## Posts tagged ‘game’

### 12 Math Games, Puzzles, and Problems for Your Holiday Car Trip

It doesn’t matter if you’re one of the 102.1 million people traveling by car, one of the 6.7 million people traveling by plane, or one of the 3.7 million people traveling by train, bus, or cruise ship this holiday season — the following collection of games, puzzles, and problems will help to pass the time, and you’ll be there long before anyone asks, “Are we there yet?”

**Games**

*1. Street Sign Bingo*

Use the numbers on street signs to create expressions with specific values. For instance, let’s say you see the following sign:

The two numbers on the sign are 12 and 3, from which you could make the following:

12 + 3 =

1512 – 3 =

912 × 3 =

3612 ÷ 3 =

4

Of course, you could just use the numbers directly as 3 or 12. Or if your passengers know some advanced math, they could use square roots, exponents, and more to create other values.

Can you split the digits within a number and use them separately? Can you concatenate two single-digit numbers to make a double-digit number? That’s for you and your traveling companions to decide.

To play as a competitive game, have each person in the car try to get every value from 1‑20. It’s easiest to keep track if you require that players go in order. To make it a cooperative game, have everyone in the car work collectively to make every value from 1 to 100.

As an alternative, you could use the numbers on license plates instead.

*2. Bizz Buzz Bang*

Yes, I’ve played this as a drinking game. No, I don’t condone drinking while driving. No, I don’t condone under-age drinking, either. Yes, I condone playing this game with minors while driving. (See what I did there?)

The idea is simple. You pick two single-digit numbers, *A* and *B*. Then you and your friends start counting, one number per person. But each time someone gets to a number that contains the digit *A* or is a multiple of *A*, she says, “Bizz!” Every time someone gets to a number that contains the digit *B* or is a multiple of *B*, she says, “Buzz!” And every time someone gets to a number that meets both criteria, he says, “Bang!”

For example, let’s say *A* = 2 and *B* = 3. Then the counting would go like this:

one, bizz, buzz, bizz, five, bang, seven, bizz, buzz, bizz, eleven, bang, buzz, …

When someone makes a mistake, that round ends. Start again, and see if you can beat your record.

You can use whatever numbers you like, or modify the rules in other ways. For instance, what if bizz is for prime numbers and buzz is for numbers of the form 4*n* + 3? Could be fun!

*3. Dollar Nim* (or any other variation)

Nim is a math strategy game in which players take turns removing coins from a pile. Different versions of the game are created by adjusting the number of coins that can be removed on each turn, the number of coins in the pile originally, how many piles there are, and whether you win (normal) or lose (misère) by taking the last coin.

A good aspect of Nim is that you don’t actually need coins. To play in the car, players just need to keep track of a running total in their heads.

A version of Nim dubbed **21 Flags** was played on *Survivor Thailand* several years ago. There were 21 flags, and each team could remove 1, 2, or 3 flags on each turn. The team to remove the last flag won. This is a good first version to play in the car, especially for young kids who have never played before. Then mix it up by changing the initial amount and the number that can be removed on each turn.

Our family’s favorite version, **Dollar Nim**, is played by starting with $1.00 and removing the value of a common coin (quarter, dime, nickel, penny) on each turn. (While discussing this post with my sons, they informed me that they much prefer **Euro Nim**, which begins with 1€, but then the coin values to be removed are 50c, 20c, 10c, 5c, 2c, and 1c. It’s essentially the same game, but they like the different coin amounts.)

**Three-Pile Nim** consists of not just one but three piles with 3, 5, and 7 coins, respectively. On each turn, a player must remove at least one coin, and may remove any number of coins, but all removed coins must be *from the same pile*.

Finally, **Doubling Nim** is played as the name implies. On the first turn, a player may remove any number of coins but not the entire pile. On every turn thereafter, a player may remove any number of coins up to double the number taken on the previous turn. For instance, if your opponent removes 7 coins, then you can remove up to 14 coins.

For every version of Nim, there is an optimal strategy. We’ve wasted hours on car trips discussing the strategy for just one variation. Discussing the strategies for all the versions above could occupy the entire drive from Paducah to Flint.

*4. Guess My Number
*

One person picks a number, others ask questions to try to guess the number.

