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        riddles >> easy >> How many Birds
        
(Message started by: seo@rivaa.in on Dec 10^th, 2012, 8:50pm)

Title: How many Birds
Post by seo@rivaa.in on Dec 10^th, 2012, 8:50pm

One bird is leading two birds, now tow birds are leading one bird... so how many birds are there.....?

Title: Re: How many Birds
Post by towr on Dec 10^th, 2012, 10:40pm

[hide] 3 [/hide]

Title: Re: How many Birds
Post by cartoonle on Dec 10^th, 2012, 11:48pm

tow is two?

[hide]than 3 birds?[/hide]

Title: Re: How many Birds
Post by seo@rivaa.in on Dec 11^th, 2012, 5:13am

[hide] ya cartoonle the total no of birds are 3[/hide] :)

Title: Re: How many Birds
Post by marlonmark on Jan 2^nd, 2013, 2:20am

answer will 3 my friend it was already said by Towr

Title: Re: How many Birds
Post by atyq on Jan 28^th, 2013, 10:00am

I thought it was two :o

Title: Re: How many Birds
Post by lizzie88 on Feb 7^th, 2013, 11:36am

If ans is 3 your question was right :)

Title: Re: How many Birds
Post by pfeinstein on Feb 17^th, 2013, 5:12am

[hide]3, of course![/hide]

Title: Re: How many Birds
Post by MpAdvisor on Feb 20^th, 2013, 4:13am

[hide]it's 3[/hide]

Title: Re: How many Birds
Post by tsitut on Feb 20^th, 2013, 12:11pm

3 obviously XD

Title: Re: How many Birds
Post by ravibhole_1 on Apr 6^th, 2013, 11:39pm

only three

Title: Re: How many Birds
Post by whizen on May 29^th, 2013, 6:04pm

Are the birds in the first part of the question the same ones as the birds in the second?
In that case, there are 6 birds... counting the bird that I have in my hand, and the 2 in the bush.

Title: Re: How many Birds
Post by alien2 on Aug 2^nd, 2013, 7:58am

Did you see them birds? (http://www.environmentalgraffiti.com/news-seven-blow-your-mind-beautiful-kinds-bird?image=1)

Title: Re: How many Birds
Post by Nursejim on Aug 24^th, 2013, 11:19am

.<:>.

.=bird

So to me there are four 4 birds. One bird "towing" two birds and "the" two birds in "tow" are leading one bird. Since the one bird is not capitalized then it's name is not One Bird or "the" one bird...so there can be four. Most likely I am over thinking things again ;)

Title: Re: How many Birds
Post by sanaya on Sep 2^nd, 2013, 5:33am

3 birds...I am sure

Title: Re: How many Birds
Post by swapnilraja1212 on Sep 17^th, 2013, 1:21am

4 birds buddies

b1-b2,b3-b4

Title: Re: How many Birds
Post by aiyanct on Nov 29^th, 2013, 11:12pm

ITS THREE "3"

Title: Re: How many Birds
Post by OmerMustafa on Dec 14^th, 2013, 3:47am

6 birds if the birds in the second part of the question are different with the birds of first part ;D

Title: Re: How many Birds
Post by AKMBorhanice on Jan 20^th, 2014, 12:26pm

Yes it is , 3 Birds

Title: Re: How many Birds
Post by progagan on Jan 23^rd, 2014, 11:26pm

There are three birds. I am sure...

Title: Re: How many Birds
Post by alien2 on Jan 30^th, 2014, 8:12pm

Five birds. Two birds ahead are leading one bird in the middle that is leading the last two birds.

Title: Re: How many Birds
Post by jordan on Feb 2^nd, 2014, 1:04am

and what is the right answer? I didn't get it :)

Title: Re: How many Birds
Post by JiNbOtAk on Feb 10^th, 2014, 8:26pm

I'd say the intended answer would be [hide]at least 3[/hide]. Of course, you can have a gajilion permutations, but this would seem to be the lowest limit.

