## Posts tagged ‘grid’

### Covering 100 Squares

There’s an old math joke that says math books are sad because they have too many problems. But I disagree; I believe that most math books — and, in particular, textbooks — are sad because they have too many *exercises*.

To try to release more true problems into the wild, I recently wrote a book called *One-Hundred Problems Involving the Number 100*. Published by NCTM, the book contains problems, suggestions for classroom use, and solutions. Some of the problems are old chestnuts, such as, “What is 1 + 2 + 3 + ⋯ + 100?” (Problem 21: Gauss and Check). Others are complete originals, and occasionally a little silly, such as, “If all the positive integers from 1 to 100 were spelled out, how many letters would be used?” (Problem 1: Spell It Out). But my favorite problem in the book, which I’ve shared twice during NCTM’s 100 Days of Professional Development (**May 14**, **Oct 7**; login required), is Problem 100: Covering with Squares.

**As shown below, a square grid with 100 smaller squares can be covered by 100 squares (each measuring 1 × 1), by 25 squares (each measuring 2 × 2), or by 13 squares (one 6 × 6, two 4 × 4, two 3 × 3, two 2 × 2, and six 1 × 1).**

**Find all values of 0 < n < 100 for which it is impossible to cover a 10 × 10 grid with n squares of integer side length.**

I’ll admit, I’m not ecstatic about the phrasing of the question at the end. The symbolic representation makes it succinct, sure, but this problem can be investigated by students who may be too young to understand that notation. It might be better to just ask, “Can you cover it with two squares? Can you cover it with 58 squares? For how many different numbers of squares can you find a covering?”

You can, of course, explore this problem using graph paper. But for a digital experience, click on the link below and explore using Google Slides:

When you click on that link, you’ll be required to “Make a copy” for your own Google Drive. This is done for two reasons: first, it’ll prohibit any editing to the original file, so that other folks who use that link will receive a similarly pristine copy; and second, it places a copy in your Google Drive that you can play with, modify, or share with your students. If you do that, be sure to change **/edit** and anything that follows at the end of the URL to **/copy**, which will require anyone you share it with to make a copy, too. Of course, you could also just share the link above with your students, and they can have a copy of the original file.

When using this problem in the classroom, start by showing a few examples — such as the ones above — and then have all students find a covering for the grid that’s different from your examples. Because every student can find at least one covering, this breeds confidence. In addition, it ensures that all students understand the problem. When all students have found a covering, have them share it with a partner. From then on, allow students to work in pairs or small groups to find other coverings.

To facilitate the discussion, I draw a blank hundreds grid on the whiteboard. As unique coverings are found, students are allowed to enter the number in the hundreds grid. Of course, I require them to first verify with a partner that they have counted the number of squares correctly; once confirmed, they can show the covering to me; and if that number hasn’t been entered in the hundreds grid yet, they get to enter it. This can be motivating for students, and it’s a good opportunity to get less participatory students engaged.

When using this problem recently with a group of students, this is how the hundreds grid looked after about 20 minutes:

You’ll notice that numbers appear in different colors; each group received a different color dry erase marker. What you might also notice is that the numbers 17, 20, 23, 26, …, 65 all appear in red. This seemed more than coincidental, so I asked about it. The group members explained:

*We realized we could cover the grid with a one 6 × 6 square and sixteen 2 × 2 squares:*

*We then divided one of the 2 × 2 squares into four 1 × 1 squares. That meant that one 2 × 2 square was replaced by four 1 × 1 squares, increasing the number of squares by three. So, the grid was now covered with 17 + 3 = 20 squares. *

*If we kept doing that — if we kept dividing the 2 × 2 squares — then we’d keep adding three more squares, so we could get 23, 26, 29, and so on, all the way up to 65.*

The realization that one configuration could be transformed into many others allowed students to find coverings for myriad numbers. For what values of *n* could the grid not be covered? That’s left as a question for you. Have fun!

### It’s Not What’s on the Outside…

Through the Academic and Creative Endeavors (ACE) program at their school, my sons participate in the Math Olympiad for Elementary and Middle School (MOEMS). While passing the door to the ACE room yesterday, I noticed a sign with the names of those who scored a perfect 5 out of 5 on the most recent contest — and my sons’ names were conspicuously absent. Last night at dinner, I asked Alex and Eli what happened, and they told me about the problem that they both missed. (What? Like we’re the only family in America that discusses math problems at the dinner table.) Here’s how they explained it to me:

Nine 1-cm by 1-cm squares are arranged to form a 3 × 3 square, as shown below. The 3 × 3 square is divided into two pieces by cutting along gridlines only. What is the greatest total (combined) perimeter for the two shapes?