The simplest version is using “greater than” and “less than” questions. Is it greater than 50? Is it less than 175? And so forth. Using this method, the guessers can reduce the number of possibilities by half with each question, so at most, it should take no more than *n* guesses if 2^{n} > *m*, where *m* is the maximum possible number that the picker may choose. For instance, if the picker is required to choose a number less than 100, then *n* = 7, because 2^{7} = 128 > 100. Truthfully, this version of the game gets boring quickly, but it’s worth playing once or twice, especially if there’s one picker and multiple guessers. And a conversation about the maximum number of guesses need can be a fun, mathy way to spend 15 minutes of your trip.

A more advanced version excludes “greater than” and “less than” questions. Instead, the guessers can ask other mathematical questions like, “Is it a prime number?” or “Do the digits of the number differ by 4?” Those questions imply, of course, that the answer must be yes or no, and that’s typical for these types of guessing games. If you remove that restriction, though, then guessers could ask questions that reveal a little more information, like, “What is the difference between the digits?” or “What is the remainder when the number is divided by 6?” With this variant, it’s often possible to identify the number with two strategic questions.

**Puzzles**

Here are three puzzles that can lead to hours of conversation — and frustration! — on a car trip. Before you offer one to your crew, though, put forth the disclaimer that anyone who’s heard the puzzle before must remain mum. No reason they should spoil the fun for the rest of you. (The puzzles are presented here without solution, because you’ll know when you get the right answer. You can find the answer to any of them online with a quick search… but don’t do that. You’ll feel much better if you solve it yourself.)

*5. Dangerous Crossing*

Four people come to a river in the middle of the night. There’s a narrow bridge, but it’s old and rickety and can only hold two people at a time. They have just one flashlight and, because it’s night and the bridge is in disrepair, the flashlight must be used when crossing the bridge. Aakash can cross the bridge in 1 minute, Britney in 2 minutes, Cedric in 5 minutes, and Deng in 8 minutes. When two people cross the bridge together, they must travel at the slower person’s pace. And they need to hurry, because zombies are approaching. (Oh, sorry, had I failed to mention the zombie apocalypse?) What is the least amount of time that all four people can cross the bridge?

And how can you be sure that your method is the fastest?

*6. Weight of Weights*

Marilyn has a simple balance scale and four small weights, each weighing a whole number of grams. With the balance scale and these weights, she is able to determine the weight of any object that weighs between 1 kg and 40 kg. How much does each of the four weights weigh?

*7.* *Product Values*

Assign each letter a value equal to its position in the alphabet, i.e., A = 1, B = 2, C = 3, …, Z = 26. Then for any common word, find its product value by multiplying the value of the letters in the word. For instance, the product value of CAT is 60, because 3 × 1 × 20 = 60.

- Find as many common English words as you can with a product value of 60.
- Find a common English word that has the same product value as your name. (A little tougher.)
- Find a common English word with a product value of 3,000,000. (Zoiks!)

**Problems**

*8. The Three of Life*

This one looks so innocent!

What’s the probability that a randomly chosen number will contain the digit 3?

But spend a little time with it. And prepare for. Mind. Blown.

*9. Hip to Be (Almost) Square*

No calculators for this one.

There are four positive numbers — 1, 3, 8, and

x— such that the product of any two of them is one less than a square number. What is the least possible value ofx?

No spreadsheets, either.

*10. Hip to Be Square (Roots)
*

Just some good, old-fashioned algebra and logic to tackle this one.

Which of the following expressions has a greater value?

or

Surprised?

*11. Coming and Going*

Potentially counterintuitive.

At the holidays, Leo drove to his grandma’s, and the traffic was awful! His average speed was 42 miles per hour. After the holidays, however, he drove home along the same route, and his average speed was 56 miles per hour. What was his average speed for the entire trip?

And, no, your first guess was most likely not correct.

*12. The Year in Numbers*

A moldy oldie, to be sure, but this puzzle is always a crowd-pleaser for those who haven’t seen it before.

Use the digits of the new year — 2, 0, 1, and 9 — and any mathematical operations to form the integers from 1 to 100. For instance, you can form 1 as follows: 2 × 0 × 9 + 1 = 1.

For an added challenge, add the restriction that you must use the four digits *in order*.

