[Guest]

Okay, my first post here, usually I'm just a lurker, so I thought I'd start with a new puzzle(which I didn't see when I searched )

You are passing the time rowing about in your canoe, in the middle of a lake that makes a perfect circle. You are about to head back to shore when you notice a very hungry ogre looking at you from the shore. After rowing a bit while watching him, you notice a few things:

- The ogre runs 4 times faster than you can row.

- You can run twice as fast as the ogre (on land, you can't run on water...)

- The distance from the center of the lake to the shore is 1 mile (i.e, the radius of the circle)

How do you escape the hungry ogre? Here's your hint:

Circumference = pi*diamater

The area of the circle isn't important to the puzzle.

[Guest]

You haven't said anything about how the ogre moves. Assuming that he tries to get as close to you as possible without going into the lake, you should go to the center and then row to the directly opposite shore (on a diameter). He won't pursue you until you leave the center, and if he can't decide which way to go around the lake, you can escape because he will be "stuck" with equally short paths on either side. (If he takes on of them, you're in trouble, because you need to row 1 mile, which gives the ogre time to run 4 miles, when he only has to run pi miles.)

Edited January 6, 2010 by BrainInAVat

[Guest]

Proposed answer follows:

The key is to get to shore before the ogre meets you, since once there you can outrun him. I believe the solution is to begin by rowing in the exact opposite direction of the ogre. He can then be expected to begin running around the lake one way or the other. Once he commits, you then alter your course to continue rowing 180 degrees away from wherever he is at the moment, and you continue this until you reach shore. If he continues running in the same direction (clockwise or counterclockwise), your path will look somewhat like an arc from the center to the shore. If he reverses direction, your path will appear more like a zigzag.

I won't bother calculating exactly how close he is when you reach shore, but he'll have run sufficiently more than 1/2 of the way around the roughly 6.28 mile circumference of the lake.

[Guest]

Option 1: Wear him out. Everyone knows an Ogre is stupid and doesn't have very good endurance. Start to one side of the lake and the stop. Once the Ogre stops start to the other side of the lake, etc. until he is worn out.

Option 2: Start going to the opposite shore of the Ogre. Assuming he realizes you are going to the opposite shore, he would start going around the lake. As soon as he starts, I would maneuver my canoe so that he is always on the opposite side of the lake (if a line was drawn through the middle of the lake, he would always be on the opposite side of the lake and I would be on that line). Mathmatically, I can do this far enough that once I get at least .2146 miles from the center [1 - (3.1416/4)] or [1 mile minus {half way around the lake divided by 4}]. Once I get past that point, I can make a straight line for shore and beat him to that spot. Then I bail out and run!

Edited January 6, 2010 by darmelpitts

[Guest]

Say v is the velocity I can paddle. Imagine a line that goes through the ogre and the center of the lake. As the ogre runs around the lake a point on the line at the center of the lake has a velocity of 0 and a point on the line at the edge of the lake has a velocity of 4v. A point on the line that is 0.25 miles from the center will have a velocity of v. So assuming I can change directions just as fast as the ogre can, and I keep at it long enough, I can get close to 0.75 miles from shore while always keeping the center of the lake between me and the ogre. And as Darmelpitts pointed out, I only need to be less than 0.7854 miles from shore before I can make a successful dash for freedom.

[roolstar]

Proposed answer follows:

The key is to get to shore before the ogre meets you, since once there you can outrun him. I believe the solution is to begin by rowing in the exact opposite direction of the ogre. He can then be expected to begin running around the lake one way or the other. Once he commits, you then alter your course to continue rowing 180 degrees away from wherever he is at the moment, and you continue this until you reach shore. If he continues running in the same direction (clockwise or counterclockwise), your path will look somewhat like an arc from the center to the shore. If he reverses direction, your path will appear more like a zigzag.

I won't bother calculating exactly how close he is when you reach shore, but he'll have run sufficiently more than 1/2 of the way around the roughly 6.28 mile circumference of the lake.

I only have one issue with these solutions,

Since there is no information about my rowing speed in the OP (usually much less then my running speed), the proposed solutions may leave me running in circles, never getting to shore!! A very tight giant spiral will be my final trajectory.

In fact, there must be some minimum rowing speed where I can reach the shore with the ogre at least a feet away from me, so that I can outrun it. This would make a nice puzzle!

[Guest]

If the lake is a circle with origin O, the Ogre is the point A in the circumference of the circle and I am the point B, which at the beginning of the problem is at the Origin O, then first, I must row in the perpendicular of the tangent of the circle in point A, which is the line segment AOB in opposite direction to B. this will get you to a spot on the shore safely. Afterwards, you can run freely and even mock the Ogre if you like

[Guest]

There is infinite number of solutions. I'll show you one. Firstly you should get to in distance of 2/9 mile from the centre of lake. In that distance you are able to do faster circles than Orge on the bank. You start paddle and you are able to reach place where you and the Orge will beone line with centre between you. At that moment you start paddle to the nearit point on shore and you will be there faster than Orge.

