## Posts tagged ‘game’

### 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?

### Two Simple Math Games

In his article “What Is the Name of This Game?” author John Mahoney discussed the mathematics of the following game:

Cards numbered 1-9 are placed face up on a table. Two players alternate picking up one card at a time. The winner is the first player who has exactly three cards with a sum of 15.

You can play this game with nine cards removed from a deck of cards, or you can play online by going to http://illuminations.nctm.org/deepseaduel. The online version is a one-player game, but it has modifications that use different numbers of cards, different values on the cards, and different required sums.

Can you find the winning strategy for this game? (Hint: The strategy is described in the linked article above.)

Here’s a modification of the game that seems interesting, too.

Use cards with the following numbers: 1, 2, 3, 4, 6, 9, 12, 18, 36. The winner is the first player who has exactly three cards with a product of 216.

The optimal strategy for this game is different than the strategy from the original game. Can you find it?

**Note:** For the original game, there are eight sets of three cards with a sum of 15:

{1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8},

{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}

For the modification, there are 36 sets of three cards with a product of 216.

Perhaps that fact will help you identify the optimal strategy.

### Swish, Swash, I Was Doin’ Some Math

I’m not sure *what* I want to be when I grow up, but I know *who* I want to be. That would be Bill Ritchie, owner and founder of Thinkfun^{®}, a company that makes and markets games and puzzles, including the wildly popular Rush Hour^{®} game.

Our house has recently become addicted to Swish^{TM}, one of Thinkfun’s newest games. *Swish* is a spatial card game played with transparent cards that can be rotated and flipped to make swishes. Swishes — named after the sound made by a perfect basketball shot going through the net — are made by layering as few as two or as many as 12 cards so that every ball swishes into a hoop of the same color.

For instance, the two cards below can be combined to make a swish. The orange dot on the left card aligns with the orange hoop on the right, and the blue hoop on the left aligns with the blue dot on the right, which forms a double swish when the cards are placed one on top of the other.

Here’s a more advanced example. These four cards can be layered to make a quadruple swish:

The examples above show the cards in perfect alignment. They just have to be placed one on top of another for the dots and hoops to align. What makes the game fun and challenging is that cards typically aren’t in perfect alignment like this. One or more of the cards will have to be flipped or rotated to make a swish. In addition, the set-up arranges 16 cards in a 4 × 4 grid, so the two complementary cards are rarely next to one another. Consequently, the game uses all three geometric transformations — reflections, rotations, and translations.

Each card contains one hoop and one dot. These two objects are placed in one of 12 regions — each card is divided into an imaginary 4 × 3 grid. Theoretically, it’s possible to make a 12-tuple swish… though I’ve yet to pull off such a feat during a game.

There are a number of interesting math questions that can be asked about the game:

- Will there always be a swish in any 4 × 4 array of cards?
- The deck contains 60 cards. Does that account for every possible combination of hoops and dots? [The answer is obviously no, but that leads to some interesting follow-up questions.] How many different cards are possible? Finding the total isn’t as trivial as it sounds, since cards can be rotated or flipped; duplicates need to be removed. Which cards were included in the deck, which ones were excluded, and why?
- How many different double swishes can be formed by the cards in the deck? How many triple swishes? … How many 12-tuple swishes?

I don’t have answers to all of those questions yet. But I look forward to discussing them (and others) when I introduce *Swish* to some colleagues at an upcoming math conference!