# The cage - a logic puzzle

One summer day John was jogging in the countryside wearing only his normal running clothes (shorts, vest, sports shoes and socks). After running for some time he saw that the lace of one of sports shoes was untied. While he was bending down to tie it up, a psychopath who was hidden nearby, approached him from behind and hit him on the head so hard that John became unconscious.

When John recovered he found that he was inside a perfectly cylindrical room about 20 meters diameter and 5 meters high. The walls, floor and ceiling of the room were perfectly smooth and symmetrical and had no visible feature so it was impossible to distinguish any particular direction inside the room.

Inside the room there was a cage. The cage itself also was in the shape of a perfect cylinder about 2 meters diameter and 2 meters high.

The cage was made of solid steel and it had only a very small spherical light bulb inside fixed exactly at the centre of the roof of the cage, and a solid door on one side. The light bulb was fixed to the roof, so that it rotated together with the cage. It was not possible to see anything outside the cage when someone was inside the cage and the door was closed (nor was it possible for someone outside the cage see inside after the door was closed)

The cage was placed on a perfectly circular platform exactly the same diameter as the cage. The platform can rotate clockwise or anti-clockwise at the discretion of the psychopath, who uses a remote control to start and stop the rotation. The platform starts and stops rotating at a very slow rate of angular acceleration, so slow that someone inside the cage cannot even feel when the platform starts to rotate and when it stops. Also the maximum speed of rotation is so slow that a person inside the cage cannot feel any centrifugal force so he doesn’t even have any idea about how fast the cage is rotating.

The psychopath then told John that he will challenge him to do a task. If he fails he will die, as have many others before. However, if he succeeds his psychopathic ego will be destroyed, John will be released and the psychopath will give himself up to the police.

The task consists of this. Surrounding the cage on the floor of the room about 10 meters from the center of the cylinder there are eight glasses each filled to the top with an orange liquid. The glasses are placed at the eight cardinal points: North, Northeast, East, Southeast, South, Southwest, West and Northwest. One of the glasses is filled up with pure orange juice. The other seven glasses have orange juice with a small amount of deadly poison mixed with the juice. It is not possible to distinguish between the glasses or their contents in any way at all except by tasting the liquid. John is not allowed to go anywhere near the glasses to examine them before the test starts.

The psychopath then tells John that the only glass without poison is the glass placed in the north direction and tells him which one it is.

The psychopath does not allow John to leave any mark or object on the floor of the room by which to indicate a particular direction.

Then the psychopath tells John to get into the cage and the psychopath closes and locks the door. The psychopath remotely turns on the little light in the cage and gives John a couple of minutes to get used to the environment inside the cage. The psychopath then slowly starts to rotate the cage for about one minute then slowly stops the rotation (which takes about another minute). The psychopath waits another couple of minutes then unlocks the door and tells John to get out and choose a glass from among the eight and to drink it all.

Before the psychopath unlocked the cage door he had already changed his own position in the room to make sure John was totally disoriented when he got out of the cage and had no possible way to tell from his surroundings in the room which direction he was facing.

John got out of the cage, quickly chose a glass of juice and drank it. The psychopath was amazed to see that John did not immediately fall down dead so he knew that John had chosen the right glass. He then told John to repeat task many more times until he realized that John had not chosen the right glass the first time just by luck.

John was released and the psychopath was put in prison for rest of his life.

How did John manage to succeed in doing the task?

NOTE: Everything that is described could actually have happened in real life on this Earth.

• Does John have anything with him when he enters the cage (e.g. his shoes/laces)? Commented Jul 21, 2014 at 16:06
• Also, I think this is more of a brainteaser than a logic puzzle. Commented Jul 21, 2014 at 16:14
• Lateral thinking answer: John had a compass in his pocket. Commented Jul 21, 2014 at 16:41
• How did John and the psychopath get inside the room? Teleportation? Commented Jul 22, 2014 at 8:53
• I think the only way to figure it out properly is to try it out. Look out joggers; here we come.... Commented Jul 22, 2014 at 13:03

I'm not sure this would actually work but the implied solution sends to involve inertia. Tie a shoelace from the light, dangle the heaviest thing you can from it, and note how it deflects when the cage rotates. The issue is how to make this system freely rotating/friction free to keep the (likely) shoe from rotating.

