This week’s puzzle has been around for many years. It is one of a family of puzzles which have varying degrees of difficulty. These puzzles usually have three common characteristics. First, they all involve getting something across a river (or pond) in a boat. Second, one or more of the things in each puzzle is in danger of being eaten if the puzzle-solver is not careful. Lastly, the puzzles are intended to be done as thought problems without any manipulatives.
By modifying this last characteristic, and allowing manipulatives to be used, many of these puzzles can be done by elementary students. I have used the puzzle presented here quite successfully with students from second grade up.
In The Frustrated Farmer, a farmer must transport a fox, a chicken, and bag of corn across a river in a row boat. She can only take one thing at a time. If the fox is left alone with the chicken (on either side of the river), it will eat the chicken. Likewise, if the chicken is left alone with the bag of corn, it will eat the corn. To solve this puzzle, you must find a way for the farmer to get the fox, the chicken, and the bag of corn safely across the river.
To make this puzzle easier for students to solve, we have included pictures of the farmer, fox, chicken, and bag of corn. We suggest that these be laminated and each one taped to a pencil or piece of dowel. A scrap piece of lumber can be made into a boat by sawing a point on one end and drilling two holes (the same diameter as the dowels or pencils) for the passengers. The river is made by cutting a piece of butcher paper in the shape of a meander. The river is glued to a piece of green butcher paper which forms the two banks.
When students have successfully solved the puzzle, they should record their solution in words and/or pictures.
In this puzzle you must help a farmer get a fox, a chicken, and a bag of corn safely across a river in a boat. The farmer may only take one thing at a time in the boat. She cannot leave the fox and the chicken together on either side of the river, or the fox will eat the chicken. Likewise, she cannot leave the chicken alone with the bag of corn or the chicken will eat the corn. How can the farmer get everything across the river without anything being eaten?
Click the arrow below to view the solution.
In The Frustrated Farmer, a farmer needs to find a way to transport a fox, a chicken, and a bag of corn across a river in a boat. She could only take one thing at a time. If the fox was left alone with the chicken (on either side of the river), it would eat the chicken. Likewise, if the chicken was left alone with the bag of corn, it would eat the corn. To solve the puzzle, you were to find a way for the farmer to get the fox, the chicken, and the bag of corn safely across the river.
One possible solution to this puzzle follows. The farmer first takes the chicken across the river and leaves it on the other side. She then comes back and picks up the fox and takes it to the other side. When she gets there, she drops off the fox and brings the chicken back with her. She then drops the chicken off and picks up the bag of corn which she takes across the river. Finally, she goes back and picks up the chicken and takes it across the river one last time. When this is done, she has safely transported everything across the river and is able to go on her way.
Three in a Row is a two-person game played on a 3 x 3 square grid using six markers. Each player needs three markers which can be easily distinguished from those of the other player. Coins, beans, buttons, checkers, math chips, or any other small manipulatives will all work well. The object of the game is to get three markers in a row horizontally, vertically, or diagonally. Players begin by taking turns placing their markers into empty spaces on the grid one at a time. Once all six markers have been placed, if neither person has three in a row, play continues by sliding the markers from one square to another. Markers may slide into any adjacent empty space horizontally or vertically, but may not move diagonally. The game ends when one player succeeds in getting all three of his or her markers in a row horizontally, vertically, or diagonally, or a stalemate is reached.
A stalemate is possible when one or both players simply move the same marker back and forth between two spots. If this begins to occur with some frequency, a rule can be made that you are not allowed to move your marker back into the spot it just vacated until one round has passed. This will prevent students from repeating the same move over and over, keeping the game from progressing.
As students play this game, they will soon discover that the strategy lies not only in how you move your markers once they are on the grid, but in the original placement of those markers. After several rounds have been played, encourage students to stop and evaluate the strategies that they are using. Which player is winning more often? What is that person doing differently? Are there certain moves/locations that will always let you win?
Six markers, three each of two kinds
To get three markers in a row horizontally, vertically, or diagonally.
1. Each player has three identical markers. Players take turns placing their markers into any empty square in the grid.
2. If neither player has three in a row once all of the markers have been placed, the game continues by sliding the markers.
3. Markers may slide horizontally or vertically into any adjacent empty space. Markers may not move diagonally.
4. Play continues until one player succeeds in getting three markers in a row.
We hope you and your students have fun playing this new game. If you have any questions or comments about this or any other Puzzle Corner activity, please feel free to leave a comment below.
How can you explain the apparent paradox of the double Möbius strips?
