Glimpses

How I approach explorations in math class

I love learning math. One element of math I love is exploration: figuring out something new and feeling the rush of realization.

I want to share those experiences with students. So do many math teachers. But the devil is in the details, and the details are where I disagree with a lot of teachers.

Lots of teachers feel the way I do. So, they say, students should learn math through exploration. We should design our classes so that students can discover the big ideas of math for themselves, and learn math while also experiencing what it’s like to be a mathematician.

I disagree. My disagreement is pretty simple: teaching is really hard. If there is an easy button out there I haven’t found it. I work hard, and I still feel like I’ve failed after lots of lessons. Teaching math while also setting students up for productive everyday explorations is even harder. It’s trying to optimize for two variables at once: optimizing learning, which is really hard to begin with, while also optimizing exploration. It just doesn’t work very well. It’s hard to optimize two things at once.

One key here is that every student has a different experience in math class. For many of my stronger students, optimizing learning isn’t very hard. They’ll be fine if we spend a ton of time on explorations. But some of my students arrive with less confidence, less motivation, less prior knowledge. These are the students who need great teaching the most. And these are the students who miss out if I try to optimize for explorations and trade away some of the learning.

I still want to show my students how cool math can be. That’s an important goal in my class. But I don’t want to do it through everyday explorations. I also don’t want to segregate students, and say that the strong students get to explore while other students don’t.

Enter “glimpses.”

Here’s an example (source):

This is fun to do with students! It’s a great puzzle, and not too hard to understand. I wrote a whole post about sharing this problem with my students last year, and I’m looking forward to doing it again this year. These are what I call “glimpses.” They’re short explorations that give students a taste of what math can be. They don’t have any specific curricular goal. And variety is key: they come in all sorts of shapes and sizes, with the goal of appealing to as many students as possible.

Here’s are the key characteristics of how I use glimpses:

• They're pretty short. Rarely a full lesson, typically 5-15 minutes.

• I use them irregularly. Glimpses are occasional, not an everyday routine.

• They’re as accessible as possible. I want every student to have a substantive experience exploring something.

• They are not connected to the curriculum. When we’re doing school math, I’m optimizing for learning. When we’re doing glimpses, I’m optimizing for exploration.

I think of these a bit like how an English teacher might think about getting students to find a book they enjoy. I want to expose students to lots of different glimpses, each a reasonable guess at something they might like. But it’s hard to predict which glimpse will get a student hooked. It might take a lot of failures to find one success. So the goal is to have a lot of irons in the fire, and hope that every student finds a few glimpses that speak to them over the year.

I don’t know the right fraction of math class that should be dedicated to this type of exploration. If you tell me it’s 0%, I will tell you I am sure that you are wrong. We should absolutely be doing this type of math with students. If you tell me it’s 50% I will also tell you I am sure that you are wrong. School math is worth learning. I want to give students glimpses of how cool math can be, but if students aren’t learning equations and percents and geometry I’m not doing my job. I would ballpark this part of my class around 5% right now? But I’m inconsistent. Sometimes it’s more, sometimes it’s less. I haven’t figured it all out. I just know I’m committed to trying these, finding fun stuff to share with students, and hoping to get more students hooked on math.

The List

Below is a very long catalog of my favorite glimpses to share with students.

There are lots of collections of math problems on the internet. What makes these different?

First, they’re probably not that different. One of my goals with these glimpses is that there are lots of links, and you could go down an endless rabbit hole following the links and references and finding more stuff. There’s plenty more out there, this isn’t an exhaustive list.

The big thing that makes these distinct is that they’re not trying to teach some curricular objective. They’re meant to be fun, accessible mathematical explorations. There’s plenty of math in there, but the primary goal is for students to enjoy exploring some math. I’m not stressed about aligning these to any standards or fitting them in to a specific place in your curriculum.

These are also pitched a bit lower than a typical set of problems you might find on the internet. I find that lots of collections of problems are designed more to appeal to math teachers and mathematicians than students. Many of these might seem trivial, like they’d be too easy for students. And maybe they are, for your students. I think there’s stuff here that could appeal to anyone from the elementary grades up through high school. There are plenty of more challenging classics further down the list, but also lots of simple, accessible stuff for younger or less confident students up top. I’ve used most (though not all) of these with students in some form, at some point in my teaching.

Finally, I would argue that there are two big types of mathematical explorations. One type is what I’ll call an insight problem. These are problems that are often opaque at the start, and require a sudden insight to figure out. A second type is what I’ll call a tinkering problem. The paper folding problem above is a good example. Anyone can start by just folding a piece of paper and seeing what happens. These are problems that can be explored gradually, making steady forward progress. The vast majority of the problems on this list are tinkering problems. There are some insight problems in there, and insight problems can be great for students who already have a lot of mathematical confidence. But in general, tinkering problems are a better way to draw students in to mathematical exploration. Some tinkering problems look like math problems. In other cases they look like games or puzzles. They can be a single prompt, or a sequence of connected prompts.