• The puzzle says the cylinder rotates and accelerates so slowly as to avoid detection. The shoelace will then have enough time to exert its torque to impart angular speed to the shoe. This does not work.
– Hans
Commented Jul 22, 2014 at 5:21
• Agree to @Hans, how could this work? Commented Jul 22, 2014 at 7:56
• I had the same missgivings but as the conversation continued it became clear that was the imtended solution. Commented Jul 22, 2014 at 12:06
• I think its the intended too, what one can do is take out one of those tiny threads from the lace or clothing
– guru
Commented Jul 22, 2014 at 14:34
• @kaine: Intended or not, explain to me how this would work as the intended solution could be wrong, too.
– Hans
Commented Jul 24, 2014 at 0:56

Since we don't have any visual cues to measure the end orientation, John needs some way to judge how much he rotated. That'd require very good timekeeping and some way to measure the angular acceleration. We'll assume that John has perfect mental and physical skills and can track time, angular displacement, etc exactly. The first idea would be to hang one of his shoes from the lightbulb and measure the deflection from vertical. There's a couple of problems with this, though. One is that there's a lot of friction, and two is that since the lightbulb is in the center of the room the shoe won't actually displace. It will rotate, but it'll be very hard to make the shoe start out perfectly still, plus at some point the torque in the shoelace will mess with the rotation calculations so that John dies at the end. So what can we do instead?

We can juggle!

John starts juggling two socks in one hand. The socks are "almost always" in free motion, meaning they'll start revolving independently of John. If he walks a little to properly catch them, the angle he makes with the center of the room and the door will roughly equal the amount the cage rotated (although in the opposite direction). He can keep doing this until the rotation stops and calculate how much the cage rotated overall in order to find true north. A cool consequence of this solution is that if the cage remained still but the rest of the room rotated, then John would be out of luck.

If John could also throw and catch objects perfectly, he can do this without juggling by throwing a shoe into the air with no angular momentum. The amount it rotates before he catches it, then, will be the amount the room rotates too.

• Alternate solution: John pulls out his cell phone and downloads a compass app. Commented Jul 21, 2014 at 20:31
• Alternate alternate solution: John has been using a compass belt for the past three years and just knows true north at all times. Commented Jul 21, 2014 at 20:31
• Alternate alternate alternate solution: John has spent the past three years building up an immunity to [poison]. Commented Jul 21, 2014 at 20:32
• Alternate alternate alternate alternate solution: John is a pigeon. Commented Jul 21, 2014 at 21:21

John enters the cage, noting the orientation North, stands in the center of the cage facing north and waits. As soon as the light comes on, he starts hopping in place, up and down.

Since the cage is spinning so slowly, and the amount of time John's feet contact the floor would be minimal, John should be able to keep his north-facing orientation.

Once the cage is opened, John knows which way is north, and drinks the glass in this direction.

Remember, John is a runner, likely in good shape, and shouldn't have any problem jumping in place for a few minutes at a time when his life is on the line.

This is the best answer I can come up with (thanks to the person who suggested juggling).

• I would only add that he takes off a shoe and puts it at the edge of the cage before jumping, the degree of movement compared to the 8 cardinal directions would be so slight that he would be able to make it back to the middle and start jumping before he lost track of the original direction. Commented May 18, 2017 at 20:36
1. John breaks off the zipper pull from his vest, which conveniently happens to be made of a ferrous metal.
2. He then rubs it on his shorts in one direction. His shorts, conveniently made of silk, do the job of magnetizing the zipper pull.
3. He rips a bit of buoyant foam from in-sole of one of his running shoes, making a circle as small as possible while still having enough buoyancy to hold up the zipper pull.
4. John places his other shoe in the center of the cage, easily found by dropping something from just below the light bulb.
5. Being a lifelong runner and very health-conscious, John is easily hydrated enough to urinate into his other running shoe, filling it up to the top and letting the liquid settle.
6. He floats the magnetized zipper pull on the body of liquid with his circle of buoyant foam, noting its orientation relative to the cup he knows is North.
7. The zipper pull will eventually return to that orientation, regardless of how the cage was rotated.

