# Ants playing tag!

$5$ ants are initially arranged on the corners of a regular pentagonal table (edges are long $L$), as showed in this picture: At some point they all start moving with a constant speed $S$ and, at any moment, pointing the previous ant in counterclockwise order (as in the picture). They keep moving until they meet in the center $O$ of the table.

How long does it take to meet in $O$?
Also, how many times each ant rotated around $O$?

Notes: Consider the ants as points. Also, I kindly ask to everyone not to close-vote this question as math problem because it features a counterintuitive solution!

• "Previous" usually means "opposite direction", so the "previous and in counterclockwise order" would be the "next and in clockwise order", the opposite of your diagram. Right? Apr 3, 2015 at 16:38
• By previous I mean the "one who precedes". Anyway, to avoid confusion, check the picture, it's pretty self-explanatory. Apr 3, 2015 at 17:28
• Sorry, I was unclear. I was hoping you could edit the question to avoid confusing people. Apr 3, 2015 at 17:34
• Wouldn't they just infinitely spiral towards the middle, never reaching it? Apr 3, 2015 at 18:17

They take time $\left(1 + \frac{1}{\sqrt{5}}\right) L/S$. They rotate infinitely many times.

Each ant moves towards the next ant, decreasing the distance between them, but the other answer is moving slightly angled away. The ants remain in a pentagon, so this angle doesn't change, and the distance decreases at a constant rate. We'll find this rate, and from it determine when the ants meet.

Each ant moves towards the next one at speed $S$. But, the chased ant is also moving, angled away. Only the component of its velocity parallel to their displacement matters; the perpendicular component doesn't change their distance. These components are $(S \cos (2\pi/5), S \sin(2\pi/5))$, so the ant is increasing its distance away at rate $S \cos (2\pi/5))$. So, in total, the distance decreases at rate $S (1 - \cos (2\pi/5))$.

Dividing their initial distance $L$ by this rate gives the time to meet. Using Wolfram Alpha to express this in radicals gives $$\left(1 + \frac{1}{\sqrt{5}}\right) L/S$$

After rotating through some angle around the center, the ants are still in a regular pentagon, but scaled down. Since the angle of rotation isn't affected by scale, they still need to rotate through the same angle as they did at the start. So, that angle must be infinite.

• This was a nice problem, but the trig for the pentagon was a bit ugly -- it wouldn't have been needed for a square.
– xnor
Apr 3, 2015 at 10:47
• Nice! Another solution: The circumcenter of the ants is also moving at speed S; the circumradius of the table is L/2sin(π/5), so this gives L/2sin(π/5)S, which is the same value. Apr 3, 2015 at 11:43
• Aravind, the circumcenter (intended as the intersection of axis) of the 5 ants isn't moving! Apr 3, 2015 at 12:45
• @leoll2 Yeah, going through my reasoning made me realize my velocity calculation was faulty. Is it right now?
– xnor
Apr 3, 2015 at 19:55

Taking a slant on xnor's idea, consider the velocity of any ant towards $O$. I'll use the standard figures for the internal angle of a regular pentagon and the radius of a circumscribed circle from wikipedia. Name the vertices $A,B,C,D,E$ starting at the top and working clockwise, and let $\theta = \angle OAB$.

By symmetry, $\angle OAE$ is also $\theta$, so $\angle BAE = 2\theta = 3\pi/5$ (internal angle of a regular pentagon), i.e. $\theta = 3\pi/10$.

The velocity $V$ of $A$ towards $O$ is $S \cos \theta$, so the time $T$ taken for $A$ to reach $O$ is ${AO \over V} = {L \over 2 \sin (\pi/5)} \times {1 \over S \cos \theta}$ (using the formula for the radius of a circumscribed circle).

$$\therefore T = {L \over 2S \sin (\pi/5) \cos (3\pi/10)} = {L \over 2S \sin^2(\pi/5)} \approx 1.45L/S$$

As xnor observes, the ants spiral through an unbounded angle.

As $T$ and $S$ are finite, the total scalar distance $D=TS$ traversed by each ant is also finite. Since $T \approx 1.45L/S$ from before, we also have $TS \approx 1.45L$, i.e. $D \approx 1.45L$.