It would be awesome if math teachers saw glimpses like a parallel curriculum that runs through school math. The goal would be to make sure students have seen some of the classic math problems and puzzles by the time they complete their math education. In the same way that we might prioritize ensuring students have read certain poems during the language arts education, we could prioritize sharing these ideas with students. I wanted to put these together as one rough draft of what that might look like.

One more note. There are a few problems that are repeated multiple times throughout this sequence. I think an underrated part of math is revisiting an old problem with new eyes. Not all problems are repeated, but there are some problems that are well-suited to an initial exploration, then a thoughtful extension on a second visit.

Well, here goes. (This list is very long. You’ll probably need to view the web version to see all the glimpses because it’s too long for most email clients.)

101 and You’re Done (link)

101 and you’re done is a group game played in six rounds. In each round, roll a die. You can either add that number to your score, or add that number times 10. For instance, if you roll a 4, you can add 4 or 40. After six rounds the closest player to 100 wins. But if you go over 100, you’re out!

Tips:

• In general the game is played one round at a time, and players must decide which number (i.e. 4 or 40) to add to their score before the next roll. But for a fun variation, you can also roll all six numbers at the start and then decide which numbers to multiply by ten and see who can get the closest to 100.

Nim (version one)

This is the first of a series of Nim-style games, eventually building up to the classic game of Nim described on Wikipedia. In this version, play with two players and a pile of 10 objects. Players take turns. On their turn, each player can either take 1 object or 2 objects. The player who takes the last object (or two objects) wins. After playing this game, you can make up variations by either starting with a different number of objects, or allowing players to take 1, 2, or 3 objects on each turn.

Tips:

• The game is fun on its own, but it’s also fun to try to figure out the ideal strategy. One way to do this is to play with different numbers of objects. What is the ideal strategy if there is 1 object? 2 objects? 3? 4? 5? Look for patterns!

Adsumudi (link)

Adsumudi is a number game with a target number in the middle, and five numbers to use around the outside. The goal is to use the numbers on the outside and the operations addition, subtraction, multiplication, and division to make the target number.

Tips:

• Can you make the target number using two numbers? Three? Four? Five? Each Adsumudi puzzle is possible with two through five numbers.

• The game also includes “Fun Ones” which are always possible with only addition and subtraction (though multiplication and division are still allowed). These are a good place to start when playing with younger students.

• A nice way to structure the game is to start by finding one or several ways to solve it with two numbers. Then, give a bit of think time and try to find a strategy using more numbers.

Tower of Hanoi (link)

The Tower of Hanoi is a puzzle about rearranging shapes. An example is below. The goal is to move the stack from one peg to another. A larger disc cannot be placed on top of a smaller disc, and only one disc can be moved at a time.

Tips:

• Wooden versions can be purchased at many stores, you can play a digital version, or play with coins (for instance a quarter, a nickel, and a penny to start, then add in a dime for a fourth object).

• Start with a small number of discs, and gradually work up to larger numbers.

• What is the fewest number of moves you can use to solve the puzzle with three discs? Four discs? Five?

Two Fold (link)

The image below says it well. With just one fold, you can make a rectangle into a nine-sided polygon. How many sides can you make with two folds?

Tips:

• One way to scaffold this problem is to start with a triangle. How many sides can you make with one fold of a triangle? Two folds of a triangle? Then, how many sides can you make with one fold of a rectangle? Then two folds of a rectangle? How are you sure you’ve found the maximum number of sides?

The Sierpinski Triangle (link)

Start with a large triangle. Draw a triangle in the center of the large triangle. You’ve created three smaller triangles. Repeat the process in each of those smaller triangles. Continue repeating this process as many times as you can. The final result should look like the image below.

Tips:

• With a class of students, have each student create their own Sierpinksi triangle, then cut them out and arrange them into another, larger Sierpinski triangle.

• How many triangles did you make with each step? What pattern do you see?

• Getting the pattern down can take a few tries. Have some extra paper handy.

Nim (version two)

This is another two-player game. Start with 8 objects in a circle. On your turn, you can remove one object, or objects that are touching. The player who takes the last object (or objects) wins.

Tips:

• This game is much harder to figure out than the first version of Nim! Consider playing with 4 objects to look for a systematic strategy, then 6, then 8.

• Why is the game different with even numbers than with odd numbers?

A Math Magic Trick

Have students begin with any number from 1 - 10. Have them perform the following calculations:

Multiply their number by 9

Add the digits of their number together

Subtract 4

Multiply by 2

No matter what number they chose, their final number will be 10.