He can repeat this process as many times as he wishes, since the would-be killer is content to keep giving him plenty of fluids throughout the process.

• Uhh. A bit. Gross. Commented Jul 24, 2014 at 2:56
• Very good. However, "His shorts, conveniently made of silk, do the job of magnetizing the zipper pull." How does this work? Note, we are not talking about building static electric charge on the zipper pull but magnetization of it.
– Hans
Commented Jul 24, 2014 at 5:13
• I highly doubt you could make an effective compass this way. (not likely to build up enough charge to overcome the mass of the rotating part) I will say, however, that at least the floating object's inertia will prevent rotation a little. Commented Jul 24, 2014 at 13:04
• @kaine: My point is the charge, however much or less built up on the metal, is irrelevant. Magnetization and charge carrying are two completely different things. You may wrap a coil around the metal and repeatedly induce a spark on one end of the coil and thus a current through the coil in one direction, by the static charge from another material. However, rubbing the metal itself will not do.
– Hans
Commented Jul 24, 2014 at 13:56
• Let's hope the cage is not made of permalloy or mu-metal which is of high magnetic permeability and thus shields against magnetic field.
– Hans
Commented Jul 24, 2014 at 14:05

John speaks a language with absolute frame of reference such as Guugu Yimithirr or Tzeltal (so he ordinarily speaks of his north foot and south foot, or his east foot and west foot). Speakers of these languages have been shown to have a strong internal compass. As such, John always knows which direction is north.

The only static element inside the room who could be used as a reference is...

The light bulb filament, so after the psychopath tells John about the glass, he can look up, check the filament's direction (which given it's particular shape can be used precisely to reference that glass) and do whatever he likes during the psychopath's master plan, and then safely drink the unpoisoned juice from the correct glass. The flaw in the psychopath system is the light bulb rotating with the entire system and not fixed or static.

• "The light bulb was fixed to the roof, so that it rotated together with the cage."
– jscs
Commented Feb 12, 2015 at 6:20

Simple... no magnet, no laces required :)

The steel cage is offset from the center of the room, thus creating a reference point to compare the distance of each glass to the nearest wall of the main room.

Exit the steel cage and just look ! (drawing has exagerated proportions to relate the idea - the op does specify room is "about" 20m diameter, and glasses are "about" 10 meters from center of cage...)

• "WTG" for thumbs down without providing a logical argument. What are you here to judge? my drawing skills ? Commented Feb 12, 2015 at 13:00

The correct tool for the task is ...

a Foucault pendulum.
http://en.wikipedia.org/wiki/Foucault_pendulum

He ties a heavy object to the lamp and let it swing. The torque on the shoelace is enough to turn the object itself, but not to change the direct

I've seen the above answers but I have another solution. If John can throw something in the air perfectly up just before the rotation starts, maybe a hankerchief parachute, if the object is perfectly symmetric and the throw is perfectly up, then if the object falls slowly enough, he can note how it rotated in relation to him, (may require a pattern on the hankerchief).

• user3559247 - It would not be possible to be in the air for 2 minutes.
– Len
Commented Feb 12, 2015 at 5:39
• @user3559247 - John could continue throwing it up in the air and offset and calculate the angular acceleration from total time in the air. Commented Feb 13, 2015 at 4:19
• I don't get why this is down voted, it's a good answer, just not a good solution. (It's in no way worse that those saying to tie a shoe lace to a light bulb and calculate by EYESIGHT millimeters and fractions of degrees.) Commented Apr 15, 2015 at 20:51

It says The psychopath does not allow John to leave any mark or object on the floor of the room by which to indicate a particular direction. i.e it only talks about doing something on the floor of the room so could John just take a drink of the unpoisoned glass and then he knows which one it is by the lower level in the glass.

• "John is not allowed to go anywhere near the glasses to examine them before the test starts." Commented Aug 22, 2016 at 4:07