**Bonus: A Book**

My sons get sick when they read in the car. That’s why I love the two books *Without Words* and *More Without Words* by James Tanton. Literally, there are no words! Each puzzle is presented using a few examples, and then students must follow the same rules to solve a few similar, but more challenging, puzzles.

Wherever you’re headed during the holiday break — driving to your relatives’ house, flying to Fort Lauderdale, or just relaxing at home — I hope your holidays are filled with joy, happiness, and lots of math!

### Our Library’s Summer Math Contest

Every summer, our local library runs a contest called *The Great Big Brain Game*. Young patrons who solve all of the weekly puzzles receive a prize. The second puzzle for Summer 2017 looked like a typical math competition problem:

Last weekend, the weather was perfect, so you decided to go to Cherry Hill Park. When you got there, you saw that half of Falls Church was at the park, too! In addition to all the people on the playground, there were a total of 13 kids riding bicycles and tricycles. If the total number of wheels was 30, how many tricycles were there?

First, some comments about the problem.

- I dislike using “you” in math problems. I believe it’s a turn-off to students who can’t see themselves in the situation described. There are enough reasons that kids don’t like math. Why give them another reason to shut down by telling them that they went somewhere they didn’t want to go or that they did something they didn’t want to do?
- Word problems are not real-world just because they use a local context, and this one is no exception. This problem attempts to show an application for a system of linear equations, but true real-world problems don’t have all the information neatly packaged like this.
- Wouldn’t the person posing this problem already have access to the information they seek? That is, if she counted the number of kids riding bikes and the total number of wheels, couldn’t she have just counted the number of bicycles and tricycles instead? It has always struck me as strange when the (implied) narrator of a math problem wants you to figure out something they already know.

All that said, this was meant to be a fun puzzle for a summer contest, and I don’t mean to scold the library. I don’t know that I’d use this puzzle in a classroom — at least, not presented exactly like this — but I love that kids in my town have an opportunity to do some math in June, July, and August.

Now, I’ll offer some comments on the solution. In particular, the solution provided by the library was different than the method used by one of my sons. Here’s what the library did:

Imagine that all 13 kids were on bicycles with 2 wheels. That would be a total of 26 wheels. But since 30 wheels are needed, there are 4 extra wheels. If you add each of those extra wheels to a bicycle, that’ll create 4 tricycles, leaving 9 bicycles. So, there must have been 4 tricycles at Cherry Hill Park.

And here’s what my son did:

If you can’t see what he wrote, he created a system of two equations and then solved it:

2a + 3b = 30

a + b = 13a + 2b = 17

13 – b + 2b = 17

b = 42a + 12 = 30

2a = 18

a = 9

That’s all well and good. In fact, it’s perfect if you want to assess my son’s ability to translate a problem and solve a system of equations. But I have to admit, I was a little disappointed. What bums me out is that he went straight to a symbolic algorithm instead of considering alternatives.

I think I know the reason for this. This past year, my son was in a pull-out math program, in which he studied math with someone other than his regular classroom teacher. In this special class, the teacher focused on preparing him to take Algebra II in sixth grade when he enters middle school. Consequently, students in the pull-out class spent the past year learning basic algebra. My fear is that they focused almost exclusively on symbolic manipulation and, as my former boss liked to say, “Algebra teachers are too symbol-minded.”

A key trait of effective problem solvers is flexibility. That type of flexibility comes from solving many problems and filling your toolbox with a variety of strategies. My worry — and this isn’t just a concern for my son, but for every math student in the country — is that students learn algorithms at the expense of more useful problem-solving heuristics. What happens when my son is presented with a problem that can’t be translated into a system of linear equations? Will he know what to do when he doesn’t know what to do?

The previous pull-out teacher said that when she presented my sons with problems that they didn’t know how to solve, their eyes would light up. They liked the challenge of doing something they hadn’t done before. I’m hopeful that this enthusiasm isn’t lost as they proceed to higher levels of mathematics.

### Σ Π :: The Sum and Product Game

This joke, or a close facsimile, has been taking a tour of email servers recently, and it’s now showing up on t-shirts, too:

…and it was delicious!

Appropriate for Pi Day, I suppose, as is the game my sons have been playing…

Eli said to Alex, “18 and 126.”

