Category Archives: Puzzle

Hatching the Egg

Hatching the Egg

This week’s Puzzle Corner activity challenges students to rearrange the nine pieces of a paper egg to produce a number of different shapes resembling birds. Doing this will exercise students’ spatial visualization abilities while building their problem-problem solving persistence. Hatching the Egg is patterned after the “Magic Egg” puzzle that appears in the Pieter van Delft and Jack Bottermans book, Creative Puzzles of the World (Key Curriculum Press. Berkeley, CA. 1993).

This puzzle is one member of a large family of challenging, multiple-solution, dissection puzzles. Unlike the penultimate dissection puzzle, tangrams, and the vast majority of other dissection puzzles, Hatching the Egg has pieces that have rounded as well as straight edges. (The pieces of most dissection puzzles are created with straight lines only.) Just like these other puzzles, however, the individual pieces can be placed together in a number of different ways to make interesting, and recognizable, shapes. In the case of Hatching the Egg, over 50 shapes resembling birds can be created. Because this puzzle has so many possible solutions, it can be used over an extended period of time.

This activity comes on two pages. To save on copying costs, the first page includes four puzzles. To make the puzzle pieces more durable, this sheet can be run off on card stock. Students can use envelopes or self-sealing sandwich bags to store the puzzle pieces between uses. The second sheet has outlines of six of the birds that can be made. Once students successfully make one of the birds pictured, they can make a record of their solution by drawing the pieces inside the outline. In the process of playing with the puzzle, students may also discover other bird shapes that are not pictured. When this happens, they should be encouraged to make records of their creations.

Students should be encouraged to work individually on this puzzle and record their solutions as they are discovered. These records can be shared at a later date after all students have had ample time with the puzzle. It is important to caution students not to share their solutions with other students ahead of time—doing so will rob these other students of the joy of discovery.

Click on the egg and carefully cut out the pieces of one of the egg puzzles and then follow the directions below.

Hatching Egg1

Use all 10 pieces of the egg puzzle to make the birds whose outlines are pictured here. When you find a solution, make a record of it. For an additional challenge, try to make some other bird shapes. Over 50 bird outlines can be made! Make a record of each of your creations.

Hatching the Egg 2

 

Solutions

Click the arrow below to view the solutions.

Hatching the Egg Solutions

Relative Reckonings

Relative Reckonings

This week’s Puzzle Corner activity comes from the field of recreational mathematics where people do math just for the fun of it. One of the areas of recreational mathematics is logic. Logic puzzles are usually challenging and are normally resistant to quick and easy solutions.The puzzle presented here, Relative Reckonings, is no exception.

This puzzle challenges you to find out how 23 roles—one grandfather, one grandmother, two fathers, two mothers, four children, three grandchildren, two brothers, one sister, three sons, one daughter, one father-in-law, one mother-in-law, and one daughter-in-law—can be represented by only seven people. It is—or should be—obvious that some of the people have to play multiple roles (e.g., a father is also a son). This answers the first question in the puzzle. The challenge comes in making sense of just who is who, a task that is not as easy as it might first seem.

Relative ReckoningsHow is this possible?

Who are the seven people and what are their relationships to each other?

I hope that you and your class find this Puzzle Corner activity both challenging and enjoyable.

Solution

Click the arrow below to view the solution.

The family relationships described in this puzzle are possible because each person plays more than one role. The son is also a father, the grandson is also a brother, etc. The people present are three grandchildren—a girl and two boys, their mother and father, and their paternal grandparents.

Check Out Our Exciting Puzzle App for the iPad

Check Out Our Exciting Puzzle App for the iPad

AIMS iPad Puzzle App

The AIMS Puzzle Play Book is great for helping you increase your students’ problem-solving skills. But, did you know that The AIMS Puzzle Play Book is also the basis for a fun iPad app?

Oh, it’s true! You can click here to install it.

But before you do, check out the video review below:

If you like the app make sure to give it a rating and review and to also check out the Puzzle Play book.

Coin Capers

Coin Capers

The Puzzle Corner this week has two versions which are variations of a common theme in manipulation puzzles. In each game the goal is to switch the position of two or more sets of coins by moving them within the spaces provided. Coins may only move into adjacent empty spaces. No jumping is allowed, but coins may move in either direction.

