Math Storytelling Day: The Hospital Floor

[Feature photo above by Christiaan Triebert via flickr (CC BY 2.0).]

Have you ever heard of Math Storytelling Day? On September 25, people around the world celebrate mathematics by telling stories together. The stories can be real — like my story below — or fictional like the tale of Wizard Mathys from Fantasia and his crystal ball communication system.

My Math Story

My story begins with an unexpected adventure in pain. Appendicitis sidewhacked my life last week, but that’s not the story. It’s just the setting. During my recovery, I spent a lot of time in the smaller room of my hospital suite. I noticed this semi-random pattern in the floor tile, which made me wonder:

• Did they choose the pattern to keep their customers from getting bored while they were … occupied?
• Is the randomness real? Or can I find a line of symmetry or a set of tiles that repeat?
• If I take pictures from enough different angles, could I transfer the whole floor to graph paper for further study?
• And if the nurse finds me doing this, will she send me to a different ward of the hospital? Do hospitals have psychiatric wards, or is that only in the movies?
• What is the biggest chunk of squares I could “break out” from this pattern that would create the illusion of a regular, repeating tessellation?

I gave up on the graph paper idea (for now) and printed the pictures to play with. By my definition, “broken” pattern chunks need to be contiguous along the sides of the tiles, like pentominoes. Also, the edge of the chunk must be a clean break along the mortar lines. The piece can zigzag all over the place, but it isn’t allowed to come back and touch itself anywhere, even at a corner. No holes allowed.

I’m counting the plain squares as the unit and each of the smaller rectangles as a half square. So far, the biggest chunk of repeating tiles I’ve managed to break out is 283 squares.

What Math Stories Will You Tell?

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Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

• How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

Reblog: Patty Paper Trisection

[Feature photo above by Michael Cory via Flickr (CC BY 2.0).]

I hear so many people say they hated geometry because of the proofs, but I’ve always loved a challenging puzzle. I found the following puzzle at a blog carnival during my first year of blogging. Don’t worry about the arbitrary two-column format you learned in high school — just think about what is true and how you know it must be so.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

One of the great unsolved problems of antiquity was to trisect any angle using only the basic tools of Euclidean geometry: an unmarked straight-edge and a compass. Like the alchemist’s dream of turning lead into gold, this proved to be an impossible task. If you want to trisect an angle, you have to “cheat.” A straight-edge and compass can’t do it. You have to use some sort of crutch, just as an alchemist would have to use a particle accelerator or something.

One “cheat” that works is to fold your paper. I will show you how it works, and your job is to show why …

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More Than One Way To Find the Center of a Circle

[Feature photo above by hom26 via Flickr.]

My free time lately has gone to local events and to book editing. I hope to put up a series of blog posts sometime soon, based on the Homeschool Math FAQs chapter I’m adding to the paperback version of Let’s Play Math. [And of course, I’ll update the ebook whenever I finally publish the paperback, so those of you who already bought a copy should be able to get the new version without paying extra.]

But in the meantime, as I was browsing my blog archives for an interesting “Throw-Back Thursday” post, I stumbled across this old geometry puzzle from Dave Marain over at MathNotations blog:

Is it possible that AB is a chord but NOT a diameter? That is, could circle ABC have a center that is NOT point O?

Jake shows Jack a piece of wood he cut out in the machine shop: a circular arc bounded by a chord. Jake claimed that the arc was not a semicircle. In fact, he claimed it was shorter than a semicircle, i.e., segment AB was not a diameter and arc ACB was less than 180 degrees.

Jack knew this was impossible and argued: “Don’t you see, Jake, that O must be the center of the circle and that OA, OB and OC are radii.”

Jake wasn’t buying this, since he had measured everything precisely. He argued that just because they could be radii didn’t prove they had to be.

Which boy do you agree with?

• Pick one side of the debate, and try to find at least three different ways to prove your point.

If you have a student in geometry or higher math, print out the original post (but not the comments — it’s no fun when someone gives you the answer!) and see what he or she can do with it.

Dave offers many other puzzles to challenge your math students. While you are at his blog, do take some time to browse past articles.

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The Linear Inequality Adventures of Ohio Jones

Last week, Kitten and I reached her textbook’s chapter on graphing linear equations, and a minor mistake with negative numbers threw her into an “I can’t do it!” funk. It’s not easy teaching a perfectionist kid.

Usually her mood improves if we switch to a slightly more advanced topic, and luckily I had saved these worksheets on my desktop, waiting for just such an opportunity. Today’s lesson:

• Some fun(ish) worksheets
“For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana’s lesser known brother)…”

I offered to give her a hint, but she wanted to try it totally on her own. It took her about 40 minutes to work through the first few rooms of the Lost Templo de los Dulces and explain her solutions to me. I’m sure she’ll speed up with experience.