Alex thought for a second, then replied, “2, 7, and 9.”

“Yes!” Eli exclaimed.

I was confused. “What are you guys doing?” I asked.

“We invented a game,” Eli said. “We give each other **the sum and product of three numbers**, and the other person has to figure out what the numbers are.”

After further inquisition, I learned that it wasn’t just any three numbers but **positive integers** only, that **none can be larger than 15**, and that they must be **distinct**.

Hearing about this game made me immediately think about the famous Ages of Three Children problem:

A woman asks her neighbor the ages of his three children.

“Well,” he says, “the product of their ages is 72.”

“That’s not enough information,” the woman replies.

“The sum of their ages is your house number,” he explains further.

“I still don’t know,” she says.

“I’m sorry,” says the man. “I can’t stay and talk any longer. My eldest child is sick in bed.” He turns to leave.

“Now I know how old they are,” she says.

What are the ages of his children?

You should be able to solve that one on your own. But if you’re not so inclined, you can resort to Wikipedia.

But back to Alex and Eli’s game. It immediately occurred to me that there would likely be some ordered pairs of (sum, product) that wouldn’t correspond to a unique set of numbers. Upon inspection, I found eight of them:

(19, 144)

(20, 90)

(21, 168)

(21, 240)

(23, 360)

(25, 360)

(28, 630)

(30, 840)

My two favorite ordered pairs were:

(24, 240)

(26, 286)

I particularly like the latter one. If you think about it the right way (divisibility rules, anyone?), you’ll solve it in milliseconds.

And the Excel spreadsheet that I created to analyze this game led me to the following problem:

Three distinct positive integers, each less than or equal to 15, are selected at random. What is the most likely product?

Creating that problem was rather satisfying. It was only through looking at the spreadsheet that I would’ve even thought to ask the question. But once I did, I realized that solving it isn’t that tough — there are some likely culprits to be considered, many of which can be eliminated quickly. (The solution is left as an exercise for the reader.)

So, yeah. These are the things that happen in our geeky household. Sure, we bake cookies, play board games, and watch cartoons, but we also listen to the NPR Sunday Puzzle and create math games. You got a problem with that?

### Don’t Believe the HIPE

Let’s get this party started with a classic word puzzle.

What English word contains four consecutive letters that appear consecutively in the alphabet?

In *Mathematical Mind-Benders* (AK Peters, 2007), Peter Winkler describes how the puzzle above served as inspiration for a word game.

I and three other high-school juniors at a 1963 National Science Foundation summer program began to fire letter combinations at one another, asking for a word containing that combination… the most deadly combinations were three or four letters, as in GNT, PTC, THAC and HEMU. We named the game after one of our favorite combinations, HIPE.

This seemed like a good game to play with my sons. I explained the game, and then I gave them a simple example to be sure they understood.

ER

They quickly generated a long list of solutions, including:

- tERm
- obsERver
- fishERman
- buckminstERfullerene

Since that introduction a few weeks ago, the boys and I have played quite a few games. It’s a good activity to pass the time on a long car ride. The following are some of my favorites:

WKW

RTWH

RTHW

(these two are fun in tandem)HIPE

(the game’s namesake is a worthy adversary)TANTAN

The practice with my sons has made me a better-than-average HIPE player, so when I recently found myself needing to keep my sons busy while I prepared dinner, I offered the following challenge:

Create a HIPE for me that you think is difficult, and I’ll give you a nickel for every second it takes me to solve it.

Never one to shy away from a challenge, Eli attacked the problem with gusto. Fifteen minutes later, he announced, “Daddy, I have a HIPE for you,” and presented me with this:

RLF

That was three days ago. Sure, I could use More Words or some other website to find the answer, but that’s cheating. Winkler wrote, “Of course, you can find solutions for any of them easily on your computer… But I suggest trying out your brain first.”

The downside to relying on my brain? **This is gonna cost me a fortune.**

For your reading enjoyment, I’ve created the following HIPEs. They are roughly in order from easy to hard, and as a hint, I’ll tell you that there is a common theme among the words that I used to create them.

