Logic Puzzle

who can solve the following puzzle ( i couldnt...)

'there are three armies, a1, a2 and b. a1 and a2 are fighting b. a1 is sitting on a hilltop, b in a valley and a2 on another hilltop. now the boss of army a1 wishes to attack army b in the valley at noon. the only way to communicate this to army a2 is a carrier pigeon. armies a1 and a2 can't see or hear each other and they can only win if they attack b simultaneously. so the boss of a1 sends the pigeon to a2 with the message 'attacking at 12, please confirm attack' - a1 now must wait for the confirmation of this message (after all, the pigeon could have been shot down or gobbled up or wandered off...) so a2 replies 'message received, roger wilco.' - a2 however knows that a1 is waiting for the confirmation and cannot attack unless they know that their confirmation has arrived at a1. so a1 has to confirm the confirmation. etcetera etcetera....

is there a way a1 and a2 can EVER attack? according to the magazine where i found this there is a way (a certain message at a certain time or something like that) - however i cant believe that. my point of view is: whoever sends the LAST pigeon NEVER knows if it arrives, so the last pigeon is meaningless, hence there can be no last pigeon... ?????????? on the other hand after 10000 confirmations, both sides should know they can attack, shouldnt they????? HELP I CANT SLEEP ANYMORE

<<< Tini >>>

Do some role play with a mate. (=friend, I've been told off about using the term 'mate' before, but can't get out of the habit!). When do _you_ know beyond any doubt that you both have the shared information? Is there a point at which you can switch to 'and I don't need confirmation'. If one side says 'I know, you know, and to hell with it, I don't need confirmation, do or die (or both)', and the second side says 'yes, of cource we both know, I'm just being polite, don't bother replying, see you at noon', then let the first side send 'nothing wrong in politeness. noon. over and out.' - now what further message is required. By the participants own terms _more_ messages have been sent than necessary! However, this did require the switch to a _new protocol_ mid-stream.

However, as it stands, the induction argument does hold.

<snip coordinated attack problem>

Let's simplify and assume that each message has a certain probability of getting through.

Now, let's pick a _really_ unlikely (and devastating) event, such as getting struck by lightning.

Figure out the probability of that unlikely event. Call that 'U'.

Now, keep sending messages saying 'We will attack at time X. You do the same' _until_ the probability of the other army _not_ getting the message is less than U.

Now, since you're not really worried about getting struck by lightning, you should be even _less_ worried about your message not getting through, since the message not getting through is _less_ likely that getting struck by lightning.

-Fred Klein's Law of Reduced Probability

Fred Klein