Marquette University

Department of Mathematics, Statistics and Computer Science

Wim Ruitenburg's Fall 2004 MATH025.1001

The Dots Problem

There are lots of variations upon the Dots Problem. There is a situation sketch in the book, pages 11-12. Some players, standing in a circle or so, may each have or not have a dot on their foreheads. Following certain additional rules, each person is to guess whether or not he or she has a dot on his/her forehead, or what the color of their dot is. The standard solutions use that one or some of the players can deduce from the behavior of other players what these other players `must' think, and so discover what kind of dot, if any, is on their own forehead. We illustrate the validity of such problems through the following sequence of examples.

In the village there live 100 smurfs. It is known to all smurfs that one of them has a black dot on his cap, but none of the smurfs knows which one. Every evening they come together for a quiet get-together during which no smurf is allowed to talk or signal to other smurfs. Once a smurf knows that he has a black dot on his cap, he will not come to the next get-together. Question: How often will the dotted smurf come to evening get-togethers?
- Answer: Once. The first time all smurfs show up. Most smurfs will see the one smurf with a dot, so they know that they themselves have no dot on their caps. One smurf will see no dots on caps. So he knows that he is the one with a dot on his cap. He will not show up for the next evening.
In the village there live 100 smurfs. It is known to all smurfs that one or two of them have a black dot on their caps, but none of the smurfs knows which one(s). Every evening they come together for a quiet get-together during which no smurf is allowed to talk or signal to other smurfs. Once a smurf knows that he has a black dot on his cap, he will not come to the next get-together. Question: How often will the dotted smurf(s) come to evening get-togethers?
- Answer: Once if there is one dotted smurf. twice if there are two dotted smurfs. Why?
  - Suppose there is one dotted smurf. The first time all smurfs show up. Most smurfs will see the one smurf with a dot, so they know that they may be the one with a possible second dot. One smurf sees no dots. He will not show up for the second evening. When the second meeting comes about, the remaining smurfs see the absence of the dotted smurf. So the dotted smurf must have known that he had a dot. This can only be because they themselves had no dots.
  - Suppose there are two dotted smurfs. The first time all smurfs show up. Most smurfs will see two smurfs with a dot, so they know that they have no dot. Two smurfs see one dot, so they know that they may be the one with a possible second dot. When the second meeting comes about, the two smurfs who saw one dot, now see that this one dotted smurf is still present. So these two smurfs know that the one dotted smurf they see, must also see a dot, or else he would not have shown up. Now they know that they also have a dot on their caps. So they will not return after this second get-together.
In the village there live 100 smurfs. It is known to all smurfs that one, two, or three of them have a black dot on their caps, but none of the smurfs knows which one(s). Every evening they come together for a quiet get-together during which no smurf is allowed to talk or signal to other smurfs. Once a smurf knows that he has a black dot on his cap, he will not come to the next get-together. Question: How often will the dotted smurf(s) come to evening get-togethers?
- Answer: As often as there are dotted smurfs. Why?
  - Suppose there is one dotted smurf. The first time all smurfs show up. Most smurfs will see the one smurf with a dot, so they know that there is either one dotted smurf, or there are two including themselves. One smurf sees no dots. He will not show up for the second evening. When the second meeting comes about, the remaining smurfs see the absence of the dotted smurf. So the dotted smurf must have known that he had a dot. This can only be because they themselves had no dots.
  - Suppose there are two dotted smurfs. The first time all smurfs show up. Most smurfs will see two smurfs with a dot, so they know that there are either two dotted smurfs, or there are three including themselves. Two smurfs see one dot, so they know that there is either one dotted smurf, or there are two including themselves. When the second meeting comes about, the two smurfs who saw one dot, now see that this one dotted smurf is still present. So they know that there are two dotted smurfs including themselves. So they will not return after this second get-together. By the third meeting, the remaining smurfs see that the dotted ones are gone. So the dotted smurfs must have seen one dotted other one outside themselves. Thus the remaining smurfs know that there were two dotted smurfs, all now gone.
  - Suppose there are three dotted smurfs. The first time all smurfs show up. Most smurfs will see three smurfs with a dot, so they know that they have no dot. Three smurfs see two dots, so they know that there are either two dotted smurfs, or there are three including themselves. If there were only two dotted smurfs, then these will not show up after the second meeting. If they are still present at the third meeting, then the three smurfs who see two dots, now know that they must have a dot too. So they will not show up for the next meeting.
In the village there live 100 smurfs. It is known to all smurfs that at least one of them has a black dot on his cap, but none of the smurfs knows which one(s). Every evening they come together for a quiet get-together during which no smurf is allowed to talk or signal to other smurfs. Once a smurf knows that he has a black dot on his cap, he will not come to the next get-together. Question: How often will the dotted smurf(s) come to evening get-togethers?
- Answer: As often as there are dotted smurfs. Why? We show this in two steps.
  - First we show that if a smurf sees m dotted smurfs, then he will not show up after meeting m+1. This is our claim. We prove this claim by induction on m. If m = 0, then the smurf sees no dots, so he knows he has the one and only dot. He will not return after meeting 1. Suppose that we have shown the claim for all values of m = 0 through m = n-1. Now we show the case for m = n. This is the induction step of our proof. Suppose a smurf sees n dots, with n at least 1. If he has no dot himself, then the dotted smurfs will see n-1 dots. So these dotted smurfs will not return after meeting n (Here we used the induction assumption.). However, if the n dotted smurfs do return after meeting n, then they must see n dots too. So the smurf knows he himself has one of the dots that they see. So he will not return after this meeting n+1. This completes the proof by induction.
  - Now the second and final step. Suppose there are n dotted smurfs. The undotted smurfs see n dots, while the dotted smurfs see n-1 dots. The dotted smurfs will not return after meeting n. That next meeting, the undotted smurfs know that they have no dot.
The above result implies that if there were 67 smurfs with a dot, then they would not show up exactly after 67 get-togethers. Such results assume that the smurfs have very profound and identical reasoning skills. The proof also suggests that there is only one way of sound reasoning. Such `hidden' assumptions may not be acceptable to all.

Last updated: December 2004
Comments & suggestions: wimr@mscs.mu.edu