Actualy you can start from any place between 1-pi/4 and 1/4 mile from the center.

[Guest]

Swim

Edited January 6, 2010 by drogernaut

[Guest]

If the guy rows in a straight line, the Ogre has to run 3.14 times as far as the boat and he can only go 4 times as fast--so a whole lot of evasion isn't required. I agree that the boat continuing on a spiraling path continuously at 90 degrees to the Ogre's current position should do the trick.

Edited January 6, 2010 by Simon Legree

[Guest]

Row to the opposite side of the lake which is 3.14 miles for the Ogre to run C= PI x D or 3.14 x 2 =6.28miles / 2 (halfway around)until you are halfway there then go 90 degrees away from the Ogre again and it will take him another 1.7 miles for atotal of 4,84 miles where you are only going 1 mile therefore beating the ogre to your shore destination

[Guest]

[spoiler=Answer... I Think ]If you start at the middle, than row straight towards the shore opposite the Ogre(what on earth is an Ogre?), then when the Ogre gets in line with you, turn of your rout at a right angle and land. Then, For Pete's Sake, RUN!!!

[Guest]

Ogres arn't real. So just row to the shore and go home.

[Guest]

I agree with Simon, swim. We have no information regarding swimming, i.e., speed, whether he will swim, etc. If ogres truly are as stupid as posted, then he will be confused once you begin to swim.

[Guest]

you can row anywhere, including directly at the ogre. because he wont eat you as long as your in the boat. The split second you get to land you start running and he will never catch you. If you guys dont like this answer then the only other answer would be to start going exactly opposite of the ogre, and the second he starts chasing you in one direction (lets say to the left), then you turn the boat atleast 87 degree to the right. Since it would take ogre 3.14 units of time to get to half the lake, if you turn 87 degrees that would be exactly 4.00 units, which is the same exact time both of you would arrive to same area. so if you turn AT LEAST 87, so lets say 88, and you already had him going one way, we would assume he would continue that path. and you can get there with just enough time to run faster than he.

[Guest]

stay in the center of the lake until the ogre gets bored...then leaves...you're free to go then.

Edited January 7, 2010 by Ghanas

[Guest]

Start off in the middle of the lake and aim for a point on the lake shore that is almost exactly opposite the ogre. The ogre now has a choice. He can take the short route or the long route. I'm assuming he'll choose the short route. The ogre starts running around the lake taking the shorter route. As soon as you see him start to run you change course. Change your heading so that you're boat is pointed to a different part of the lake shore in such a way as the ogre is now taking the long route around the lake. I'm going to assume that the ogre is smart enough to determine which is the shortest way to run around the lake based upon your rowboat's current bearing. So the ogre realises that he's taking the long route and changes his direction because he thinks he will get to you faster. As soon as he stops and changes direction you quickly change course again. The ogre stop, turns around and you change course again. And so this repeats. In effect you would be following a pattern somewhat like a zig-zag. Each time you change course the ogre stops and starts to run around the other way. You're in effect keeping him rooted to the spot because he keeps changing his mind about which way to run. It would be slow progress but you would get there in the end.

I think this would work because the ogre is not going to attack you in the water but on land. So the ogre wants to catch you when you get ashore. It makes sense to me that the ogre would always try and take the shortest route to get to you. I don't think he would be smart enough to know that he can easily outrun you and then wait for you to land.

[Guest]

I like this puzzle, and in fact it caused me to join this website...An interesting solution my wife came with is this--From the center of the lake, get out of the boat on the side opposite the ogre. At 1 mile away the ogre would probably not understand what occurred. Swim toward the shore, always keeping the boat between you and ogre. He would probably never see you and you could swim safely to shore and run away. On your next trip to the lake, bring a shotgun to dispatch him, if it is ogre season.

BasicPoke

[Guest]

There is infinite number of solutions. I'll show you one. Firstly you should get to in distance of 2/9 mile from the centre of lake. In that distance you are able to do faster circles than Orge on the bank. You start paddle and you are able to reach place where you and the Orge will beone line with centre between you. At that moment you start paddle to the nearit point on shore and you will be there faster than Orge.

Actualy you can start from any place between 1-pi/4 and 1/4 mile from the center.

This is the answer I was looking for. Now, of course there are more answers, and I should have been clearer with the description of how the Ogre reacts. What I assumed, and in turn thought everyone else would assume the same thing, would be that the ogre will always run around the lake using the route with the least amount of distance, trying to keep himself in a line with you with the shortest possible distance.

Thanks for all the replies!

[roolstar]

I've been sitting here reading all these posts about ogre being stupid and I've had it!

I think you all owe my wife an apology!

In her own words: "Omf kuf tuug gsut gout!"