Title: Re: How many Birds
Post by rloginunix on Mar 15^th, 2014, 11:51am

If we reword the problem statement slightly:

How many birds are in a flock in which it's always possible for one bird to lead two and two birds to lead one?

then let's assume a steady state straight line constant velocity flock. In such a flock all the birds have identical velocity vectors - same length, same direction.

Since we need a metric to work with I propose a Flight Perpendicular (FP for short) - a perpendicular to a velocity vector through a bird. A bird is a leader then iff its, possibly forward-extended, velocity vector does not intersect any other FPs. In figure 1) B is the leader:

http://www.ocf.berkeley.edu/~wwu/YaBBAttachments/rlu_3birds.png

Leader's FP is the leadership line - line L in figure 1). Any bird on L leads. Any bird behind it - trails. A bird is behind if its, possibly forward-extended, velocity vector intersects at least one other FP. So in figures 1) and 2) one bird (B) is leading two (A and C) or two birds (A and C) are trailing.

When does a solo leadership of one bird end? If we keep rotating the velocity vector, counterclockwise in figure 2), we'll see that at some point at least one non-leading FP will merge with the leading FP - figure 3). In plain speak the condition is compound. When a straight line between any two birds is perpendicular to their velocity vectors and no other bird is in front of them then two birds lead.

If in figure 3) you mentally flip the velocity vectors 180 degrees the birds A and B are on the straight line that's perpendicular to their velocity vectors but they are not in the lead. In that case one bird (C) is leading two (A and B) or two birds (A and B) are trailing.

Let's see what will happen if we subtract and add one bird from/to the flock.

Subtraction. With a 2-bird flock we remain in 2D. Possible configurations are 1) two birds lead zero birds and 2) one bird leads one bird. Both configurations violate the problem statement.

Addition. With a 4-bird flock we may or may not leave 2D. Let's say we are still in 2D. Possible configurations are 1) three birds lead one, 2) two birds lead two and 3) one bird leads three. All three configurations violate the problem statement also.

To extend the 4-bird flock into 3D all we have to do is replace the perpendicular line with the perpendicular plane.

Finally, we have to exclude two colinear special cases. If the velocity vector is perpendicular to the common line then all 3 birds lead. If the velocity vectors coincide with the common line then in either direction only one bird can lead two and two can not lead one.

So with the above problem statement and definitions 3 birds, cocircular or colinear barring two special cases, is a unique solution.

If you fix the direction but allow individual velocities to take on different values during the flight then I think the answer should still be the same.

If both - value and direction - of individual velocities can change at any given time in an arbitrary way along an arbitrary curve/surface then it gets more complicated.

[edit]
Moved the drawing file to this forum.
[/edit]

Title: Re: How many Birds
Post by rmsgrey on Mar 17^th, 2014, 5:19am

on 03/15/14 at 11:51:10, rloginunix wrote:

A bird is a leader then iff its, possibly forward-extended, velocity vector does not intersect any other FPs.

I object to this definition.

I would define leading as a relationship between pairs of birds - for any pair of birds, either A leads B or B leads A or neither leads the other.

With three birds: A, B, C, it's tempting to add the rule that if A leads B and B leads C, then A leads C, but extending that to higher numbers of birds rules out the scenario where the birds fly in a circle - in that situation, A would be leading B but B would also be leading A...

Title: Re: How many Birds
Post by rloginunix on Mar 17^th, 2014, 4:41pm

I see your point.

I guess I didn't put down explicitly what was always on my mind, keeping it to myself instead - my interpretation of the verb or concept of leadership. In my mind I treated it as leadership of the flock as a whole. So my definition should've been "A bird is a leader of a flock as a whole ...". My bad.

You're proposing a much more narrower interpretation. Doesn't it limit the scope of your definition to just two birds? What does it mean then to say that "two" birds are leading one since there's no such definition. What am I missing? Also, how would one geometrically distinguish "A leads B" from "B leads A". Are you keeping geometry or defining it in terms of algebra alone, like "L" is a binary operator with these properties, etc? Thanks.