Why?

Tips:

• Feel free to modify the magic trick. The key is to multiply a single-digit number by 9, and then add the digits of the resulting number.

Magic Squares (link)

A magic square is a grid of numbers, where every row, column, and diagonal adds up to the same number. An example is shown below, where each row, column, and diagonal adds to 3.

Tips:

• A good place to start is to explain the idea of a magic square, then give a partially completed square. 2, 5, and 8 from the example above is often enough to solve a full square. Then, see how many different magic squares students can find in a 3x3 grid using the numbers 1-9. Do the rows and columns always add up to the same number? Why? Do certain numbers have to be in certain positions? Why?

Teeko (link)

Teeko is an old but very fun game that is a bit hard to explain concisely. The link above walks through the full game. There are a few versions of the game out there but the version above is simple to learn and fun to play.

Tips:

• The Wikipedia article describes a few other variations. The version described above is my favorite, but the others are worth playing as well.

Number Snakes (link)

Below is a number snake puzzle and its solution. The goal is to write the numbers 1 to 20 so consecutive numbers are in adjacent squares (no diagonals!). The link has 200 of these fun puzzles.

Tips:

• The first 100 puzzles are 5x4 grids and are simpler. The next 100 are 6x5 grids and are much harder.

• Remember that numbers must be orthogonal, meaning they are connected left to right or up and down, not diagonally.

A Math Magic Trick (number two)

Take any three-digit number. Multiply it by 7. Then by 11. Then by 13. The answer will be two copies of the original number. Why?

Tips:

• Can you create a version of this puzzle for two-digit numbers? Four-digit numbers?

Sprouts (link)

Sprouts is a two-player game, with an example game shown below. Start with a few dots (in the game shown below, there are 2). The first player connects two dots (the line can curve, but not cross any other line). Somewhere on the line they drew, they add an additional dot. Dots can have a maximum of three lines coming out. You can see that after the fourth step, all but two dots have three lines coming out and are “dead.” Since the last two lines cannot be connected, the game is over. The player who makes the last move wins.

Tips:

• Two dots is a good number to learn the game initially, and three is a good number for more interesting and varied play.

• Can you figure out the perfect way to play for two dots?

Inaba Puzzles: Doko Equations Puzzles (link)

This is the first of several puzzles from the prolific Japanese puzzle creator Naoki Inaba. A sample puzzle is shown below. You must write an addition equation with a 6 somewhere in the column indicated, and a 5 somewhere in the row indicated. Numbers can only be placed in boxes. The puzzles begin simply, and gradually get more complex.

Tips:

• Puzzles like these can be a nice routine at the start of a class: solve two or three each day, and gradually work from simpler puzzles to harder puzzles.

101 and You’re Done, Revisited (link)

Revisit the 101 and You’re Done game, described above. This time, roll the dice six times first. Then, decide which numbers you would like to use as tens and as ones. See which scores are possible. What patterns do you see? Why?

Tips:

• This could also be connected to the first math magic trick above

Dots and Boxes (link)

Dots and Boxes is a classic game that can be played with pencil and paper. Two players begin with a grid of dots. The game is described in the link above and is easy to learn, but has a lot of great strategy for students to explore.

Tips:

• Start with a small grid, and work up as students become comfortable with the game.

• Make sure students understand that if they complete a box they get to keep going until they don’t complete a box.

Half Fraction Snake (link)

Can you divide the snake below up into any two sections, where exactly half of each section is a certain color? This is a fun puzzle for thinking about fractions, but the only fraction involved is ½, making it accessible to a range of students.

Tips:

• The puzzle is tricky to explain. It’s helpful to begin by modeling a few examples of chunks that are half one color, and chunks that are not half one color.

• Make sure students understand the puzzle well before trying to make their own!

Jump Step (link)

The picture below says it all. You can step forward one space, or jump and multiply your space by 3. What is the fastest way to get to 1000?

Tips:

• Here are a bunch of other questions to explore: Which numbers take the longest to reach? Which numbers are easiest to reach? What if you can jump by multiplying by 2, or 4, or another number? What if you can step backwards as well as forwards? What other mathematical ideas is this problem connected to?

Shikaku Puzzles (link)

Another round of puzzles from Naoki Inaba. An example puzzle is below on the left, with the solution on the right. Each puzzle is an irregular shape with several integers. The goal is to divide the shape up into rectangles of the given areas. In the puzzle below, the result is divided up into areas of 3, 5, and 6.

Tips:

• I am partial to Inaba’s Shikaku puzzles but there are plenty of other Shikaku puzzles online if you search around.