*The answer to this problem appears below. Pause here if you’d like to solve it before reaching the spoiler.*

[Ed. Note: I didn’t see the actual exam, so the presentation of the problem above is based entirely on my sons’ description. Apologies to MOEMS for any substantive differences.]

This problem epitomizes what I love about math competitions.

**The answer to the problem is not obvious.**This is the case with many competition problems, unlike the majority of problems that appear in a traditional textbook.**The solution does not rely on rote mechanics.**Again, this differentiates it from a standard textbook problem or — shudder! — from the problems that often populate the databases of many skill-based online programs.**Students have to get messy.**That is, they’ll need to try something, see what happens, then decide if they can improve the result.**Students have to convince themselves when to stop.**Or more precisely, they’ll need to convince themselves that they’ve found the correct answer. For instance, let’s say a student divides the square into a 3 × 2 and a 3 × 1 rectangle. The combined perimeter is 18 units. Is that good enough, or can you do better? This is different from, say, a typical algebra problem, for which students are taught how to check their answer.

The problem also epitomizes what many people hate about math competitions.

**There’s a time limit.**Students have 26 minutes to solve 5 problems. Which means that if students spend more than 5 minutes on this one, they may not have time to finish the other four. (There was a student of John Benson who, when asked about his goal for an upcoming math competition, replied, “I hope to solve half the problems during the competition and all of them by the end of the week.” That’s the way mathematicians work.)**It’s naked math.**Sorry, nothing real-world about this one. (But maybe that’s okay, because real may not be better.)**The problem is presented as a neat little bundle.**This is rarely how mathematics actually works. True problems often don’t present themselves all at once; it’s through investigation and research that the constraints become known and the nuances are revealed.

All that said, *I believe that the pros far outweigh the cons*. Benjamin Franklin Finkel said, “Many dormant minds have been aroused into activity through the mastery of a single problem.” I don’t remember the last time a mind was aroused by the solution to *x* + 7 = -3, but I’ve witnessed awakenings when students solve problems like the one above.

And here’s the tragedy in all of this: **Many teachers believe that only kids who participate in math competitions can handle — or appreciate — math competition questions.** No! Quite the opposite, in fact. Students who have tuned out have done so because they’ve never been challenged and, worse, have never felt the thrill of solving a problem on their own.

What I really love about the problem, though, is it made me think about other questions that could be asked:

- How many ways are there to divide a 3 × 3 square into two pieces that will yield the maximum total perimeter?
- What is the maximum total perimeter if a 3 × 3 square is divided into three pieces?
- What is the maximum total perimeter if a 4 × 4 square is divided into two pieces? …a 5 × 5 square? …a 6 × 6 square?
*The answer to the problem about the 4 × 4 square appears below. Pause here if you’d like to solve it before reaching the spoiler.*

- (wait for it) What is the maximum total perimeter if an
*n*×*n*square is divided into two pieces?

It was that last question that really got the blood pumping.

Here’s a solution for how to divide a 3 × 3 square to yield the greatest total perimeter:

And here’s a solution for how to divide a 4 × 4 square to yield the greatest total perimeter:

What’s the solution for an *n* × *n* square? That’s left as an exercise for the reader.

### A Gridiculously Clever Blog Post

Do you know what the following graph represents?

*Sine on the dotted line.*

If you tell that joke to the right audience, you’ll likely hear a *triggle*. (If you tell it to the wrong audience, you’ll likely hear the sound of tomatoes whizzing past your head.)

*Triggle* is a portmanteau, a combination of two or more words and their definitions.

trigonometry + giggle= triggle

In a similar vein, when the expression

13 + 5 · 0 – 4

is simplified to

13 – 4,

you might say that it has suffered from *zerosion* — the removal of a term because of multiplication by zero.

The following portmanteaux may be useful for your next math discussion.

**bi·sect·u·al**

*adjective*

attracted to both halves of an angle

**grid·ic·u·lous**^{1}

*adjective*

inviting derision on the coordinate plane

**cha·rad·i·us**

*noun*

a segment from the center to the circumference based on false pretenses

**bi·zarc**

*noun*

an unusual curve

**graph·ish**

*adjective*

diagrammatically disreputable

**sub·line**

*adjective*

inspiring awe in only one dimension

**trig·a·ma·role**

*noun*

a complicated and annoying trigonometric process, such as verifying that

cot *x* + tan *x* = sec *x* · csc *x*

^{1} It came to my attention after the publication of this post that Gridiculous is (a) a trivia game developed for Windows 8 and (b) an HTML5 responsive grid boilerplate (though the link to the site seems not to be working).