The Möbius loop is a topological surface first discovered by August Ferdinand Möbius in 1858. Möbius was a mathematician and professor of astronomy whose work in topology revolutionized the field of non-Euclidean geometry. A Möbius loop can be constructed by connecting two ends of a strip of paper after giving one end a half twist. This results in a baffling surface which has only one side and one edge. The Möbius loop has been immortalized by artists like M.C. Escher, who used it in his print Moebius Strip II, which depicts ants marching in an endless line around a Möbius loop. It also has practical applications in the industrial world, where the large belts in some machinery have been designed with a half twist so that both sides get equal wear. This puzzle presents a fascinating variation of the Möbius loop in which two apparently disconnected loops turn out to be joined together. Students will be challenged to explain this phenomenon as they explore topology using the Möbius loop.
1. This puzzle works best if you construct a model in front of the class, move the pencil between the two loops to show that they are not connected, and then try to pull them apart, showing that they are, in fact, connected.
2. When moving the pencil between the two loops, you will find that after one rotation the pencil will be facing the opposite direction than it was when you started. It is necessary to make two complete rotations to return the pencil to its original orientation. This realization is an important part of explaining the puzzle, and students should be allowed to make the discovery for themselves without having it pointed out to them.
3. Once you have demonstrated the puzzle for the class, give students the necessary materials and have them construct their own version of the puzzle. It is better if the paper students are using is plain so that it is more of a challenge to distinguish between front and back.
Cut two identical strips of paper that are about 11 inches long and one inch wide.
Place one strip on top of the other, holding at the end between your thumb and first finger.
Give the strips a half twist and bring the end together.
|Tape the ends, together -top to top and bottom to bottom. You should now have two Möbius loops nested right next to each other.||Take a pencil and place it between the two loops.||Move the pencil around the loop one time until it returns to the place you began.|
Answer these questions after you have made your loops and followed the directions of the first worksheet.
1. What direction is the tip of the pencil facing now?
2. Is this the same or different than the direction it was facing when you began?
3. Move the pencil around the loop one more time. Now what direction is it facing?
4. Pull the two loops apart. What happens?
5. How can you explain this?
Coloring each side of each strip of paper a different color before the band is assembled can help students see which strips are being attached to each other.
Click the arrow below to view the solution.
In order to understand what is happening with the two strips of paper, it is important to examine how they are taped together. Before the two strips are twisted and taped together, they are placed one on top of the other. Each strip in the beginning configuration has a left and right end. (It might help to label these ends beforehand: TL, TR, BL, & BR..) When the strips are given a half twist and their ends joined, the right end of the top strip ends up being taped to the left end of the bottom strip. Likewise, the left end of the top strip ends up being taped to the right end of the bottom strip. In this way the finished product appears to be two separate Möbius loops nested within each other when in actuality they form one large loop with two half twists. This means that when the pencil is inserted between the two pieces of paper, it must travel twice around the loop to return to its original orientation.
This puzzle comes from a rich historical tradition that dates back to the 19th century when matches were first manufactured. Invented in 1827 by the British chemist John Walker, matches soon replaced the tinder boxes and flints that people had formerly used to light fires. As matches grew in popularity and became ubiquitous later in the century, they spawned a new form of entertainment—matchstick puzzles—which became quite popular when several match companies printed these puzzles on their boxes. Capitalizing on this interest, publishers began to print books of matchstick puzzles. Near the end of the 19th century, many people had developed a personal repertoire of these puzzles that they used to challenge friends and acquaintances. The toothpick puzzle presented here is modeled after the classical matchstick puzzles. For safety reasons, these puzzles use ﬂat toothpicks instead of matches. (Round tooth-picks are not recommended, as they tend to roll.)
This puzzle has six challenges, each of which starts with 36 toothpicks arranged to form 13 small squares. Students then either move or remove a given number of toothpicks to form the numbers of geometric shapes stated. Once students have a solution, they should use the dot paper to record it. The dot paper can also be used to solve the challenges if toothpicks aren’t available—students can simply draw the figure and then erase lines instead of removing or moving toothpicks. Several of the challenges have multiple correct answers. Students can be encouraged to find all of the possible answers for each challenge.
The challenges presented here require patience and persistence to solve. However, they tend to be a bit easier for students who have well-developed spatial-relationship skills. Often, these students are not the top students, and their ability to solve these puzzles faster than their peers is a great esteem builder. Conversely, this type of puzzle often frustrates those students who usually do well at traditional school tasks and provides them with a real challenge. This role reversal is often beneficial for both sets of students.