The counter-intuitive result hinted at by leoll2 is then the finite time taken to traverse an infinite spiral, where the total length of the spiral is a constant factor of about 1.45 times the length of one side of the initial pentagon.

Note: thanks to leoll2 for pointing out the trig simplification.

• So we're getting different answers?
– xnor
Apr 3, 2015 at 22:48
• @xnor Looks like it - cross check? Apr 3, 2015 at 22:53
• Correct! The final formula is a bit rough (did you realize that cos(3pi/10)=sin(pi/5)? If you can, please edit it the formula to make it look slightly more elegant :-) Apr 4, 2015 at 16:01
• @leoll2 Thanks! I've made the change and propagated the correct evaluation to the rest of the answer. Please excuse my rusty trig. Nice problem and interesting end result. Apr 4, 2015 at 16:17

Each ant is 11 pixels in diameter. So, given a bounding radius of 5.5 pixels and a start radius of 130 pixels (I approximated the diagram), they travel 3.82 radians before running into each other, at a rate of 2 pixels per iteration and assuming Float32 as the maximum accuracy.

Here is a live demo I created in javascript.

View the Demo https://jsfiddle.net/FlavorScape/7tawcgfw/

As you can see, after it completes, they spin off into infinity (because the bounding box does not stop them from approaching the line below the Float32 accuracy, is considered zero and all points gravitate to -1,1 from the "fake" center at 200 until they hit actual pixel 0,0 then do some other stuff, i'm not sure why.

<!DOCTYPE html>
<html>
<meta charset="UTF-8">
<title></title>
<body>

<script language="javascript">

var ants = [];
var center = [200,200];
var step = 2;
var interval = 33;
var done= false;
var t = 0;
var startRotation4;

var angle = 0;
for( var i = 0; i < 5; i ++ )
{
angle += 2 * Math.PI / 5;

var ant = document.getElementById( 'ant-' + i);

ant.style.left = Math.sin(angle ) * radius + center + 'px';
ant.style.top = Math.cos( angle)  * radius + center + 'px';

var dx = parseInt( ant.style.left) - 200;
var dy = parseInt( ant.style.top ) - 200;
startRotation4 = Math.atan2( dy, dx);

ants.push( ant );
}

var travelled = 0;

setInterval(function(){

document.getElementById( 'readout').innerText = 't=' + t + ' a=' + travelled ;

//chase the tail
for( var i = 0; i < 5; i ++ )
{
var target = ants[ i - 1];
var ant = ants[i];
if( ! target )
{
target= ants;
}

var dx = parseInt( target.style.left) - parseInt( ant.style.left );
var dy = parseInt( target.style.top ) - parseInt( ant.style.top );

var angle = Math.atan2( dy, dx );

var velX = Math.cos( angle ) * step;
var velY = Math.sin( angle ) * step;

ant.style.left = parseInt( ant.style.left ) + velX + 'px';
ant.style.top = parseInt( ant.style.top ) + velY + 'px';

//just detect if they're touching (center plus radius collision)
var ctX = parseInt( target.style.left) + 5.5;
var ctY = parseInt( target.style.top )+ 5.5;

var cX =  parseInt( ant.style.left ) + 5.5;
var cY = parseInt( ant.style.top ) + 5.5;

if (  i == 4 )
{
var endDx = parseInt( ant.style.left) - 200;
var endDy = parseInt( ant.style.top ) - 200;
travelled = Math.abs( Math.atan2( endDy, endDx) - startRotation4);
console.log( startRotation4 , travelled )
}

if( (Math.abs( ctX - cX ) <= antRadius ) && (Math.abs( ctY - cY ) <= antRadius))
{

if( !done && i == 4 )
{

document.getElementById('done').innerText = 'done in ' + t + ' steps. angle travelled = ' + travelled + ' radians';
done = true;
}

}

}

t++;
}, interval );

};

</script>

<div id="ant-0" style="background-color:#f00; width:11px; height:11px; position:absolute"></div>
<div id="ant-1" style="background-color:#f00; width:11px; height:11px; position:absolute"></div>
<div id="ant-2" style="background-color:#f00; width:11px; height:11px; position:absolute"></div>
<div id="ant-3" style="background-color:#f00; width:11px; height:11px; position:absolute"></div>
<div id="ant-4" style="background-color:#f00; width:11px; height:11px; position:absolute"></div>