M, military ruler of Logistan, has deferred to international pressure and agreed to hold an election in which he will run against his arch-nemesis, B. Both M and B, being politicians, are liars. You have been engaged as an independent consultant and charged with devising a representative voting procedure, i.e., your mission is to tally the true preference of each citizen who has a consistent, determinable opinion, and no other.

The chief complication relates to the fact that the Logistani electorate is composed of five (to your eyes) indistinguishable ethnic groups, each of which have a distinctive relationship to the truth. When expressing their voting preference:

• Knights respond honestly.
• Liars negate their true view.
• Subversives consider how a a knight with the same views would respond, then say the opposite.
• Revisionists admire knights and liars, and despise subversives. A revisionist will copy the most recent knight or liar to have voted, unless a subversive has voted more recently. In this latter case, the revisionist will vote for the opposite of that subversive.
• Contrarians reverse the answer of the most recent voter.

A contrarian or revisionist would respond randomly if he were the first voter queried.

You are to hold the election at the national stadium, to which the entire Logistani electorate has been invited. After some thought, you decide you can conduct the vote by asking members of the assembled electorate a single yes/no question. This is an open ballot, so each voter will call out his/her answer to the question for all to hear.

Suggest a viable question and any procedural arrangements, explaining how they enable you to fulfill your mission.

Hi Ed,

I enjoyed your response once I realised it was in part facetious(?)

Believe it or not, this problem was subject to very extensive critique and many revisions prior to publication. Nonetheless, I can see that issues remain meriting clarification. You asked some good questions!

1) There are Logistanis whose preference could be determined if only you were to ask them the right question. For instance, you could determine the opinion of a liar by asking "Do you chose to vote for B". (Of course, you would have to negate his opinion to discover what he truly wishes and for that you first would need to know whether or not he is a liar). 

Then there are other Logistanis whose opinion you can never determine no matter what question you ask, because their response is never based on personal beliefs, but changes with the wind (so to speak). The votes of members of this second category are to be discarded.

In fact, this distinction is very simple. It's only hard to express. But I hope it's clear now.

2) You also asked whether the Logistanis vote sequentially or would acclaim their choice simultaneously (perhaps with a decibelmeter on hand?). The answer to this is that you should tell me. The problem states that you can describe all such procedural issues.

3) Certain Logistanis rely exclusively on the opinion of others to form their own expression of preference. You can assume that members of this group are able to distinguish between each of the ethnic groups.

General comment (to all)

This problem is one of those which is relatively simple once you "get it" (honestly, it's a lot easier than it looks!). But it could also be devilishly difficult, especially if you make the wrong assumption. Some of the remarks made so far by Ed and other readers demonstrate important insights in the right direction, and I'd like to encourage people to persevere in developing their ideas. I also promise to turn up in a day or two with some hints, so please don't give up and do look in again from time to time!