In addition to using their spatial skills, students can also utilize logical thinking when working on these puzzles. While each challenge may eventually be solved by trial-and-error, taking a few minutes to think logically about the problem will often reveal the solution. Another key puzzle-solving trait that students will need to develop when working on these puzzles is persistence—students can’t solve a puzzle if they give up. You will need to encourage students to be persistent and to keep trying until they solve the puzzles.
As the teacher, you are encouraged to try the puzzles yourself before giving them to your students. You may find them difficult, as do many adults who are linear thinkers, but don’t assume that your students will experience this same level of difficulty. You may be surprised at how well some of them do with these puzzles. Good luck!
Arrange 36 flat toothpicks to form 13 small squares as shown below. Use the toothpicks to solve the challenges.
For readers of this post, the above title may not be an oxymoron, but for the majority of the population, finding recreation in mathematics is beyond comprehension. Most can’t begin to imagine doing math “just for the fun of it.” This negative attitude towards math is common in our culture and is worn as a badge of honor by those who boldly state, “I was never any good at math and it didn’t hurt me.”
In math circles, the dislike or fear of math is not an issue. Math people enjoy mathematics and try to pass on this attitude to others. Jim Wilson, a colleague and mentor here at AIMS, encourages teachers to foster positive feelings about math by having interesting math-related things or objects in their classrooms. Math conferences are great sources of these types of items. The art of the late Dutch graphic artist M. C. Escher is a staple at these events. His tessellations and impossible objects appear on posters, coffee mugs, and t-shirts. These items are sold out quickly as attendees buy them to display in their classrooms or on their persons.
One of my favorite Escher prints—and one that adorned by classroom wall for many years as a math-related interesting object—is Ascending and Descending. This 1960 dated print features an impossible staircase that defies logic.
In this print, monks going clockwise around the staircase never reach the top and are locked in an infinite ascent. Likewise, the monks going counterclockwise are mired in a less taxing, but no less monotonous, never-ending descent.
People familiar with Escher’s art often assume that he invented this illusion. While Escher did indeed make the impossible staircase widely known to the world with this print that has sold millions of copies since 1960, the original impossible staircase came from the field of recreational mathematics.
In 1958, L. S. Penrose, a geneticist, and his son, Roger, a mathematician, co-authored an article published in the British Journal of Psychology describing impossible objects. One of these was the impossible staircase.
This staircase is drawn in such a way that when the eye moves around it in a clockwise direction it appears to have no bottom step. If the eye follows it in an anticlockwise (the British term for counterclockwise) direction, there is no top step. This father and son team invented this interesting mathematical object just for the fun of it, and wanted to freely share it with others through the article.
Escher, who was keenly interested in impossible worlds and objects (his earlier works are full of examples of these types of things), came across the Penrose article shortly after it was published. He then used the Penroses’ mathematical creation—the impossible staircase—to produce Ascending and Descending.
There is some irony in this story. The inventors of the impossible staircase got no remuneration for their creation (there is no compensation for being published in peer-reviewed journals), while the great success of the print added considerably to Escher’s wealth. Even more ironic, in my opinion, is that although Escher credited the Penrose article for his inspiration, most people (including most math people) don’t know the backstory and believe that the artist invented this unique mathematical object.
What motivated the Penroses was not money, but the enjoyment they got from creating interesting mathematical objects. Both were deeply involved in the field of recreational mathematics—doing math just for the fun of it.
Click on the link for more information on Sir Roger Penrose and his involvement in recreational mathematics.
In an effort to promote recreational mathematics, I am sharing the AIMS version of the Penrose impossible staircase from the Puzzle Play book. The puzzles and interesting mathematical objects in this book, which I co-authored with my daughter, could be an entre into the field of recreational mathematics for you or your students. Enjoy!
The Puzzle Corner this week comes from the great American puzzle genius of a century ago, Sam Loyd, and was originally published with the name “The Royal Road to Mathematics.” Shape It Up, as we have renamed it, is similar to tangrams, but uses only five pieces that are all different from each other, unlike the seven tangram pieces which include several duplicate shapes. In this activity, students are challenged to cut out the five shapes and use them to make eight geometric figures. (In order to help you conserve paper, each student will need only half of the first student page.)