Circles are tricky. Velocity vectors will be tangent to a circle. And I can always find a formation where each vector (extended) intersects another FP making it not clear at all where does the flock begin and where does it end.

Title: Re: How many Birds
Post by rmsgrey on Mar 18^th, 2014, 8:00am

on 03/17/14 at 16:41:52, rloginunix wrote:

I would interpret "two birds lead one" as "there are two birds, A and B, and a third bird, C, such that A leads C and B leads C" (equivalently, "there is a bird, for which there are another two birds, each of which is leading it")

I've (deliberately) not fully specified the "leading" relationship - in particular, I've left out how you'd decide which of the three possibilities applies to any given pair of birds. That's partly because there are several plausible schemes - I'd probably settle for "A is leading B if the direction of the vector BA is within a certain angle, theta, of both the direction of A's motion and the direction of B's motion". As a rough idea, I'd set theta as 60 degrees, but that's open to debate. The obvious hole in this definition is the lack of any proximity criterion - if there are two birds thousands of miles apart who both happen to be flying in the right direction, is one really leading the other? The other plausible improvement would be some idea of correlation of motion over time - though that's debatable.

Title: Re: How many Birds
Post by rloginunix on Mar 19^th, 2014, 7:06pm

By carefully delineating the boundaries of my definition - 2D, constant velocity, straight line - I purposefully avoided going into flux, flow, gradient, curl, divergence and all other things vector calculus which I studied more than two decades ago (relative to the time stamp of this message) but never touched since so I can't be immediately productive there. But that's where this issue should really land, I think, in its most generic case since flying birds and vector calculus have common core: items moving through a surface over time.

My thinking was: "is it possible to create such a formation of birds that will satisfy the problem statement but will amount to more or less than a 3-bird flock as a whole".

I would define my greedy flock thus. Spill N points at random over a plane. Drive nails into each point. Stretch a rubber band a lot, pull it over the nails and then let it go. Smallest perimeter convex polygon. In 3D: throw N points at random, freeze them, stretch a rubbery ball with a hole in it over the points, let go, cover the hole. Smallest surface area convex polyhedron.

But in any case I'm still staying in a rigid formation and the definitions apply to all the birds in a flock.

on 03/18/14 at 08:00:12, rmsgrey wrote:

would interpret "two birds lead one" as "there are two birds, A and B, and a third bird, C, such that A leads C and B leads C" (equivalently, "there is a bird, for which there are another two birds, each of which is leading it").

Playing devil's advocate here. While we know what's going on between A and C and between B and C we are not really sure what's going on between A and B themselves. Is it important? Not sure.

From your definition it follows (algebraically):

A L C && B L C

Conversly, one bird leads two:

C L A && C L B

Do a simple substitution: A = C', C' = C and we get: one bird leads itself and the other bird behind it, so there are just 2 birds here.

In the following infinite two, one, two, one, two ... formation flying left to right:


* * * *
 * * *
* * * *

your pair-based definition stands: A leads C and B leads C, while C leads D and C leads E and so on ad infinitum. How can this be prevented?

Lastly, a question about geometry. I assume that you assume that vectors A and B are not parallel (something I avoid altogether). But if we fix the angles for an arbitrary formation when A is in front then we can always find a different arrangement when B is in front but the angles are the same. Don't you think angle alone is not enough?

Thanks.

Title: Re: How many Birds
Post by rmsgrey on Mar 20^th, 2014, 8:33am

If A is in front, then the vector BA is going forward (and the angles are acute); if B is in front, the vector BA is going backward (and the angles are obtuse).

I'm talking about the difference in bearing between the two vectors, not the angles at which the lines representing them would meet if you drew them on the same diagram in the obvious way.

Title: Re: How many Birds
Post by rloginunix on Mar 20^th, 2014, 9:24am

I see now, makes sense.