- MPL
- XPR
- YMM
- MSCR
- MPT
- ITESI
- NSV
- RIGON
- OEFF
- CTAH
- THME
- SJU
- TRAH (bonus points for finding more than one)

Winkler tells the story of how HIPE got him into Harvard. He wrote “The HIPE Story” as the essay on his admissions application, and four years later, he overheard a tutor who served on the admissions committee torturing a colleague with HIPEs and **calling them HIPEs**.

I can’t promise that HIPEs will get you into college, but hopefully you’ll have a little fun.

### The Game of POP

No one knows how to live a funky life more than Prince:

Life, it ain’t real funky

Unless it’s got thatpop

Need a little extra pop in your life? Here’s a game you can play.

Create a game board consisting of *n* adjacent squares. Here’s a board for *n* = 10:

Still with me? Good.

The rules of POP are rather straightforward.

- Players alternate turns, placing either an O or a P in any unoccupied square.
- The winner is the first player to spell the word POP in three consecutive squares.

I first learned this game using O’s and S’s and trying to spell SOS, but for young kids, O’s and P’s are much better… the accidental occurrences of POO and POOP add a certain *je ne sais quoi*. (But not as much as foreign phrases add to a sentence about feces.)

Alex and Eli played this game tonight on the board shown above. After six turns, the game was decided. (As you can see, an accidental POO occurred in squares 6‑8. I mean an accidental occurrence of the *word* POO, not an actual occurrence of POO itself. If the latter had happened, the game would have ended immediately, and I wouldn’t be writing about it now.) It was Alex’s turn, and he realized that he lost: playing either an O or a P in squares 3‑4 would give Eli the win, and playing either an O or P in squares 9‑10 would just delay the inevitable.

“So, what’re you gonna do?” I asked.

Alex added an O to the third square, shrugged, and handed the pencil to Eli.

*A coward dies a thousand deaths; the valiant die but once.*

In that game, Alex went first and lost. So an immediate question:

- Will the second player always win when
*n*= 10?

This then leads to follow-up questions:

- Are there other values of
*n*such that the second player has a winning strategy? - Are there any values of
*n*such that the first player has a winning strategy? - Are there values of
*n*for which neither player has a winning strategy?

If you’d like to play a game of POP, then head over to **The Game of POP spreadsheet on Google Drive**, email the link to your friend, and start adding O’s and P’s. Feel free to change the size of the game board, too! Just please be a sweetie — when you finish, clear all your letters, reset the size of the game board to 10 squares, and be sure all the directions are retained at the top of the page.

Enjoy!

### Think of a Number

I love to create math games almost as much as I love to play them.

My favorite professional project was leading the development of Calculation Nation. And my favorite game on the site is neXtu, though other games on the site may promote more sophisticated mathematical thinking.

I have many reasons to love my wife, not least of which is her creation of the game Dollar Nim. While I can’t take credit for the rules, I will take credit for its analysis and its popularization. (What do you call a wife who makes up a game that gets you a publication credit? A **keeper**!)

Recently, I’ve been frustrated by the lack of games for teaching algebra. I’ll give props to the good folks at Dragonbox, which uses a game environment to teach algebra. But I’m not yet convinced that it leads to deep algebraic understanding; even they admit “to transfer to pencil and paper, children must be explained how to rewrite equations line by line.” They also claim that “in-house preliminary tests indicate a very high level of transfer to pencil and paper,” but that’s the fox watching the henhouse.

So I’ve been thinking about games I can play with my sons that will allow them to engage in algebraic thinking. But I don’t want them to know they’re engaging in algebraic thinking. I have two criteria for all math games:

- The game mechanics depend on mathematics. The math is not tangential to the game; it
**is**the game. - Kids don’t realize (or at least they don’t care) that it’s a math game, because it’s fun.

It pains me to write that second criterion, because math **is** fun. But I know not everyone shares that opinion. So I do my best to disguise any math learning in the game and then, when they least expect it — BOOM! — I drop the bomb and show them what they’ve learned.

So here’s a game I recently devised.

- Player A chooses a number.
- Player B chooses two operations for Player A to perform on the number.
- Player A performs those operations and then tells the result to Player B.
- Player B then tries to identify Player A’s number.

These rules leave something to be desired, since Player B could simply ask A to “multiply by 1” and then “add 0,” in which case finding A’s number would involve no work whatsoever. To be a stickler, an additional rule could impose that either addition or subtraction can be used exactly once and that no operation can involve either 0 or 1. In a middle school classroom, I suppose I would state such a rule explicitly; for playing this game with my seven-year-old sons, I opted not to.