So far, she’s enjoying it much more than the textbook lesson. It’s fascinating to me how the mere hint of fantasy adventure can change graphing equations from boring to cool. Thanks, Dan!

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2014 Mathematics Game

[Feature photo above by Artis Rams (CC BY 2.0) via flickr. Title background (right) by Dan Moyle (CC BY 2.0) via flickr]

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza!

For many years mathematicians, scientists, engineers and others interested in mathematics have played “year games” via e-mail and in newsgroups. We don’t always know whether it is possible to write expressions for all the numbers from 1 to 100 using only the digits in the current year, but it is fun to try to see how many you can find.

Rules of the Game

Use the digits in the year 2014 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable by age: Young children can start with looking for 1-10, or 1-25.

• You must use all four digits. You may not use any other numbers.
• Solutions that keep the year digits in 2-0-1-4 order are preferred, but not required.
• You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
• You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
• You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.
• You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.
• You may use a double factorial, but we prefer solutions that avoid them. n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n.

[Note to students and teachers: If you want to take part in the Math Forum Year Game, be warned that they do not allow repeating decimals.]

By Denise Gaskins Posted in Puzzles

Algebra for (Almost) Any Age

Fawn Nguyen’s Visual Patterns website just keeps getting better and better. Check it out:

In addition to the 115 puzzle patterns (as of this writing), the site features a Gallery page of patterns submitted by students. And under the “Teachers” tab, Fawn shares a form to guide students in thinking their way through to the algebraic formula for a pattern.

How can you use these patterns to develop algebraic thinking with younger students? Mike Lawler and sons demonstrate Pattern #1 in the YouTube video below.

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Things To Do with a Hundred Chart #30

Here’s one more entry for my 20+ Things to Do with a Hundred Chart post, thanks to David Radcliffe in the comments on Monday’s post:

(30) Can you mark ten squares Sudoku-style, so that no two squares share the same row or column? Add up the numbers to get your score. Then try to find a different set of ten Sudoku-style squares. What do you notice? What do you wonder?

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

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Puzzle: Algebra on Rectangles

Gordon Hamilton of Math Pickle recently posted these videos on how to make algebra 1 puzzles on rectangles. As I was watching, Kitten came in and looked over my shoulder. She said, “Those look like fun!”

They look like fun to me, too, and I bet your beginning algebra students will enjoy them:

Logic Puzzle: Imbalance Problems

Kitten and I have been slogging through the decimals chapter in AoPS Pre-Algebra. She hates arithmetic, so I tried skipping ahead to the algebra puzzle in the exercises, but she refused to be taken in: a decimal problem with an x in it is still a decimal problem.

So I let her off early and pointed her toward these logical “algebra” puzzles instead:

Puzzle by Paul Salomon

By Denise Gaskins Posted in Puzzles

2013 Mathematics Game

feature photo above by Alan Klim via flickr

New Year’s Day

Now is the accepted time to make your regular annual good resolutions. Next week you can begin paving hell with them as usual.

Yesterday, everybody smoked his last cigar, took his last drink, and swore his last oath. Today, we are a pious and exemplary community. Thirty days from now, we shall have cast our reformation to the winds and gone to cutting our ancient shortcomings considerably shorter than ever. We shall also reflect pleasantly upon how we did the same old thing last year about this time.

However, go in, community. New Year’s is a harmless annual institution, of no particular use to anybody save as a scapegoat for promiscuous drunks, and friendly calls, and humbug resolutions, and we wish you to enjoy it with a looseness suited to the greatness of the occasion.

For many homeschoolers, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the 2013 Mathematics Game!

By Denise Gaskins Posted in Puzzles

Sample The Moscow Puzzles

Dover Publications is offering a free sample chapter from The Moscow Puzzles.

Cat and Mice
Purrer has decided to take a nap. He dreams he is encircle by 13 mice: 12 gray and 1 white. He hears his owner saying: “Purrer, you are to eat each thirteenth mouse, keeping the same direction. The last mouse you eat must be the white one.”

More Free Math from Dover Publications

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Olympic Logic

I love logic puzzles! Nrich Maths offers four fun Olympics Logic puzzles. And be sure to check out the rest of their Nrich Olympics Math as well.

Medals Count

Given the following clues, can you work out the number of gold, silver and bronze medals that France, Italy and Japan got in this international sports competition?

• Japan has 1 more gold medal, but 3 fewer silver medals, than Italy.
• France has the most bronze medals (18), but fewest gold medals (7).
• Each country has at least 6 medals of each type.
• Italy has 27 medals in total.
• Italy has 2 more bronze medals than gold medals.
• The three countries have 38 bronze medals in total.
• France has twice as many silver medals as Italy has gold medals.

Go to Nrich Maths and try all four puzzles!