The Tower of Hanoi and the Sierpinski Triangle (link)

If your students enjoy thinking about the Tower of Hanoi and the Sierpinski Triangle, the linked video with Ayliean MacDonald is a ton of fun to watch to think about connections between the two.

Tips:

• It’s just a video, though it’s fun to try and recreate some of what Ayliean does. This could also be a gateway to a rabbit hole of math on Youtube.

All Ten (link)

If your students enjoyed Adsumudi, this is a logical next step. All Ten is a daily puzzle produced by Beast Academy. Each day you are given four different numbers. The challenge is to use all four numbers and the operations addition, subtraction, multiplication, and division to make the numbers 1 through 10. The additional challenge here is the requirement of using all four numbers each time. The game can be played online at the link above, or on paper.

Tips:

• This puzzle is a good time to review the use of parentheses when doing arithmetic.

Subtracting Digits

Take any number. Subtract the digits from the number. Now do it again. And again. What is always true of the answers? Why?

Tips:

• This works best starting with two-digit numbers, but the pattern is true for any number.

Seven Ate Nine (link)

I can make the number 5 as a sum of consecutive numbers: 2 + 3. I can make the number 10: 1 + 2 + 3 + 4. I can make the number 15 two different ways: 1 + 2 + 3 + 4 + 5 and 4 + 5 + 6. But some numbers are impossible: no matter how hard you try, you can’t make 4 as a sum of consecutive numbers. Which numbers are possible, and which numbers are impossible?

Tips:

• What patterns are there in consecutive sums of two numbers? Three numbers? Four numbers?

Cows and Fields (link)

The problem sounds simple: fill in the circles and boxes below with digits 1-9 without repeating (you will have one left over). The boxes w, x, y, and z must equal the sum of the adjacent circles, so for instance A + B = x.

Tips:

• Can you find a solution? How many solutions can you find? What patterns can you find in the solutions?

• A triangular version of this problem, with three fields and three fences, can be a nice place to start.

Nine Dots Puzzle (link)

Take nine dots in a square grid as shown below. Using four straight lines and without lifting your pen, connect all nine dots.

Tips:

• This puzzle requires a stroke of insight, literally “thinking outside the box.” It is best suited for someone with some mathematical confidence, and can be frustrating for students who put effort in and can’t make any progress.

The Bridges of Königsberg (link)

The map below shows the city of Königsberg, and the river flowing through it. There are seven bridges connecting the two sides of the river and the two large islands in the middle of the river. Can you devise a walk that crosses each bridge exactly once?

Tips:

• The link above takes you to a nice interactive lesson exploring this idea, along with a few other examples of similar problems to explore.

Nim (version three)

This is another version of the Nim games described above. Start with two piles of objects, one with 5 objects and the other with 3. On your turn you can take as many objects from one of the piles as you want. The person who takes the last object (or objects) wins. For a second version of the game, start the same way. In this game, on your turn you can take either one object from one pile, one object from the second pile, or one object each from both piles.

Tips:

• Consider starting with smaller numbers to help students see patterns and figure out the ideal strategy.

Möbius strip (link)

Make a Möbius strip! Take a strip of paper, give it a half twist, and tape the ends together. This object only has one side and one edge – try drawing a line along it, and you will see it only has one side. For an interesting challenge, try to figure out what will happen if you cut a Möbius strip lengthwise, down the middle of the strip. What will you get? What if you cut it again?

Tips:

• Make the tip pretty wide to try the cutting activity. It’s also just fun to explore the Wikipedia page linked above for more related ideas.

Pascal’s Triangle (link)

Begin by writing out as much of Pascal’s triangle, shown below, as you can. Each number is the sum of the two numbers below it. Then, look for patterns. What patterns can you find?

Tips:

• A few patterns you might not notice: what is the sum of each horizontal row of the triangle? Which numbers in the triangle are even? Which are multiples of three? What patterns can you find in the diagonals of the triangle?

Exploding Dots (link)

Exploding dots is a large exploration in its own right, fun for young children but also connecting to deep ideas into algebra. This isn’t as easy to use as a short puzzle, but it’s a fascinating rabbit hole to go down.

Tips:

• I recommend focusing on the 2 → 1 machine with younger students as a place to start exploring.

The Fibonacci Sequence (link)

The Fibonacci sequence begins 1, 1, 2, 3, 5… Each number is the sum of the two previous numbers. How far can you go? What patterns can you find? What can you learn about the Fibonacci sequence? The sequence is connected to several future problems…

Tips:

• Here are a few interesting problems: Which Fibonacci numbers are even? Odd? Why? What about multiples of three? Can you extend the Fibonacci numbers backwards?

Gauss

Legend has it that as a young math student, Karl Gauss was asked by his teacher to add the numbers 1 to 100. The teacher thought it would take Gauss a long time, but he came back with the answer moments later. How did he do it?