In Version One there is a row of 13 squares, with two additional squares above the middle square. In this version, four pennies are placed at the four far-left squares, and four dimes are placed in the four far-right squares. The challenge is to switch the positions of the dimes and the pennies, and to do this in the fewest possible moves. Coins may be moved by sliding them into an empty space adjacent to the one they are in. Coins may move in either direction, but you may not jump coins at any time.
FinalVersionOne

Once you have solved the puzzle, attempt to do it in the fewest possible moves. Describe your solution.

In Version Two there are two rows of 11 squares connected by a single square in the middle. Version Two has two games, both of which require four types of coins. Quarters go in the upper left corner, nickels go in the lower left corner, dimes go in the upper right corner, and pennies go in the lower right corner.
Coin Capers Version Two Image
Game One:
Use quarters, nickels, dimes and pennies. Place four quarters in the upper left squares, four nickels in the lower left squares, four dimes in the upper right squares, and four pennies in the lower right squares. The object is to have the quarters and nickels trade places and the dimes and pennies trade places. The same rules apply as before, coins may move into any adjacent empty space, and no jumping is allowed.
Game Two:
Set up the board the same as in game one, only this time the challenge is to switch the quarters and the pennies and then switch the dimes and the nickels.
You may decide to use both or only one of these versions with your students.

Other objects may be substituted for coins, however, they must be small enough to fit in the spaces on the student sheets, and be of four different types. Colored buttons, different types of beans, or centicubes would all work equally well.

Solutions

Click the arrow below to view the solution.

Students were challenged to switch the position of coins in various arrays using the fewest number of moves each time. For the sake of simplicity, one move is considered to be any time a coin moves, regardless of the number of spaces it moves.

In version one, the pennies and dimes can be made to switch places in 20 moves.

In version two, both games can be solved in a total of 44 moves – 22 moves to switch each set of coins.

If you or your students are able to solve any of the versions in Coin Capers in a fewer number of moves, please send us a description of your solution(s) in the comment section below.

Tangled Hearts

Tangled Hearts

This week’s activity is a disentanglement puzzle. These puzzles range from simple to difficult and most appear, at first glance, to be impossible. Once they are carefully studied, however, solutions usually present themselves.

Since this puzzle is easily made from inexpensive materials, each student should have one. Make a sample copy of the puzzle beforehand to familiarize yourself with the construction process. This not only enables you to give students guidance as they make the puzzle, but it also provides you with an opportunity to try the puzzle yourself!

The hearts below will make two puzzles. Copy the page on lightweight cardstock or oak tag. Each puzzle requires an 80 centimeter length of string, some tape, and two pennies (or other similar-sized objects). Follow the directions below to construct the puzzle.

Large HeartSmall Heart

 

Each puzzle needs a large heart and a small heart.

(For a sturdier puzzle, we recommend that the puzzle be made out of cardstock or oak tag.)

 

1. Cut out the two hearts from above and punch out the holes.

2. Cut an 80 cm length of string and lay it across the large heart as shown.

Step1

3. Thread the two ends of the string through the bottom hole in the large heart from underneath.

Step2

4. Thread both ends through the hole in the small heart. Thread one end of the string through the right hole in the large heart and the other end through the left hole.

Step3

5. Finish you heart by taping a penny, or other similar-sized object, to each of the two loose ends of the string.

Step4

Once the puzzle is assembled, the challenge is to remove the smaller heart without cutting the string or untaping the ends.

If you are successful, the next challenge is to join the two hearts once more without cutting the string or retaping the ends.

Note: If your string becomes too tangled, it’s okay to untape the ends and reconstruct the puzzle according to the directions. 

Solutions

Click the arrow below to view the solutions.

TangledHeartsSolutions

Leprechaun on the Loose

Leprechaun on the Loose

This week’s puzzle activity is another disentanglement puzzle. These puzzles range from simple to difficult and most appear, at first glance, to be impossible. Once they are carefully studied, however, solutions usually present themselves.

Since this puzzle is easily made from inexpensive materials, each student should have one. Make a sample copy of the puzzle beforehand to familiarize yourself with the construction process. This not only enables you to give students guidance as they make the puzzle, but it also provides you with an opportunity to try the puzzle yourself!