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Skit: Knights and Knaves Logic Puzzles

photo by puuikibeach via flickr

Our homeschool co-op held an end-of-semester assembly. Each class was supposed to demonstrate something they had learned. I planned to set up a static display showing some of our projects, like the fractal pop-up card and the game of Nim, but the students voted to do a skit based on the logic puzzles of Raymond Smullyan.

We had a small class (only four students), but you can easily divide up the lines make room for more players. We created signs from half-sheets of poster board with each native’s line on front and whether she was a knight or knave on the flip side. In the course of a skit, there isn’t enough time to really think through the puzzles, so the audience had to vote based on first impressions — which gave us a fair showing of all opinions on each puzzle.

By Denise Gaskins Posted in Puzzles

Raymond Smullyan Excerpts at Dover Publications

To celebrate their re-release of his classic puzzle books, the Dover Math and Science Newsletter featured an interview with Raymond Smullyan, as well as several extended excerpts from his books. (For my math club students: Professor Smullyan invented the Knights and Knaves puzzles.) Enjoy!

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2012 Mathematics Game

photo by Creativity103 via flickr

For our homeschool, January is the time to assess our progress and make a few New Semester’s Resolutions. This year, we resolve to challenge ourselves to more math puzzles. Would you like to join us? Pump up your mental muscles with the 2012 Mathematics Game!

Rules of the Game

Use the digits in the year 2012 to write mathematical expressions for the counting numbers 1 through 100.

Bonus Rules
You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.

You may use multifactorials:

• n!! = a double factorial = the product of all integers from 1 to n that have the same parity (odd or even) as n.
• n!!! = a triple factorial = the product of all integers from 1 to n that are equal to n mod 3

[Note to teachers: Math Forum modified their rules to allow double factorials, but as far as I know, they do not allow repeating decimals or triple factorials.]

By Denise Gaskins Posted in Puzzles

Giveaway: Hexa-Trex Puzzle Book

Bogusia Gierus, host of this month’s Math Teachers at Play blog carnival, is offering to give away her First Book of Hexa-Trex Puzzles for just the cost of shipping. How generous!

My math club had fun with several of these puzzles a few years ago, and the “Easy” ones (like the sample shown here) were just right for my 4th-5th grade students. One girl enjoyed them enough that she took home extra copies to share with her father.

It’s a thin book, just the right size for a stocking-stuffer. To see the full range of difficulty levels, look over the puzzles on Bogusia’s Daily Hexa-Trex page. To get your own copy of the book, read the giveaway instructions on Bogusia’s blog.

Object of the Puzzle

The object of the puzzle is to find the equation pathway that leads through ALL the tiles.

Forming Equations

• Two or three (or four or five etc.) digit numbers are made up of the individual tiles in the particular order as the equation is read. For example 5 x 5 = 2 5 is correct, but read backwards 5 2 = 5 x 5 is incorrect.
• The equation must be continuous (no jumping over tiles or empty spaces).
• Each tile can be used ONLY ONCE.
• Order of operations is followed. Multiplication and division comes before addition and subtraction.
• The tile “-” can be used as both a subtraction operation or a negative sign in front of a digit, making it a negative number.

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What to Do with a Hundred Chart #27

[Photo by geishaboy500.]

It began with a humble list of 7 things to do with a hundred chart in one of my out-of-print books about teaching home school math. Over the years I added a few new ideas, and online friends contributed still more, so the list grew to its current length of 26. Recently, thanks to several fans at pinterest, it has become the most popular post on my blog:

Now I am working several hours a day revising my old math books, in preparation for publishing new, much-expanded editions. And as I typed in all the new things to do with a hundred chart, I thought of one more to add to the list:

(27) How many numbers are there from 11 to 25? Are you sure? What does it mean to count from one number to another? When you count, do you include the first number, or the last one, or both, or neither? Talk about inclusive and exclusive counting, and then make up counting puzzles for each other.

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

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More Than One Way to Solve It, Again

photo by Annie Pilon via flickr

We continue with our counting lessons — and once again, Kitten proves that she doesn’t think the same way I do. In fact, her solution is so elegant that I think she could have a future as a mathematician. After all, every aspiring novelist needs a day job, right?

If only I could get her to give up the idea that she hates math…

Permutations with Complications

How many of the possible distinct arrangements of 1-6 have 1 to the left of 2?

More Than One Way to Solve It

Photo by Eirik Newth via flickr.

In a lazy, I-don’t-want-to-do-school mood, Princess Kitten was ready to stop after three math problems. We had gotten two of them correct, but the last one was counting the ways to paint a cube in black and white, and we forgot to count the solid-color options.

