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I've got a bit of a puzzle here that sits at the intersection of mathematics and technology. Hopefully this doesn't fall into brainteaser territory - I'm not sure a neat solution is possible!

I have two devices. Each one has a short token they would like to share. I am happy for either device to get the other's token; they don't both need both tokens.

Each device can transmit or receive locally, but it can't do both at the same time. A transmission takes 1-2 seconds. Both devices know the time (very accurately) but have no other knowledge of each other.

I can see a flakey solution here - start in receive mode for a random number of seconds, then enter transmit mode for one transmission, then repeat with a new random number. Sooner or later (probably sooner) a full transmit window will be captured in the other device's receive window; there is some risk of transmit collisions and partial receives, but the bigger the range of that wait window the less likely this becomes (and the token is a known length + has a check digit; partial receives are easily discarded).

My solution works, but feels very inelegant. It occurs to me this problem probably exists in many domains and may even have a name / common approach. Could anybody steer me in the right direction? Obviously it would be better if the devices could have a known master / slave relationship, or if one could transmit on odd seconds and one on evens; but there are dozens of these devices and no way of knowing which two will end up next to each other trying to pair.

I've tried to abstract away a lot of the details but happy to provide more where it would be helpful.

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