To make the puzzle, copy the next page onto a lightweight cardstock or oak tag material. Carefully cut out the Leprechaun and use a hole punch to make holes in the three spots indicated. Cut a one-meter length of string and fold it in half. Take the loop formed by the last step and insert it into one of the top holes. Take the loop underneath and bring it back up through the other top hole. Hold the loose ends of the string together and thread them through the loop and then through the bottom hole. Securely tape these loose ends to a penny or other object that cannot pass through the three holes.

Now that the puzzle is made, the first challenge is to get the string off without cutting it or removing the penny. If you successfully get the string off, the second challenge is to get the string back on the puzzle.

Leprechaun DownloadStudent instructions:

  • Cut out the Leprechaun and use a hole punch to make holes in the three places indicated.
  • Take a one-meter piece of string and fold it in half to make a loop.
  • Insert the loop into one of the top holes and bring it back up through the other top hole.
  • Hold the loose ends of the string together and thread them through the loop and then through the bottom hole.
  • Tape the loose ends to a penny.

Challenge:

  1. Remove the string without cutting it or removing the penny.
  2. If you are successful, try to get the string back on the Leprechaun.

The solution to the Leprechaun on the Loose puzzle is very similar to the solution for the Tangled Hearts puzzle. Click here to see the Tangled Hearts solutions.

Square Off

Square Off

This week’s Puzzle Corner is a modification of a kind of puzzle that can be found on the Nikoli puzzle website. These puzzles consist of square or rectangular grids that have numbers in some of the spaces. The puzzles are called “Nurikabe,” which means, “painting walls.” The object is to fill the grid by “painting walls” between the numbers so that each number is left in a white area containing that many squares. Look at the example below of a simple game in progress.

Square Off

As you can see, each digit is left in a space containing that number of squares. The puzzles on the Nikoli website are designed so that there is only one possible solution for each. They also have additional rules about the filled-in spaces that do not apply to our version. In the puzzles presented here, multiple solutions are possible for all of the grids, and manipulatives are provided to facilitate the solution-discovery process. Our grids are also fairly small—five by five and six by six—which greatly reduces the difficulty.

Distribute the student pages and some kind of small manipulative to students. The manipulatives can be Area Tiles, Math Chips, or any other small objects that will fit in the squares on the page. The use of manipulatives allows students to try finding solutions without leaving a record of their mistakes. It also facilitates the discovery of multiple solutions, as they can easily rearrange the “walls” in different configurations.

Students should work on these puzzles individually until they have found at least one solution for each. You may wish to have them work on the puzzles for small amounts of time each day for several days. At the end of this time, a class discussion should be held during which students share their solutions. At this time, you may wish to compile a master list of solutions for each puzzle and try to determine if all possibilities have been discovered.

If your students are interested in a more difficult challenge, have them try designing their own puzzles.

They can try to design puzzles that have multiple solutions and those that have only one possible solution. Students can also be encouraged to visit the Nikoli website and try its version of the puzzle. Remember, they use bigger grids, and there are some different rules that make finding the solution more difficult.

Click here to download puzzle.

Solutions

Click the arrow below to view the solutions.

Students were challenged to enclose the numbers in square grids in white areas containing that number of squares. Two possible solutions for each grid are given here.

SquareOffSols

Tower Trade

Tower Trade

Tower Trade is a paper adaptation to the traditionally wooden Towers of Hanoi puzzle. In the classic version, a wooden base supports three equally spaced dowels, usually aligned in a row, although the earliest version of this puzzle is reputed to have had the dowels arranged in an equilateral triangle. Six – this number varies — circular disks of differing diameters are stacked on one of the three posts. The disks are arranged by size with the largest diameter one on the bottom and the smallest one on top. The challenge in the puzzle is to transfer all six disks to another post while following two rules:

  • only one disk may be moved at a time and
  • a larger diameter disk may never be placed on top of a smaller diameter one.

While this is a difficult task, most people can do it if they are persistent. The real challenges when working with the Towers of Hanoi are learning its patterns and finding a way to transfer all the disks in the minimum number of moves.