For my perfectionist daughter, one mistake was excuse enough to quit. She leaned her head against me as we sat together on the couch and said, “We’re done. Done, done, done.” If she could, she would have started purring — one of the most manipulative noises known to humankind. I’m a soft touch. Who can work on math when there’s a kitten to cuddle?

by tanjila ahmed via flickr

Still, I managed to squeeze in one more puzzle. I picked up my whiteboard marker and started writing:

DONE
DOEN
DNOE
DENO
DNEO
ONED
ODNE

The (Mathematical) Trouble with Pizza

Photo by George Parrilla via flickr.

Kitten complained that some math programs keep repeating the same kind of problems over and over, with bigger numbers: “They don’t get any harder, they just get longer. It’s boring!”

So we pulled out the Counting lessons in Competition Math for Middle School. [Highly recommended book!] Kitten doesn’t like to compete, but she enjoys learning new ideas, and Batterson’s book gives her plenty of those, well organized and clearly explained.

Today’s topic was the Fundamental Counting Principle. It was review, easy-peasy. The problems were too simple, until…

Pizzas at Mario’s come in three sizes, and you have your choice of 10 toppings to add to the pizza. You may order a pizza with any number of toppings (up to 10), including zero. How many choices of pizza are there at Mario’s?

[The book said 9 toppings, but I was skimming/paraphrasing aloud and misread.]

• Can you figure out the answer?

2011 Mathematics Game

[Photo from Wikipedia.]

Two of the most popular New Year’s Resolutions are to spend more time with family and friends, and to get more exercise. The 2011 Mathematics Game is a chance to do both at once.

So grab a partner, slip into your workout clothes, and pump up those mental muscles!

Here are the rules:

Use the digits in the year 2011 to write mathematical expressions for the counting numbers 1 through 100.

• All four digits must be used in each expression. You may not use any other numbers except 2, 0, 1, and 1.
• You may use the arithmetic operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), and ! (factorial). You may also use parentheses, brackets, or other grouping symbols.
• You may use a decimal point to create numbers such as .1, .02, etc.
• Multi-digit numbers such as 20 or 102 may be used, but preference is given to solutions that avoid them.

Bonus Rules
You may use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal.

You may use multifactorials:

• (n!)! = a factorial of a factorial, which is not the same as a multifactorial
• n!! = a double factorial = the product of all integers from 1 to n that have the same parity (odd or even) as n
• n!!! = a triple factorial = the product of all integers from 1 to n that are equal to n mod 3

[Note to teachers: The bonus rules are not part of the Math Forum guidelines. They make a significant difference in the number of possible solutions, however, and they should not be too difficult for high school students or advanced middle schoolers.]

By Denise Gaskins Posted in Puzzles

A Football Puzzle

[Photo by rdesai.]

The MIT Mathmen got the ball on their own 20-yard line for the last drive of the game. They were down by 2 points, so they needed at least a field goal to win the game.

If quarterback Zeno and his offense advanced the ball halfway to the opposing team’s end zone on each play…

Logic Games at Blogging 2 Learn

Image via Wikipedia

For the rest of NaBloPoMo (National Blog Posting Month), my other blog is featuring a logic game or puzzle every day. So far, I’ve shared three of my online favorites:

And there’s plenty more fun to come. Drop in every day until December to see a new puzzle or game:

Don’t miss any of “Let’s Play Math!”:  Subscribe in a reader, or get updates by Email.

A Couple of Chess Puzzles

Image via Wikipedia

Chess is a favorite game for recreational mathematicians — not to play it, but to play around with it. Many puzzles and challenges are based on the moves of chess pieces.

Stretch your brain with these puzzles:

• Can you go on a Knight’s Tour? Start your knight on any square, and try to hop around to all the rest.
• Or, how many queens can you place on the board so that no queen can capture another?

Lewis Carroll’s Logic Challenges

Image via Wikipedia

Symbolic Logic Part I was published in 1896. When Lewis Carroll (Charles Lutwidge Dodgson) died two years later, Part II was lost. Because they couldn’t find the manuscript, many people doubted that he ever wrote Part II. But almost eighty years after his death, portions of Part II were recovered and finally published. The following puzzles are from the combined volume, Lewis Carroll’s Symbolic Logic, edited by William Warren Bartley, III.

These puzzles are called soriteses or polysyllogisms. Carroll began with a series of “if this, then that” statements. He rewrote them to make them more confusing, and then he mixed up the order to create a challenging puzzle.

Given each set of premises, what conclusion can you reach?

How to Start an Argument: The Monty Hall Problem

[Photo by MontyPython.]

You can get a good argument going in almost any group of people with the infamous Monty Hall problem:

Imagine you are on a TV game show, and the host lets you choose between three closed doors. One of the doors hides a fancy sports car, and if you pick that door, you win the car.

You pick door #1.

The host opens door #3 to reveal a goat. Then he gives you a chance to switch your door for the unopened door #2.

Should you switch?

What if you say you’re going to switch, and then the host offers to give you \$5,000 instead of whatever is behind door #2?

Try the game for yourself at the Stay or Switch website.