Tips:

• Can you add the numbers 1 to 200? 1 to 1,000? 1 to 1,000,000?

Hexaflexagons (link)

Can you make a hexaflexagon?

Tips:

• The video above moves a bit fast, so it might be helpful to search online for more detailed guides to making a hexaflexagon.

• This is tricky! I recommend practicing a few times before introducing it to students.

• I have never been able to make the more advanced version in the video with six “sides” but it seems fun if you can make it!

Postage Stamp Problem (link)

You have unlimited postage stamps for 3 cents and 4 cents. Can you combine them to make 13 cents? 12 cents? 11 cents? 10 cents? What is the largest number you can’t make? What if you try the same problem with different values of postage stamps? What patterns can you find?

Tips:

• This is one problem where the general rule can be hard to find. A good place to start is to try the problem for 3 and 4, 4 and 5, 5 and 6, etc, and see what patterns pop out.

Zero Puzzles (link)

Another Inaba puzzle. An example is below. For each puzzle, add zeros to some or all numbers shown to make a true statement.

Tips:

• The puzzle start simply and get gradually harder.

Persistence (link)

The problem is described below. Can you find a number with a persistence of 4? 5? 6?

Tips:

• The additive version is interesting as well.

The Collatz Conjecture (link)

The Collatz Conjecture is a famous problem in math. Start with any positive whole number. If the number is even, divide by two. If the number is odd, multiply by three and add one. Try starting with 10, 15, 27. What happens? What is the general rule? Are there any exceptions?

Tips:

• The Collatz Conjecture says that all numbers will eventually end up at 1. One way to frame this problem is to challenge students to find a number that doesn’t end up at 1, and later tell them that if they do so, they will become famous in the math world.

Prime Spirals (link)

Create a grid of numbers spiraling outward, as below. Go as far as you can. Then, shade the prime numbers. What patterns do you see? Why? How far can you go? The video linked above explains some of the patterns, though they’re also fun to explore independently.

Tips:

• This takes some patience: more and more patterns will appear as the spiral continues outward.

• Try shading other numbers – even numbers, multiples of 3, triangular numbers, and more. What other patterns can you find?

11s and Pascal’s Triangle

What is 11×11? 11×11×11? 11×11×11×11? Write these numbers in order vertically. What patterns do you see? Where have you seen these patterns before?

Tips:

• If students are comfortable with exponents, this problem could also be framed as 110, 111, 112, 113, 114.

• Why does the pattern change at five 11s?

Eight Queens (link)

A queen on a chessboard can move horizontally, vertically, or diagonally, as far as it wants in each direction. Below is an example of four queens on a 4x4 chessboard, where none of the queens can “see” each other (none of the queens can reach another queen in a single move. Can you do the same for eight queens on a regular 8x8 chessboard?

Tips:

• One way to frame this problem is to start with 4x4, then 5x5, 6x6, and 7x7. These problems are much easier to solve than 8x8.

Climbing Stairs (link)

Imagine you are climbing a staircase. With each step, you can climb one or two stairs. How many ways are there to climb 3 stairs? 4? 5? 6?

Tips:

• Make sure students understand the constraint that they can only climb one or two stairs at a time before beginning.

• Try to solve at least up to 6 stairs before looking for patterns.

• It is often helpful to give students a few hints to organize their work and make sure they don’t miss any possibilities.

• If students find the pattern, how far can they extend it? Can they explain why this pattern holds?

Space Race (link)

Space Race is another variation on Nim-style games. The directions are shown below.

Tips:

• Consider starting with a smaller board at first to make the strategy easier to find.

Maximaze (link)

In the maze below, each step adds to, multiplies by, or subtracts from your current score. Without repeating any segments, what is the highest score you can get?

Tips:

• Beyond guess-and-check, what strategies can be used to maximize the score?

• Can you make your own maze? What can you do to make the maze tricky?

The Josephus Problem (link)

Imagine a large group of people standing in a circle. Pick a place to start. The first person stays in, the next person is out and leaves the circle. The third person stays in, the fourth is out. Continue until only one person is left. How can you predict which person will be left at the end?

Tips:

• The original framing of the problem is a bit dark. An alternate framing is “Duck Duck Die” (a play on duck duck goose).

• Playing with numbers like 2, 4, and 8 can be a helpful place to start.

Rice on a Chessboard (link)

Picture a chessboard. A chessboard is an 8x8 grid, with 64 total squares. Imagine 1 grain of rice is placed on the first square, then 2 grains on the second, then 4, then 8, doubling every time. How many grains would be on the chessboard at the end?

Tips:

• An interesting pattern can be found by first looking at the number of grains of rice on each successive square, and then looking at the total grains of rice on the board after each successive square.