This activity will challenge students’ spatial visualization abilities as they learn to see how the five puzzle pieces can be put together to form each of the eight figures. Students can be told that puzzle pieces may be flipped over however, there is at least one solution for each shape which does not require this. For older students there is also the potential to discuss geometry as they identify the characteristics of the figures they create. Once students have solved the eight figures given, they can be challenged to create their own figures and trade them with classmates to solve. In fact, if your students come up with any especially creative or unusual figures using the five puzzle pieces, send us copies of their work.
Make each of the following figures with your five pieces. Each figure must use all five pieces. Make a record of each solution you discover. (Some figures may have more than one solution.)
Extra challenge: Create some more irregular shapes like the Seahorse and the Rabbit. Make a picture of each, and trade them with your classmates to solve.
Click the arrow below to view the solutions.
Here are some of the possible solutions for this puzzle. Others are possible.
This week’s Puzzle Corner activity is an adaptation of a classic puzzle from recreational mathematics. It is traditionally posed as a thought problem to be worked out in your head; as such, it is moderately difficult. However, I have found that many elementary school children can solve this puzzle -if they have manipulatives to make it concrete.
Most traditional versions of this puzzle show a picture of four separate pieces of chain, each with three links. The reader is challenged to find a way to join the four pieces into a circle by opening and closing only three links. Although this might seem impossible at first glance, persistent puzzle solvers are usually able to find a solution.
While the above thought problem is too difficult for many elementary students, they can often solve it if they are given plastic or metal snap rings (I used binder rings purchased in a stationary store) with which to model the problem. Since I rarely have a large enough supply of snap rings to do the puzzle as whole-class activity, I usually set it up at a center. I explain the challenge to the entire class on Monday and then let students go to the center at various times during the week. I usually set up two sets of rings so that more than one student can be at the center at a time. I ask students who have successfully solved the puzzle not to give the answer to others. On Friday, I let those who have solved the puzzle share their problem-solving processes with others in the class in a whole-class session.
Your mother has four separate pieces of gold chain, each with three links, that she wants to have made into a bracelet. The jeweler charges $10 for each link she has to open and solder closed. What is the minimum amount it will cost to have the bracelet made?
Hint: Use 12 snap rings to model the problem.
Click the arrow below to view the solution.
In It’s A Snap students were asked to find the minimum cost of making a necklace out of four sections of chain, each with three links, if it cost $10 to open and solder each link.
While the answer might seem to be $40 at first glance, the necklace can actually be made for $30. This is done by opening all three links in one of the sections and using each link to connect two of the remaining sections of chain. This way the necklace can be made by opening and soldering only three links.
This week’s Puzzle Corner was inspired by a puzzle in the March 2006 Games magazine. The original version used the digits one through nine in a nine-by-nine grid. Our version uses five-by-five and six-by-six grids, making it much simpler than the original. The challenge is similar to that of the popular Soduku puzzles. Numbers must be arranged in a square grid so that each number appears only once in each row and column. However, the square grid is divided into smaller rectangles, and each rectangle contains a small number. That small number indicates the sum of the numbers that fill the squares making up the rectangle. Thus, unlike a Soduku,
which is mostly logic and process of elimination, Connect Squares involves the use of arithmetic.
To make the problems easier for students to solve, it is recommended that they use small slips of paper marked with the appropriate numbers. These can be moved around in the grids until the solution is reached. Once the numbers are in the correct places, the solution can be written on the paper to make a permanent record.
When students have solved the problems, encourage them to share the processes they used to solve them. It is important for students to be able to communicate mathematically, and they may be surprised to discover the variety of strategies used by their classmates. A sample problem and a discussion of a strategy that can be used is given here as an illustration.
Look at the top row. The two and five are already in place. The next two squares must sum to four, which means they must contain the one and the three. (At this point, it is not clear which spot is the one and which is the three.) This leaves the four to go in the top right corner. With the four in place, the number below it can be easily identified as a three (4 + 3 = 7).
Now examine the far right column. With the four and three in place, you know that the remaining numbers must be one, two, and five. Of those, only one and five can total six. That means the bottom right corner must be two, and the square to the left of that must be three.
The only two-number combination (using the numbers one to five) that can have a sum of eight is five and three. (Four and four cannot be used because the same number cannot appear twice in any row or column.) Therefore, the bottom left corner must be five because there is already a three in that row, which means the number above it is 3 (3 + 5 = 8).
When using the numbers one to five, there is only one two-number combination that has a sum of nine (4 + 5), so you know that the second row, second column must be a four and the second row, third column must be a five. That means that the first column can be completed as well.
The second row can be completed with a two by a process of elimination. The top row can be completed because you know the three cannot be in the fourth column; therefore, it must be in the third column.