The best way to work with the Towers of Hanoi is to have a wooden version of this puzzle for every student. My wife did this by having her father, a retired educator who enjoys woodworking, build 33 of the puzzles for her class. He made them quite inexpensively out of particle board. Richard Thiessen, AIMS President and our resident woodworking expert, makes his towers out of medium density fiberboard (MDF). MDF, which is used to make cabinets, is readily available and is only slightly more expensive than particle board. MDF has a couple of great advantages over particle board – it saws cleanly and it doesn’t leave the rough edges the particle board does. To make the construction process much faster and easier, both Richard and my father-in-law cut square, rather than circular, disks. If your local high school has a woodshop, it might be worth seeing if you can get a class set of the puzzles made for you. (See * at the end of this post for the dimensions to Richard Thiessen’s version of the Towers of Hanoi.)

Since many of you do not have access to a woodshop, I am indebted to Richard Thiessen for the viable idea of a paper version which is presented here. Each student needs a set of six different-sized rectangles which represent the six disks viewed from the side. The second sheet (base) contains four sets of “disks” which can be copied on construction paper or tagboard and then cut out.

This puzzle can be modified for different grade levels simply by changing the number of disks. With six disks, it takes a minimum of 63 moves to transfer the tower to another post. This might be fine for older students, but too frustrating for younger ones. Four disks (15 moves) or five disks (31 moves) might be more appropriate for younger students. As mentioned earlier, every student needs to have a Towers of Hanoi puzzle. Provide lots of time for students to work with this puzzle and challenge them to transfer the disks in the fewest moves possible.

Download and copy this sheet on construction paper or tagboard. Each student will need to cut apart one set of rectangles for the Tower Trade puzzle.

Rectangles Tower Trade

Using your rectangles, build a tower on the left post as in diagram below. The largest rectangle must be on the bottom and the others stacked on top of it, ordered by size with the smallest on top.

Tower Trade 2

Obtain Base Here

The challenge in this puzzle is to transfer all the rectangles to a different post while following these rules:

  1. Move only one rectangle at a time.
  2. A larger rectangle can never be placed on top of a smaller one.

*Base – 12 inches by 3 3/4 inches by 3 3/4 inches.
Disk – Six, ranging from 1-inch square to 3 1/2-inch square, all 1/2 inch thick.
Posts – three 3/8-inch dowels 4 inches long in holes drilled at, 2, 6, 10 inches along the center line of the base.

Linking Loops

Linking Loops

This week the Puzzle Corner activity utilizes concepts from one of the most recent fields in mathematics – topology. Topology is the study of geometric properties that are not affected by changes in size and shape. This includes the study of knots, inside and outside, networks, and the transformation of shapes and surfaces.

Linking Loops is a classic disentanglement puzzle that has been around for many years. If any of your students are familiar with it and know how to solve it, be sure to ask them not to give away the answer to their classmates.

Students will need to work in pairs for this activity. It is best if you pair boys with boys and girls with girls. Each student will need a piece of string about 150 cm long. The string should be at least the thickness of kite string. Students will tie a loop at both ends of their string through which they will slip their hands. These loops should fit loosely. Check to see that all students are properly connected to their partners, as shown in the diagram on the student sheet.

At first, this puzzle tends to be fairly difficult for students, and they will likely spend a lot of time going between legs and over heads in attempts to separate themselves. At some point, however, students should realize that no amount of acrobatics or contortions will separate them from their partners, and begin to think about the problem logically. You will probably need to spread this activity over the course of a week or more to give all pairs a chance to discover the solution for themselves. Encourage students who do discover the solution to keep it a secret so that each group can experience the satisfaction of solving the puzzle.

loopsInstructions
1Take your piece of string and tie a loop in each end so that your hand can fit easily through the loop, with room to spare.
2. Put your left hand through one of the loops.
3. Link your string with your partner’s string, and place your right hand through the remaining loop as illustrated.

loopgrls

 

 

Challenge: You and your partner should now be joined by your strings. The challenge in this puzzle is to detach yourself from your partner without cutting either string or removing your hands from the loops at any time. Good luck!

 

 

I hope you and your students have fun as you try to discover the solution to this topological challenge. In fact, you may want to bring a camera on the day that you assign this puzzle to capture some of the interesting and often humorous positions that students will get themselves into.

Solution

Click the arrow below to view the solution.