• About how many grains of rice are there after 10 squares? 20 squares? 30 squares? Where does this pattern come from?

• An alternate framing: place pennies on the board as above. Would you rather have $1,000,000 or the total pennies on the chessboard?

Kaprekar’s Number (link)

Take any four-digit number. Write its digits in order from greatest to least, then from least to greatest. Subtract. Repeat. What happens? Why?

Tips:

• What happens? Will this always happen for all starting numbers?

• Try it with other numbers of digits as well.

Arranging Dominoes (link, beginning on page 11)

How many ways can you arrange dominoes into rectangles? This simple problem yields fascinating patterns. The details are described in the link above.

Tips:

• This problem connects to the climbing stairs problem mentioned previously. The connections are explored in the linked lesson plan.

Egyptian Fractions

Th Egyptians only had symbols for unit fractions, meaning fractions like ½, ⅓, ¼, etc. (Strictly speaking they also had a symbol for ⅔ but we’ll ignore that for now.) They wrote other fractions as the sum of unit fractions. For instance, the fraction that we today call ¾ they would write as ½ + ¼. They would write ⅖ as ⅓ + 1/15 (check that this works!). To write these sums, they always chose the largest possible fraction at each step. ⅓ is the largest unit fraction less than ⅖, and then 1/15 is the largest unit fraction for what is left. How many fractions can you rewrite as Egyptian fractions? Which fractions are easiest to write, and which are hardest? Why?

Tips:

• Depending on the students, visuals may be helpful for this task.

Ultimate Tic Tac Toe (link)

Tic tac toe can get boring. In Ultimate tic tac toe, each square has a smaller tic tac toe board inside of it. The rules are tricky: on your turn, you make a move in one of the small squares. On your opponents turn, they most move in the large square corresponding to the small square you moved in. The rules are explained in more detail at the link above.

Tips:

• Ultimate tic tac toe is a ton of fun, but can also be tricky to learn at first. A fun variation for beginners is to get rid of the constraints about the next move. On your turn, you can move in any of the small squares. While this game has less complex strategy, it can still be a ton of fun and can be a good on-ramp to the full version.

Area Mazes (link, link, link)

A final Inaba puzzle. Naoki Inaba has several books dedicated solely to area mazes, but above are a few links to get you started. These puzzles lead to fascinating geometric strategies using only simple rectangles and lengths.

Tips:

• The puzzles aren’t as carefully sequenced as previous puzzles, so don’t be afraid to skip around. For the curious, Inaba’s books are excellent!

Hilbert’s Hotel (link)

Fun fact: there are different sizes of infinity. Hilbert’s Hotel is a thought experiment from David Hilbert exploring different sizes of infinity, and considering why infinity + 1 is still the same size as infinity. The video above is a nice introduction to the idea, though there are plenty of other variations online.

Tips:

• A series of fun followup questions: how would you rearrange the guests of Hilbert’s Hotel to illustrate that infinity + 5 = infinity? What about infinity x 4 = infinity? Infinity x 3 + 2 = infinity? And so on.

• It’s worth looking at other sources to better understand the broader meaning of the second half of the video.

Square Chomping (link)

Imagine a rectangle, with dimensions 24 by 9. Or, better yet, draw that rectangle on grid paper. Now, chomp off (get rid of) the largest square you can from one side of the rectangle. In this case, that square will be 9x9, leaving 15x9. Now do it again. And again. Keep chomping squares. How large is the last square you chomp? Now try this for 12x8, 30x10, 9x4, 16x14, and more. What is always true about the last square you chomp?

Tips:

• If you’re having trouble seeing the patterns, considerFINISH.

Seven Ate Nine, Revisited (link)

Revisit this problem from earlier: I can make the number 5 as a sum of consecutive numbers: 2 + 3. I can make the number 10: 1 + 2 + 3 + 4. I can make the number 15 two different ways: 1 + 2 + 3 + 4 + 5 and 4 + 5 + 6. But some numbers are impossible: no matter how hard you try, you can’t make 4 as a sum of consecutive numbers. Which numbers are possible, and which numbers are impossible? To add on to the last version, in how many ways can you write each number as the sum of consecutive numbers? Which numbers can be written one way? Two ways? Three ways? Why?

Tips:

• What patterns are there in consecutive sums of two numbers? Three numbers? Four numbers?

Split 25 (link)

What is the largest product you can make using numbers that add to 25?

Tips:

• It can be helpful starting with smaller numbers to look for patterns. 8, 9, and 10 are a nice place to start.

• What if decimals are allowed?

Eightfold (link)

Print out the paper at the link above. Can you fold it so that the numbers are in sequential order, from 1 through 8?

Tips:

• This is harder than it looks, and requires some creative folding!