We played this game three times on the car ride to school yesterday. One game went like this:

- I thought of a number (14).
- Eli asked me to add 3 to my number.
- Alex asked me to multiply by 3.
- I told them the result: 51.

Eli then guessed that my number was 16. He had subtracted 3, then divided by 3.

“No!” said Alex. “You added 3 first, so you need to subtract 9.”

“Why 9?” Eli asked. “Daddy only added 3.”

“But he multiplied by 3, so if you subtract first, you have to subtract 3 × 3.”

Eli then realized that my number was 14.

He thought for a second. “Oh,” he said. “I should have divided by 3 **first**, then subtracted.”

Wow, I thought. This is going even better than I hoped.

Though they didn’t use the proper terminology, the boys had a great discussion about “undoing” operations by performing inverse operations in reverse order. In 10 minutes, they taught themselves how to solve a two-step equation:

3*x* + 3 = 51

Grace Kelemanik once said that she knew she was being effective when she didn’t have to say a word. She’d watch from the back of the room as students carried the conversation and guided one another to correct mathematical thinking.

I will never claim to be half the educator that Grace Kelemanik is. But yesterday morning, I was pretty darn effective.

**I’d love to hear about math games you’ve played with kids, whether you invented them or not.**

### Improving a Math Game

It was 7:02 a.m. on a Saturday morning. Alex ran into my bedroom and woke me from an incredible dream — I was speaking to Riemann, Newton, Pascal, and several other dead mathematicians, and they were just about to reveal an odd perfect number.

“Deedy!” he yelled — somehow *daddy* has been transformed to *deedy* in my house — and I sat bolt upright.

“What?” I asked, rubbing the sleep from my eyes.

“Do you know what 58 × 46 is?”

“I have no idea,” I told him. “What is it?”

“I don’t know, either,” he said. “But it was one of the questions Eli gave me on this morning’s math quiz.”

A few minutes later, he had the answer to that exercise and several others that appeared on the quiz that his brother had created for him.

This is what my twin six-year-olds do. They give each other math quizzes. With two-digit multiplication exercises and slightly more complex combinatorics problems (“How many two-digit numbers don’t have a 3 in them?”). For fun.

So when they recently brought home a math game from school called *One Less* — in which each player rolls a die and has to place a token on a number that is “one less” — my only thought was, “Really?”

Verbatim, here are the directions to the game:

Each player gets 10 counters. Players take turns rolling a die and placing a counter on a number that is one less than the number rolled. The game ends when one player has placed all 10 counters.

Upon reading the directions, I had one question: **RUFKM?**

Let’s review.

- Kids who perform multi-digit multiplication
*for fun*are asked to do single-digit subtraction*for homework*. - The game ends when one player uses all his counters. Mind you, no one actually
*wins*— the game just*ends*.

Well, this will never do.

I opted not to send a note to the teacher about how they need to increase the rigor of their mathematics curriculum. Doing so would just make me **that guy**.

Instead, I decided to turn a bad game into a good game. So we modified the rules as follows:

On a turn, a player rolls a die and places a coin on a space with a value one less than the number rolled. Players alternate turns. A player earns a point each time she gets three of her coins in a row. Game ends when one player has used all 10 coins. The winner is the player with the most points.

This allowed for all kinds of interesting questions:

- What’s the maximum possible score in a game?
- What’s the best arrangement of numbers on the game board?
- Will the first player always win?
- How does the game change if points are awarded for two-in-a-row or four-in-a-row?
- How does the game change if scoring gives 1 point for one-in-a-row, 3 points for two-in-a-row, 6 points for three-in-a-row, 10 points for four-in-a-row, and so on?
- How much wood could a woodchuck chuck if a woodchuck could chuck wood?

We determined the answer to the first question (8 points), and we agreed that we didn’t much care to know the answer to the last question. It seems like the first player shouldn’t always win; but he did in all of the games that we played.

As for the best arrangement of the game board, I have no idea. But if you’d like to explore, several game boards are included in the PDF link below.

What modifications have you made to games to improve them or to make them more mathematically robust?