### Easiest KenKen Ever?

Saying that I like KenKen^{®} would be like saying that Sigmund Freud liked cocaine. (Too soon?) ‘Twould be more proper to say that I am so thoroughly addicted to the puzzle that the length of my dog’s morning walks aren’t measured in miles or minutes but in number of 6 × 6 puzzles that I complete. (Most mornings, it’s two.) Roberto Clemente correctly predicted that he would die in a plane crash; Abraham de Moivre predicted that he would sleep to death (and the exact date on which it would occur… *creepy*); and I am absolutely certain that I’ll be hit by oncoming traffic as I step off the curb without looking, my nose pointed at a KenKen app on my phone and wondering, “How many five-element partitions of 13 could fill that 48× cell?”

I am forever indebted to Tetsuya Miyamoto for inventing KenKen, and I am deeply appreciative that Nextoy, LLC, brought KenKen to the United States. How else would I wile away the hours between sunrise and sunset?

I am also extremely grateful that the only thing Nextoy copyrighted was the name KenKen. This allows Tom Snyder to develop themed TomToms, and it allows the PGDevTeam to offer MathDoku Pro, which I believe to be the best Android app for playing KenKen puzzles.

The most recent release of MathDoku has improved numerical input as well as a timer. Consequently, my recent fascination is playing 4 × 4 puzzles to see how long it will take. A typical puzzle will take 20‑30 seconds; occasionally, I’ll complete a puzzle in 18‑19 seconds; and, every once in a while, I’ll hit 17 seconds… but not very often.

Today, however, was a banner day. I was in a good KenKen groove, and I was served one of the easiest 4 × 4 puzzles ever. Here’s the puzzle:

And here’s the result (spoiler):

The screenshot shows that I completed the puzzle in just **15 seconds**. And it’s not even photoshopped.

This puzzle has several elements that make it easy to solve:

- The [11+] cell can only be filled with (4, 3, 4).
- The [4] in the first column dictates the order of the (1, 4) in the [4×] cell.
- The (1, 4) in the [4×] cell dictates the order of the (1, 2) in the [3+] cell.

After that, the rest of the puzzle falls easily into place, because each digit 1‑4 occurs exactly once in each row and column.

What’s the fastest you’ve ever solved a 4 × 4 KenKen puzzle? **Post your time in the comments.** Feel free to post your times for other size puzzles, too. (I’m currently working on a 6 × 6 puzzle that’s kicking my ass. Current time is 2:08:54 and counting.)

### SudoClue for a Cold Winter’s Night

The holiday break is nigh, which means you need to be careful not to catch a cold.

For many people, the time preceding a holiday, vacation, or spring break is busy — finishing up a term paper, completing holiday shopping, or getting things off your desk so you can enjoy your trip to Tahiti. During that time, your immunity kicks into high gear. It helps to fend off germs while you’re pushing yourself to get stuff done. When your break finally comes, though, you relax, and your body thinks, “Oh, cool, the stress is over.” And BAM! No more immunity, and your body succumbs to infection. Sniffles, headache, and a cough ensue.

How can you prevent this?

Easy. **Do puzzles.**

That’s right. You can trick your body into thinking that you’re still stressed by doing crosswords, sudoku, nurikabe, battleship, nonogrids, or whatever you like. Your mind is working hard, so your body keeps your immunity up. Doing something you enjoy fends off disease. Win-win.

Well, MJ4MF is here to help. The following **SudoClue** puzzle will provide a half-hour of much-needed stress. The idea is rather simple. Use the clues to fill in the corresponding squares of the 6 × 6 grid, then fill in the remaining squares like a sudoku puzzle.

The puzzle below, as well as an easier version, are available in PDF format.

**SudoClue Puzzle – Easy** **SudoClue Puzzle – Hard**

Enjoy!