Painted Cube (link)

Imagine painting a 4x4x4 cube blue on all sides, and then taking it apart. Some of the small cubes won’t have any paint on them. Some will have paint on one side, or two sides, or three sides. How many of each type of cube are there? What about for a 3x3x3, or a 2x2x2? Can you come up with a general rule?

Tips:

• Physical cubes are a nice place to start, though even with physical cubes it can be tough to visualize the paint.

• The 1x1x1 cube is interesting. Does the pattern continue?

The Dragon Curve (link)

The dragon curve is a fascinating fractal. You can make a small version of it with a strip of paper. Fold the paper repeatedly in the same direction, as shown in the photo below. Then, unfold it, and make sure each angle is 90 degrees. See how many folds you can make!

Tips:

• A long, thin strip of paper – for instance from lightweight poster paper – works best for this.

• It’s hard to make a decent dragon curve with paper because it’s hard to fold repeatedly. It’s worth trying this out, then exploring the Wikipedia article above and seeing where it leads.

76 = 24 (credit to Sunil Singh in his book Pi of Life)

Take the statement 76 = 24. That statement is not true. Imagine that you can move each symbol around, as if they are refrigerator magnets. Can you rearrange the given statement into a true statement?

Tips:

• This is an “insight” problem that won’t make much sense until a stroke of insight by the solver. That stroke of insight is much more likely if the solver has experience with exponents.

Kazu Sagashi puzzles (link)

The goal of these Inaba puzzles is to find a square with the given number of piece of fruit in it. There is always exactly one solution!

Tips:

• Focus on the number in each puzzle; the Japanese characters are not necessary to understand the puzzle.

• The puzzles get gradually harder, see how far you can get!

Triangulations (link)

The problem is in the image below. How many triangulations can you find?

Tips:

• It’s helpful to start by thinking about how to organize the triangulations to avoid duplicates.

• How many triangulations are there for a square? Pentagon? Heptagon? What patterns can you find in those numbers?

Mountain Ranges (link)

The problem is below. How many mountain ranges can you find?

Tips:

• Triangular grid paper is very helpful for this problem.

• How many mountain ranges are there that are 2 unit wide? 5? 6? Can you find a pattern?

• Where else do these numbers show up?

1, 3, 4, 6 puzzle (link)

Using the numbers 1, 3, 4, and 6, can you make all the numbers from 1 to 24? Rules: you must use each number exactly once for each calculation. You can only use addition, subtraction, multiplication, division, and parentheses. For instance, (3 + 4 - 6) / 1 is allowed, but 4 + 3 - 1 is not allowed because it doesn’t use 6, and 3 + 3 + 6 - 4 - 1 is not allowed because it uses 3 twice.

Tips:

• There are a few variations on this puzzle, for instance using four 4s. Feel free to try a different version, but this one is particularly interesting.

The Locker Problem

Imagine a row of lockers, numbered 1 to 100. Every locker is closed. Now someone walks down the row, starting at #1, and opens every locker. Then, starting at #2, they close every 2nd locker (so 2, 4, 6, 8, etc). Then, starting at #3, they open or close every 3rd locker (so if it is closed, they open it, and if it is open, they close it). Then every 4th 5th, etc. When they are finished, which lockers are open? Why?

Tips:

• It might be exhausting to try and do this for every number up to 100. You could start by going up to every 20th locker and seeing what patterns you can find, and making a prediction before going further.

Nim (link)

The other versions of Nim have built up to this much more challenging version. You can start with different numbers in each pile, but the classic version starts with four piles: one object, three objects, five objects, and seven objects. On your turn you can take as many objects as you want from one pile. You must take at least one object. The player who takes the last object wins.

Tips:

• The patterns in this game are much harder to find than previous games, but it’s still a ton of fun!

Twin Puzzles (link)

Another Inaba puzzle. In each pair of twin puzzles, the goal is to put arithmetic operations and parentheses to make two true statements, using the same operations and parentheses in both puzzles.

Tips:

• If you’re stuck, try solving each puzzle one at a time first!

Weighing Coins (link)

You have nine coins. Eight of the coins are genuine, but one is fake and is heavier than the rest. You have a balance scale, so you can put coins on both sides of the scale and see which side is heavier. However, the scale cannot tell you the exact weight of any coin. What is the smallest number of weighings you can use to determine the counterfeit coin?

Tips:

• There are many variations on this puzzle, which you can explore at the link above.

Knights and Knaves (link)

You are on an island where every individual is either a knight, who always tells the truth, or a knave, who always lies. You walk up to an individual. What is one question you can ask to determine whether they are a knight or a knave?

Tips:

• There are many variations on this puzzle, which you can explore at the link above.