The fourth column can be completed with a four by a process of elimination. The bottom row can also be completed because you know the four cannot be in the second column; therefore, it must be in the third column.
Look at the fourth row. The only three-number combination that totals seven is one, two, and four. You know the one cannot go in the second column because there is already a one there. The row can be completed with the two in the second column, the one in the third column, and the five in the far right column.
The third row can now be completed. The three must go in the second column, the two in the third column, and the one in the far right column.
Click the images below to download the student pages.
Click the arrow below to view the solutions.
Connect Squares challenged students to fill square grids with numbers so that the same number did not appear more than once in any row or column. They also had to make sure that the small numbers in the grid corresponded to the sums of the numbers in the connecting squares. The solutions are shown here.
Solution 1 Solution 2 Solution 3
This week’s Puzzle Corner activity comes out of a rich historical tradition that dates back to the 19th century when matches were first manufactured. Invented in 1827 by the British chemist John Walker, matches soon replaced the tinderboxes that people had formerly used to light fires. As matches grew in popularity and became ubiquitous later in the 19th century, they spawned a new form of entertainment—matchstick puzzles—that became quite popular when several match companies printed these puzzles on their boxes. Capitalizing on this interest, publishers began to print books of match-stick puzzles. By the turn of the 20th century, many people had developed a personal repertoire of these puzzles and used them to challenge friends and acquaintances. The toothpick puzzle presented here is modeled after these classical matchstick puzzles, but for safety reasons it uses flat toothpicks instead of matches.
This puzzle may require patience and persistence to solve. It will be a bit easier for any students who have well-developed spatial-relationship skills. Often, these students are not the top students and their ability to solve puzzles like this one faster than their peers is a great esteem builder. Conversely, this type of puzzle often frustrates those students who usually do well at traditional school tasks and provides them with a real challenge. This role reversal has the potential to be beneficial for both sets of students.
Your challenge in this puzzle is to move exactly 3 toothpicks in the following arrangement to make 5 triangles. Good luck!
Click the arrow below to view the solution.
The challenge was to move only three of those toothpicks to create a total of five triangles. This can be accomplished by moving the triangle on either the left or right above the other two triangles. This gives you four small triangles and a fifth large triangle.
The Puzzle Corner this week is a new puzzle which I developed while “piddling around” with an activity that I happened to be looking at.
The problem I was working on is called Arranging Rectangles. The puzzle involves cutting out seven shapes and arranging them to form a rectangle. The pieces in Arranging Rectangles have a total area of 18 square units, therefore it is possible to create two different rectangles –a 2×9 rectangle and a 3×6 rectangle. As I considered this puzzle, the wheels began to turn. “How boring that there are only two possible solutions,” I thought, “there should be at least four!” With this idea in mind I set about to create my own puzzle which would have at least four possible rectangles that could be made from one set of shapes. After piddling around with a sheet of graph paper and a pair of scissors, I arrived at this puzzle which has eight pieces that can be arranged to form a 2×18 rectangle, a 3×12 rectangle, a 4×9 rectangle, and a 6×6 rectangle (square).
In Recreating Rectangles, students are challenged to cut out the eight pieces on the page and put them together to create a rectangle. If they are careful in their search for solutions, they will discover that there are four different sized rectangles which can be made using the eight pieces. As the facilitator of the activity, you should not provide students with this information, but allow them to discover it on their own. The extra challenge that is presented simply makes the problem a little more difficult by taking away the freedom to flip the puzzle pieces. All four rectangles are still possible with this restriction.
Once students discover a solution, they should record it on grid paper. Depending on how many solutions an individual student discovers, he or she may need more than one sheet of grid paper, so have some extra copies of this page on hand. You may also want to make colored pencils or markers available for students to use when they record their answers so that they can easily distinguish between the different pieces in each solution. Students should work alone to discover and record as many different solutions as they can. If desired, this problem can be spread over several days so that they can work on it in sections rather than spending a lot of time on it all at once.
Carefully cut out the shapes below and put them together to make as many different sized rectangles as you can. Each rectangle should use all eight pieces. You can flip pieces over if necessary. Draw a picture of each solution you discover on a sheet of grid paper.
Extra Challenge: Once you have discovered a few solutions using the rules above, try to get a solution without flipping any of the pieces over. Record your discoveries on grid paper.
Click the arrow below to view the solutions.
One possible solution for each rectangle is shown below. None of the pieces were flipped over to reach solutions.