- Number of unique tetrominoes
- Product of all single-digit divisors of 143
- Number of English words that end with –
*dous* - Half of π
^{2}, approximately - Number of tetrominoes that can be drawn without lifting your pencil from the paper
- 0!
- Number in the title of the greatest math joke book ever
- Integer between
*e*and π - Circumference divided by radius, to the nearest whole number
- Smallest number of colors sufficient to color all planar maps
- V
- Side length of a square whose area (in square units) is equal to its perimeter (in units)
- A perfect number
- A hat trick
- For integer values of
*n*, the smallest prime divisor of*n*^{2}+*n* - Number of total handshakes when four people shake hands with each other

Don’t feel like thinking too much on holiday break? Fine, **here’s a hint**. And if you’ve already fully entered holiday break mode, **here’s the solution**.

### Word Squares Revisited

On December 22, I posed the following puzzle in the Square Deal post:

Create a 4 × 4 grid composed of common English words (you can use the list of official Scrabble four‑letter words as a reference) such that the sum of the point values of the 16 letters is as high as possible.

In that post, I mentioned that I had created a grid worth 62 points, and many readers have asked about it. Here it is:

Z | I | Z | Z |

I | D | E | A |

Z | E | S | T |

Z | A | T | I |

All of the words are acceptable when playing Scrabble. While IDEA and ZEST are very common, the other two are not:

- ZIZZ = a nap
- ZATI = a species of macaque

I am certain that a better solution exists, but I have neither the time nor the energy to search for it. Anyone out there want to write a computer program to solve it?

** UPDATE (1/14/11): Check out the comments section from the previous post. Veky Edgar wrote a computer program and found the maximum possible score (92 points), and Scandinavian manually created a grid worth 85 points.

### Square Deal

Recently, my twin sons Alex and Eli have taken a shine to crossword puzzles. They’re only 3½, but they love letters and words, so I started making up crosswords for them using a free online crossword puzzle maker. I construct clues based on things they know — for instance, REMY is the answer to the clue OUR DOG, and IDAHO is the answer to the clue STATE NAME WE CAN SPELL WITH OUR BATH TUB LETTERS. (They have a set of foam alphabet letters, but each letter A‑Z occurs only once, so it’s not possible to spell any words with repeated letters.) They don’t quite have the motor skills to write the letters, so they read the clues and spell the answers aloud as I fill in the grid.

Tonight, I asked if they’d like to help me make a crossword puzzle. I drew a 3 × 3 grid, and I asked, “To make a crossword puzzle, you have to fill in words both down and across. Can you give me a word with three letters?” Eli suggested TOW, which I used in the first row of the grid:

T | O | W |

Then I asked, “Okay, so the word in the first column starts with a T. Can you think of a three‑letter word that starts with a T?” Never one to overlook the obvious, Eli suggested TOW again. I filled it in, and we moved to the middle column. “Can you think of a three‑letter word that starts with an O?” Alex suggested ONE. Eli immediately realized that we could now add an E at the end of the middle row to make ONE. At that point, eight of the nine squares were filled. I pointed to the third column. “Can you think of a three-letter word that starts with a W and an E?” Eli shouted WET, and the grid was complete:

T | O | W |

O | N | E |

W | E | T |

Eli then pointed out that ELI contains three letters. “Let’s make another one!” he said. So we did:

W | E | T |

E | L | I |

T | I | E |

Alex then requested that we make a 4 × 4 grid that included his name, and I was very impressed with the grid that they concocted:

I | O | W | A |

O | V | A | L |

W | A | V | E |

A | L | E | X |

I told you that story mainly because I like talking about my sons, but also because it leads me to a cool puzzle that I think you’ll enjoy.

Use the point values for each letter as in the word game SCRABBLE:

- 1 point: A, E, I, O, L, N, R, S, T, U
- 2 points: D, G
- 3 points: B, C, M, P
- 4 points: F, H, V, W, Y
- 5 points: K
- 8 points: J, X
- 10 points: Q, Z

Create a 4 × 4 grid composed of common English words (you can use the list of official Scrabble four‑letter words** as a reference) such that the sum of the point values of the 16 letters is as high as possible.

The 4 × 4 grid that my sons created would be disqualified, because IOWA and ALEX are proper nouns. That aside, it contains four A’s; two O’s, W’s, V’s, L’s and E’s; one I; and, one X. If it were acceptable, it would be worth 35 points.

I was able to create a grid with a sum of 62 points. I’m sure that better grids are possible. What’s the best that you can do?

** Special thanks to Veky Edgar, who pointed out that the list of four-letter words appearing on my web site was incorrect. The list has been updated, and I’ve stolen it from a more credible source this time, so I believe it is now correct.