Wim Ruitenburg's Fall 2011 MATH 1300-101

Marquette University

Department of Mathematics, Statistics and Computer Science

Wim Ruitenburg's Fall 2011 MATH 1300-101

Last updated: November 2011
Comments and suggestions: Email wimr@mscs.mu.edu

Mathematical puzzles are more than just recreation, just as child's play is more than just fun. Mathematical puzzles are excellent training for problem solving skills.

Mathematical puzzles

There is a light bulb in a distant room of our home. All we see before us in our current room are three light switches labeled A, B, and C. Exactly one of these controls the light bulb. It seems that in general (and assuming we are not lucky) that we need two trips to determine which one of the switches controls the light bulb. Find a way to solve the problem with guaranteed success, in just one trip.
This problem of the three switches and the bulb, is not purely a mathematics problem; its solution needs extra principles from physics. It is usual for a solution to a puzzle, to need scientific principles from outside the ones stated in the problem, but they should be kept to a bare minimum, and be obvious from the context of the problem.
Here is one variation of a classic word problem:
- Two women are sitting at a roadside cafe, talking about family. One mentions that she has 3 daughters the product of whose ages equals 36. Remarkably, she mentions, the sum of their ages equals the number on the house across the street. The other replies that this is not enough information to find the ages of the 3 daughters. True, says the one, but note that my oldest daughter has beautiful eyes. Ah, says the other, but then I know your daughter's ages. What are the ages of the 3 daughters?
At the river's edge stand three cannibals and three missionaries. There is a boat that carries at most two people across at a time. For the well being of all, at no time are cannibals allowed to outnumber missionaries on either side or in the boat. How can we get all six across the river?
Here is a variation: A farmer wishes to move a wolf, a goat, and a cabbage across the river. The farmer plus boat can carry only one of the 3 items at a time. If left alone, the wolf will eat the goat, and the goat will eat the cabbage. How does the farmer get all 3 across safely?
Consider the two pieces of paper below, outlined by solid lines. The dotted lines are only an auxiliary grid by which we can see that the first piece of paper is a 4 by 6 rectangle, and the second is a 5 by 5 piece with two opposite corners clipped. Problem: Show how to cut one piece into two parts by cutting with a scissors, so that the other paper can be obtained by putting the two parts together in another way. We seem to make the puzzle above harder by asking to solve the same clip and paste question for the following two pieces of paper: Did we really make it harder? The more complicated picture on the right leaves so few possibilities to try, that it should be easier.
The row of coins puzzle is one of the homeworks, so we skip it here.
We have a cup with 5 ounces of tea, and a pitcher with 5 ounces of cream. We pour some cream into the tea cup. Then we pour some of the liquid from the tea cup back into the pitcher. It happens that both tea cup and pitcher again contain 5 ounces each, although maybe not pure tea or pure cream. Question: Did more tea move from cup to pitcher, or did more cream move from pitcher to cup, or did equal amounts move?

We easily cover an 8 by 8 chess board with 32 dominoes each the size of exactly two fields. For example, we can line up the dominoes in rows. There are very many ways by which we can cover the 8 by 8 chess board with 32 such dominoes. Now for the more complicated puzzle: From the blue chess board below we cut the top left field and the bottom right field. We also throw away one of the dominoes. Question: Can we cover the damaged blue chess board below with 31 dominoes each the size of exactly two fields?

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The `smurfs' puzzle.
- 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 assembly 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 assembly. Question: How often will the dotted smurf come to evening assemblies?
  - 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 next time.
- 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 assembly 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 assembly. Question: How often will the dotted smurf(s) come to evening assemblies?
  - 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 all hey 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 the only 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 assembly.
- 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 assembly 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 assembly. Question: How often will the dotted smurf(s) come to evening assemblies?
  - 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 assembly. 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 assembly 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 assembly. Question: How often will the dotted smurf(s) come to evening assemblies?
  - 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 assemblies. 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.