN0.0000E10.0000
352&448@+blN10E10
658&...@P+blN0.1519E8.2635
352&448@+blN10E10
644&...@P+blN0.6031E6.5798
352&448@+blN10E10
621&...@P+blN1.3397E5.0000
352&448@+blN10E10
590&...@P+blN2.3396E3.5721
352&448@+blN10E10
551&...@P+blN3.5721E2.3396
352&448@+blN10E10
507&...@P+blN5.0000E1.3397
352&448@+blN10E10
458&...@P+blN6.5798E0.6031
352&448@+blN10E10
406&...@P+blN8.2635E0.1519
352&448@+blN10E10
352&...@P+blN10.0000E0.0000
352&448@+blN10E10
297&...@P+blN11.7365E0.1519
352&448@+blN10E10
245&...@P+blN13.4202E0.6031
352&448@+blN10E10
196&...@P+blN15.0000E1.3397
352&448@+blN10E10
152&...@P+blN16.4279E2.3396
352&448@+blN10E10
113&...@P+blN17.6604E3.5721
352&448@+blN10E10
82&...@P+blN18.6603E5.0000
352&448@+blN10E10
59&...@P+blN19.3969E6.5798
352&448@+blN10E10
45&...@P+blN19.8481E8.2635
352&448@+blN10E10
51&448@Agent->+blN19.65E10
352&463@Rabbit Begins Here+rdN10E10.5
352&448@P+bkN10E10
661&...@P+blN0.0381E10.8716
599&...@L+blN2.0381E10.8716
597&475@Agent Stops Here on 2nd Loop+blN2.1E10.88

Copy to a text file, save the text file as "Name.plt" with quotation marks,
and the file will run in KBH Scratch Plot...

And the rabbit path is a small-cap handwritten-style "e"...
Click to expand...

I've totally lost track of what you are trying to do. Can you recap,
and possibly post the graph on the web and give us the link?

Dave
 
L

legg

Okay the circular pond has North and East coordinates at the radius point of
10 , 10 and the radius is 10 units . The agent is at North and East
coordinates of 20 , 10 and moves clockwise in 1-degree steps. The rabbit at
the center of the pond moves away from the agent at each step on a line
between the agent and the rabbit. And the rabbit moves 1/4 the distance that
the agent moves.

The rabbit gets out at North and East coordinates of 1.3437 , 4.9694 when
the agent is at an azimuth of 232 degrees on the circle edge. But the rabbit
is out at an azimuth of 210 degrees or so. In other words the rabbit doubled
back while the agent kept going. Before the rabbit doubled back it looks
like a slow outward spiral but I will write a ScratchPlot file later.

Of course this is one systematic method with no answer for what the agent
does when the rabbit is within one unit of shore and near the agent.

Here is the KBH code:
Click to expand...
<snip>

GIGO.

RL
 
C

[email protected]

Assuming a pond's radius = 1 and the rabbit's speed = 1: If the
agent's speed is v, the rabbit can position himself 1/v from the
center on the opposite side of the pond from the agent. He then heads
radially toward the shore and watches to see which way the agent runs.
When the agent has committed himself to a direction, the rabbit turns
90 degrees in the opposite direction and heads in a straight line
toward the shore. If the agent reverses direction, the rabbit will
again head out on a radius until the agent, the rabbit, and the center
of the pond again are collinear or the agent reverses again. At this
point, the rabbit will be closer to the shore before and the agent
will find himself in a worse position; thus, once he commits to a
direction, the agent cannot afford to reverse it. The rabbit must swim
sqrt(1 - 1 / v^2) while the agent runs pi + arccos(1 / v). So solving

sqrt(1 - 1/v^2) = ( pi + arccos(1 / v) ) / v

for v gives the number in question.

Dave
Click to expand...

I was thinking about generalizations...

Setting maximal accelerations seems difficult. Generalizing to more
dimensions is pointless. Other shapes seem ugly. But here are two
that may be interesting and feasible:

1) The agent has different maximal counterclockwise and clockwise
speeds (perhaps he must run backwards when going clockwise).

2) There are n agents.
 
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