Replicated!

A spaceman is captured by space aliens and is told that he will be released only if he passes a certain test. The aliens tell him that he will be placed in a replication chamber. After one hour, the replication chamber will be used to make an atom-for-atom copy of everything in the chamber, including the spaceman. The copies will appear in an identical chamber located somewhere else on the alien's spaceship. On the wall of each chamber will be a set of buttons, a red button and a green button. Above the set of buttons will be two lights, a red light and a green light. The lights indicate the color of the button that is being pressed in the other chamber. An hour after replication, the spaceman and his copy will be required to press one of the two buttons. If they both press the same color button, the spaceman and his copy will be recycled as bantha-fodder. If they press different color buttons, the spaceman will be rejoined with his copy (using an un-replicator) and he will be released. Neither light will glow until a button is held down in both chambers. The spaceman will be given no indication of whether or not his chamber contains the originals or the copies. Until his time is up (indicated by a countdown timer on the chamber walls), the spaceman and his copy can practice using the buttons. The spaceman will be given a 'quantum coin', which generates heads or tails with equal probability, and will produce results that are uncorrelated with his copy's coin. The spaceman could use his coin to select a button, and have a 50 percent chance for survival, but is there anything that he can do to increase his odds?

Carl G.

. . . . . . . . . . . . . . . . . . . . . . . . . two simple solutions assuming each copy knows everything that the other knows, then make a deal with yourself. (chemical encoding of memory should be preserved in 'exact' copies)

solution 1 each of you will press/hold buttons at random until you press different buttons... then you will always press that button.

Solution 2 press buttons... on a heads/tails decision you wait 5 seconds, or immediatly start tapping a predetermined code on the other button. if you both switch... try again. eventually one will switch and the other won't... and a dialogue can begin.

I don't understand what the purpose of the lights is, because they only light up when both buttons are pressed. Neither can I see what possible purpose there could be in practising using the buttons.

Surely, since they are exact replicas of each other, there is no problem at all. They both just press any button, and since the conditions in the other chamber are identical, they can be certain that they both press the same button.

Unless, of course, there are quantum phenomena at work which affect thir decision making process. Then they might be out of luck! If the spaceman has a natural preference say for green over red, then he should obviously choose green, because he can then be almost certain that his copy will do the same for the same reason.

Personally, I plan immediately to memorise an order of preference of colours, just in case I am ever unfortunate enough to land up in this situation. Thanks for the warning!

Derek Holt.

He fixes the following plan in his mind, to be performed (by both of him, of course) after replication.

[1] Repeatedly flip the quantum coin until it comes up heads 6 times in a row, or until the green light comes on.

[2] If the coin comes up head 6 times first, then he has Role A, and indicates this by pressing the green button. If he sees the green light first, then he has Role B, and indicates this by pressing the red button.

[3] When A sees the red light, he signals this by pressing the green button twice. When B sees the green light flash twice, he confirms this by pressing the red button twice.

[4] When the hour is up, A presses green and B presses red.

There are two possible problems.

[A] There is a small but finite probability (the exact number depends on the speed of flipping), that heads will never come up 6 times in the whole hour. In that case each man should press the button according to his last flip: heads green, tails red.

Possibly the algorithm can be improved by reducing the number 6 to a smaller number when there have been enough flips. On the other hand, the smaller the number, the greater the chance of problem [B] occurring.

[B] If heads comes up 6 times at the same time for both men, they may reach for the button together. Even while reaching for the button in step [2], the man who thinks he's about to be A should be watching for the green light to come on, and if it does, he should switch to Role B.

If he sees the green light coming on just as he presses the button, he should keep pressing it. If neither man presses the red button to switch to Role B, or if both do it together, they should start over at step 1.

Note that any solution where the man decides to change his behavior when a certain amount of time has passed is bogus, since both of him will estimate the time the same. He must use the randomizer.

And I missed that. Sorry.

Push the red button for 5 seconds. Note the color of the light. If it isn't red, you're dead meat because the aliens are cheating. Otherwise flip the coin; heads stay red, tails go green. Push the button for another 5 seconds. If the color of the light and the button differ, remember your button and push that one when the time comes. Otherwise flip the coin again and repeat until they are different. The chances of losing are negligible assuming the aliens aren't cheating.

The spacemen and his copy don't have to press the same button in order to see what the other is pressing (e.g., one could press red and the other green), but they do have to hold down a button to see which button the other is pressing.

The spaceman only survives if the buttons are *different*.

Carl G.

I am not sure if I have read the problem right. If I have, it is easy.

When the practice period starts, each spaceman presses the green button, and sees the green light come on. They realise that this won't do, and each tosses his quantum coin. If a coin comes up heads, its owner then presses red; and if it comes up tails, green.

If they find they are now pressing the same button again, they repeat the procedure. But if they find they are pressing different buttons, they just stand there keeping them pressed, to establish what they will do at the final showdown.

puzzle is spoiler...

: A spaceman is captured by space aliens and is told that he will be released : only if he passes a certain test. The aliens tell him that he will be : placed in a replication chamber. After one hour, the replication chamber : will be used to make an atom-for-atom copy of everything in the chamber, : including the spaceman. The copies will appear in an identical chamber : located somewhere else on the alien's spaceship. On the wall of each : chamber will be a set of buttons, a red button and a green button. Above : the set of buttons will be two lights, a red light and a green light. The : lights indicate the color of the button that is being pressed in the other : chamber. An hour after replication, the spaceman and his copy will be : required to press one of the two buttons. If they both press the same color : button, the spaceman and his copy will be recycled as bantha-fodder. If : they press different color buttons, the spaceman will be rejoined with his : copy (using an un-replicator) and he will be released. Neither light will : glow until a button is held down in both chambers. The spaceman will be : given no indication of whether or not his chamber contains the originals or : the copies. Until his time is up (indicated by a countdown timer on the : chamber walls), the spaceman and his copy can practice using the buttons. : The spaceman will be given a 'quantum coin', which generates heads or tails : with equal probability, and will produce results that are uncorrelated with : his copy's coin. The spaceman could use his coin to select a button, and : have a 50 percent chance for survival, but is there anything that he can do : to increase his odds?

The basic plan is this:

During the practice period, flip the quantum coin. If it is heads, hold down the red button. If it is tails, count to twenty and then press the red button briefly.

If you are holding down the red button and after a count of twenty it briefly lights up, you are RED.

If you count to twenty then press the red button and it lights up briefly, then the other spaceman was already holding it down, and he is RED

Well first of all I must object to your choice of the term 'quantum coin' as there appears to be nothing quantum about it

I'm not sure that I'm reading the question correctly, but I think that if both people are pressing buttons before the hour is pressed, they will see what buttons are being pressed, but that it won't determine their fate. The trouble is that the puzzle then seems too easy?

I think you're misread the problem. Either that, or you see 'no problem' in becoming 'bantha fodder.'

S P O I L E R

Actually, my answer is too easy, and I suspect that I'm misunderstanding. So maybe spoiled space is not called for.

Each person presses a button at random (using the quantum coin). If they end up pressing different colors, they are done. They remember what they just pressed, and then press the same button when the hour is up. If they end up pressing the same button, then they wait ten seconds to do new coin flips, and then come back and each press a random button. If they keep pressing the same button as each other, they keep reflipping. But whenever they have pressed opposite buttons, they stop and wait to do that when the hours is up. If they repeat at ten second intervals, they only have a (1/2)^(360)=10^(-108) chance of failure.