LoopSolutions

Transforming Toothpicks by Two

Transforming Toothpicks by Two

This week’s Puzzle Corner is an original toothpick puzzle which will challenge students’ problem-solving and critical-thinking skills while helping improve their geometric vocabulary. This puzzle, and others like it, also stretch students’ spatial-visualization and manipulation abilities. As you present puzzles of this nature to your class, you will find that your students possess a wide range of spatial abilities. Often, students who are strong spatially are not the top students in the class. Puzzles like this give them a chance to excel while some of the traditionally “good” students may struggle a bit more. As with any puzzle, persistence is the key, so encourage students to persevere until they find a solution.

Each student will need 14 flat toothpicks and a copy of the student sheets. (Flat toothpicks are preferable to round toothpicks because they will not roll.) There are two presented, a set of warm-up challenges and a set of main challenges. Depending on the age and spatial development of your students, you may wish to present only one of these.

The warm-up challenges (student sheet one) use only 12 toothpicks, and all involve removing two toothpicks to leave specified shapes. Most of these warm-up challenges have multiple solutions. The warm-ups are fairly simple and few in number, so it should only take students one class period to discover at least one solution for each challenge.

The main challenges (student sheet two) use 14 toothpicks, but the basic 12-toothpick arrangement is the same as in the warm-ups. The difference is that for each challenge, two toothpicks from the original arrangement must be moved, and two must be added to create the specified shapes. Again, most of the challenges have multiple solutions. Because there is a total of 12 challenges, some of which may prove quite difficult, the second student sheet should be worked on over a period of several days to give students time to solve each one. Having several days to work on the problem will also reduce the frustration students feel if they are not able to discover solutions immediately.

All of the challenges use geometric terms such as triangle, parallelogram, trapezoid, irregular, congruent, etc. Be sure to introduce any vocabulary or concepts your students may not be familiar with before beginning the activity.

Students should be encouraged to sketch each solution that they discover for a particular challenge. These solutions can be sketched on the back of the student pages or on sheets of scratch paper. At the end of the puzzle-solving time, students can compare the different solutions they were able to discover for the various challenges. You may even wish to use space on a wall or bulletin board to compile a master list of the different solutions discovered by the class.

This activity can easily be extended by modifying the rules in a variety of ways (move three toothpicks and add two, move one toothpick and add three, etc.), or by having students create their own puzzles using the rules given. Encourage students to create their own challenges and explore beyond the restrictions of the puzzle.

Arrange 12 toothpicks on your desk to form the shape shown below.

transform2

Warm-Up Challenges

  • Remove two toothpicks to leave exactly four congruent triangles.
  • Remove two toothpicks to leave three congruent triangles and two congruent parallelograms.
  • Remove two toothpicks to leave two congruent triangles and two congruent parallelograms.
  • Remove two toothpicks to leave exactly three congruent triangles and a parallelogram.

Make a sketch of each solution you discover. There may be more than one way to solve a challenge.

Take 14 toothpicks, and arrange 12 of them to form the shape shown below.

transform3

Challenges
Move two toothpicks in the arrangement and add the two extra toothpicks to make:

  • Two congruent triangles, two congruent parallelograms, and an irregular pentagon.
  • Two congruent triangles, two congruent parallelograms, and a trapezoid.
  • Two congruent triangles, two congruent parallelograms, and two congruent hexagons.
  • Three congruent triangles and two congruent trapezoids.
  • Three congruent triangles and two non-congruent parallelograms.
  • Three congruent triangles, two congruent hexagons, and a trapezoid.
  • Four congruent triangles and an irregular hexagon.
  • Four congruent triangles and two congruent parallelograms.
  • Five congruent triangles and a parallelogram.
  • Exactly six congruent triangles.
  • Six triangles, not all congruent, and two congruent parallelograms.
  • Seven triangles, not all congruent, and a trapezoid.

Sketch each solution you discover. There may be multiple solutions for several of the challenges.

Solution

Click the arrow below to view the solution.

In Transforming Toothpicks by Twos, students were challenged to take a hexagonal array of toothpicks and create various geometric shapes by removing or moving and adding toothpicks. Click here to download a solution for each challenge. Keep in mind that there may be several solutions possible for a given challenge. The dashed lines indicate the toothpicks that were moved/removed.