Giant Cat Army (link)

Just a goofy fun number riddle. The only tools you have are +5, +7 and a square root. You start with the number 0. You need to make 2, 10, and 14, in that order, without repeating any numbers, getting a number above 60, or getting a non-whole number. Can you do it?

Tips:

• Fluency with small square roots is helpful for this problem, though it’s also possible to give students a table since only numbers up to 60 are allowed.

Systems Puzzle

Can you figure out what number each shape represents? There are lots of strategies you can use, and this puzzle hints at some ideas related to systems of equations.

Tips:

• If students are stuck, point them toward the second row as a place to start.

The Monty Hall Problem (link)

The Monty Hall problem is a famous problem in probability and comes from the television show “Let’s Make a Deal.” The contestant must choose from three doors. Behind two of the doors are goats (and in this case, the contestant is not a fan of goats). Behind the third door is a fancy sports car. The contestant chooses one door – let’s say door 3. The host responds by opening a different door – let’s say door 2. Behind that opened door is always a goat. Now the host makes an offer. Would the contestant like to stay with door 3, or switch to door 1?

Tips:

• The example from the video, with 100 doors rather than 3, is a nice way to get your head around some of the counterintuitiveness of this problem.

Noah’s Ark (link)

I originally learned about this problem from Fawn Nguyen but I’m having trouble finding the original source and I’m not sure where the linked Google Doc above comes from.

Here’s the prompt. The animals below all need to board Noah’s ark. To sail smoothly, the ark must be balanced so the weight on the left matches the weight on the right. All like animals weigh the same (so all kangaroos, all elephants, etc each weigh the same). How many seals must go in the place of the question mark to balance the ark?

Tips:

• Start by focusing on polar bears and zebras!

The Bear in the Moonlight (link)

These are a set of fables about probability. While there are some questions and a few pieces of math that can be done, they are also just lovely stories that teach valuable lessons about probability.

Tips:

• These are accessible to young students but also share lesson worth exploring with high schoolers.

Sequences and Infinity (link)

Be warned that the linked video is controversial. It claims that if you add up all of the natural numbers, 1 + 2 + 3 + 4 + … going to infinity, the sum is equal to -1/12. It’s a fascinating exercise in doing algebra with infinity. It might also lead you down a bit of rabbit hole…

Tips:

• It’s worth framing this video as one perspective on infinity, rather than some absolute truth of mathematics.

13 Trick (link)

You have to watch the video linked above. Can you figure out this magic trick?

Tips:

• This one will take a few watches, though it’s not too hard to recreate yourself!

Visual Patterns (link)

Below is one example of a visual pattern, and the linked website has hundreds. What will the next step look like? How do you see it growing? Can you come up with a general rule?

Tips:

• Visual patterns are very different depending on the age of the students. High school students can try to write an algebraic expression for the pattern, while younger students might be challenged by just drawing what the next step looks like.

Happy Numbers (link)

Take a number. Square its digits and add those together. Repeat. Which numbers end up at 1? For instance: 19 -> 82 -> 68 -> 100 -> 1. How many happy numbers can you find?

Tips:

• If students are comfortable with other bases, exploring happy numbers in bases other than 10 can be fun.

Continued Fractions

Simplify the following fractions:

Keep going as far as you’d like. What patterns do you notice?

Tips:

• How is this connected to some patterns from other problems earlier?

A Tricky Logic Puzzle (link)

Here is the classic description of the puzzle: A census taker approaches a woman leaning on her gate, number 14, and asks about her children. She says, “I have three children and the product of their ages is seventy–two. The sum of their ages is the number on this gate.” The census taker does some calculation and claims not to have enough information. The woman enters her house, but before slamming the door tells the census taker, “I have to see to my eldest child who is in bed with measles.” The census taker departs, satisfied. It seems as if there is not enough information to solve the problem, but there is.

Tips:

• If you’re stuck: the census taker knows the number on the gate, knows all the possible combinations that multiply to 72, yet still does not know the answer. That is an important hint to narrow down the possibilities.

Dividing a Circle (link)

Put two dots on a circle and connect them. The circle is divided into 2 sections. Now put three dots on a circle and connect all of them. Now the circle is divided into 4 sections. What about four dots? Five? Six? What is the pattern?

Tips:

• Make sure you get to six dots!

• In this problem the dots should not be symmetrical, and no more than two lines should ever meet at a point.

7.11

A customer walks into a 7-11 and chooses four items. When the cashier rings the items up, the total cost is $7.11. Remarkably, the product of the four items when they are multiplied together is also exactly 7.11 What are the costs of the four items?

Tips:

• This is a very challenging question. One helpful hint: convert all numbers involved to whole numbers, so find four three-digit numbers that sum to 711, and four three digit numbers that multiply to 711,000,000.