You guys should be ashamed of yourselves.

[Guest]

Thanks for consuming a nontrivial chunk of my life. But it was an enjoyable diversion.

I wrote a computer program to come up with an example of a solution. There are many other similar solutions that are possible.

The strategy for the canoe is to go to the center of the

lake, and then head directly for the side of the lake

opposite to where the ogre is at that time. As the ogre

runs around the lake to intercept, the canoe will sprint

a short distance, and then correct course to again head

for the point on the shore opposite where the ogre is at

that time, and then repeat that process segment by

segment until it gets close enough to the shore to beat

the ogre. Once the canoe reaches a point that is less

than PI / 4.0 miles from the destination point, the canoe

can safely sprint for the shoreline, secure in reaching

it before the ogre can get there.

The center of the lake is: ( 0.0, 0.0)

The west side of the lake is: (-1.0, 0.0)

The north side of the lake is: ( 0.0, 1.0)

The east side of the lake is: ( 1.0, 0.0)

The south side of the lake is: ( 0.0, -1.0)

The problem is symmetric.

Assume the ogre starts on the west side.

Assume the canoe starts in the center of the lake.

Canoe sprints for 165.8761 feet, and then corrects course.

On the graph, the ogre's course is shown in red, and the

canoe's course is shown in green.

Sprint Canoe Position Ogre Position Canoe Mi Ogre Mi

0.0000 (0.0000, 0.0000) (-1.0000, -0.0000) 0.0000 0.0000

1.0000 (0.0314, 0.0000) (-0.9921, -0.1253) 0.0314 0.1257

2.0000 (0.0626, 0.0041) (-0.9686, -0.2487) 0.0628 0.2513

3.0000 (0.0929, 0.0123) (-0.9298, -0.3681) 0.0942 0.3770

4.0000 (0.1218, 0.0245) (-0.8763, -0.4818) 0.1257 0.5027

5.0000 (0.1487, 0.0408) (-0.8090, -0.5878) 0.1571 0.6283

6.0000 (0.1729, 0.0609) (-0.7290, -0.6845) 0.1885 0.7540

7.0000 (0.1938, 0.0843) (-0.6374, -0.7705) 0.2199 0.8796

8.0000 (0.2108, 0.1107) (-0.5358, -0.8443) 0.2513 1.0053

9.0000 (0.2236, 0.1394) (-0.4258, -0.9048) 0.2827 1.1310

10.0000 (0.2316, 0.1698) (-0.3090, -0.9511) 0.3142 1.2566

11.0000 (0.2347, 0.2011) (-0.1874, -0.9823) 0.3456 1.3823

12.0000 (0.1874, 0.9823) (0.1981, 0.9802) 1.1282 4.5129

The ogre ran 4.5129 miles while canoe paddled 1.1282 miles.

The ogre is 57.8556 feet away when the canoe beaches. You better run, sucker!

[bonanova]

This has been posted and solved at least twice, in different settings, and

Searching for the previous posts is difficult owing to different settings for the puzzle.

There are two solutions of interest - the one that gets the person to safety the quickest,

and the one that keeps the greatest distance between the person and the danger.

I posted the second solution, previously, but I can't find the reference.

If anyone's interested, I can reproduce it.

- bn

[Guest]

You could row the oppisite dirction of the orge, then, when the orge reaches the shore your paddleing twoards, stop paddling when your just out of the orge's reach. then, "Pretend" your paddling the opposte direction, THen the orge with reaches the opposite shoreline, paddle the rest of the way to shore in the opposite direction of the shore the orge is standing on.

[Guest]

anyone can swim faster then they can row a boat. you simply jump out of your boat, swim to shore, and run away.

[Guest]

oh and sorry for not using the spoiler, this is the first time i've ever posted on this site. it wont happen again.

[Guest]

Well, I'm sure this is already said somewhere above me but I'm making it a bit simpler.

The ogre will have to travel at least 2pi miles around the lake. During that time, I can row a distance of (1/2)pi miles. So if I'm going in a circle, I can form my path about the centre of the lake with a radius of a bit less than 1/4 mile (r=d/(2pi) ) and still be completing one round faster than the ogre. So till that point, I can make sure I'm always directly opposite of the ogre across the centre. Then I can make a direct dash for the shore 'cause I would have only 3/4 mile left to cover. Ogre would have traveled (3/4)*4= 3miles but he needs at least pi miles to reach the place where I'd land so I'll be ahead of him by more than 0.14 of a mile.

If I have a long rope and rock, I can drop the rock tied to the rope at the centre of the lake. Then I can row tangential to the shore of the lake and slowly releasing the rope so that I'm always opposite the ogre. So it'll be a loose spiral at first but slowly getting tighter and tighter as the radius gets bigger till at radius 1/4mi, I can't release any more rope cause ogre can complete one round the same as I am. So that's when I'll dash for the shore. (Just an idea I have)