Puzzling Week Returns: Coxeter's Pippin

Peeled & equatorially cut apples ready to be cored for making apple slab.
January 24, 2011

John and I peel dozens of apples for pie, sauce, and slab. Depending on the size and quality of the apples, we will either peel and core by hand (with or without a corer) or by crank. John recently taught me that cutting apples equatorially and then coring them takes less time then the traditional cutting and coring. I have often wondered if there was an even easier or better way than all of the methods we have tried. I found a way tonight! But this is a puzzle. Your clue is the photo I took (above) of equatorially cut apples from January when I made two huge slabs for the weekend. By the way, I no longer use the King Arthur slab recipe. I double a pie crust recipe for a double 9” pie, use a 13x9 Pyrex pan, and just make an apple pie (butter, flour, brown sugar, cinnamon, nutmeg, ginger, cloves are all measured by feel and sight) in the rectangular Pyrex.

New Scientist Magazine December 21, 1961

There are many ways of eating an apple. At one extreme is the child who met the request, "Can I have the core of your apple?" with a flat, "There ain't going to be no core!". Various procedures with various implements are designed to remove the core, but probably not one apple in a million is cored in the most efficient fashion possible. How in fact should an apple be cored, to remove all the core with the least possible waste?

Spoiler Alert! The answer is the first comment to this post.

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Puzzler: Kenken Puzzles

Play online at kenken.com or at the New York Times (free but registration required).
Technorati Tags: kenken, Will Shortz
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A Number Game by Will Shortz

The computer lady gave me this puzzle today and I can't do a thing until I solve all of the equations. This is the text that accompanied the puzzle:

This test gauges your mental flexibility. Few have been found who could solve more than half of the questions on the first try. The puzzle originally consisted of 24 "equations" by Will Shortz, printed in the May-June 1981 issue of Games Magazine [to which I subscribe!], with an acknowledgement to Morgan Worthy.

Instructions: Each equation contains the initials of words that will make it correct. furnnish the missing words. (example: 60 = M in an H. Answer: 60 = Minutes in an Hour.) Good luck!

1 = W on an U 2 = Number it T to T 3 = B M (S H T R) 4 = Q in a G 4 = Q in a G (Note: this Q in a G is a different answer than the previous one.) 5 = D in a Z C 6 = L on an I 7 = W of the A W 8 = S on a S S 9 = P in the S S 9 = L a C has 10 = D in a T N with the A C 11 = P on a F T 12 = K of the R T 13 = S on the A F 18 = H in a G C 20 = F and T 21 = D on a D

26 = L of the A 29 = D in F in a L Y 32 = D F at which W F 40 = D and N of the G F 52 = W in a Y 52 = C in a D 54 = C in a D (with the J) 57 = H V 64 = S on a C 80 = D to G A the W 88 = P K 90 = D in a R A 99 = B of B O the W 200 = D for P G in M 1000 = W that a P is W 1001 = A N 2001 = a S O 20,000 = L U the S

I have solved some of these and I have found a spoiler (which I am not using!). My solutions and the spoiler URL are in the comments. Add your own solutions!

Technorati Tags: Will Shortz, A Number Game
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A Math Problem For You

Click on the image to read it full-size. Have fun!

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Set Duel: 2nd Win!

I can't believe it — twice in one month! My time? 1:35. Not good, but good enough. Of course, I couldn't even do today's USA Today five-star Sudoku puzzle. Next Saturday, go play Set and post your time at Deb's blog!

Technorati Tags: Set Weekly Challenge
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I Won the Set Duel!

I have played Set since July of 1999 when I was introduced to it in a discrete mathematics course at Rutgers University. I played every single day. That's a lot of games. It is the first thing I do when I wake up in the morning. I have learned that a good time for my Set game is a good brain day. A bad time will be a slow brain day. Then I stumbled across the Set Weekly Challenge at Debra's deblog.com. The times of the Set players there astounded me. I wanted so badly to win the badge for a week but it seemed impossible. Until yesterday! My time of 48 seconds beat everybody (and that's saying a lot because I use a laptop with no mouse, I still play it first thing in the morning without caffeine, and my competition have had times as low as 20 seconds).

I can only claim this award until next Saturday night when another player will walk off with it. I wish I could keep it in my sidebar and list the dates that I won (there may be more!). But alas, I can't. So stop by deblog — Debra is an amazing woman involved with TwitterLit, KidderLit, and BAFAB.

Technorati Tags: Set Weekly Challenge
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Unplugged Project: Mancala

Update here.

Cut egg cartons apart to make your own Mancala game. Paint them or cover with pretty papers and you have a custom made and worthwhile game that two people can play. I am making mine for my classroom. I have cut apart the egg cartons and will have students at school paint them with paints that the art teacher feels would work best. The only other thing I need for the games are 24 little stones for each game. I may use beans or poker chips that I have at school because stones are rather difficult to find under a few feet of snow.

I piled them up in front of Possum's little countertop bed and she isn't sure if she likes this. (I had to use this opportunity to post a couple of the few good photos I get of this girl.)

What is Mancala? It is an ancient game from Africa that is mathematically significant for your children to learn. I am not even going to bother any of us with the mathematics but if you like, you can have your children talk about strategies the will increase their chances of winning. It is a rather complex game to learn, but once you learn you become addicted. I used to know how to play years ago but I had to learn all over again in order to write this post. You cannot overemphasize the importance of game playing for mathematics learning and for family relationships.

If you click the Mancala Snails image above, you will be taken to an online, free version of Mancala. It is the easiest way to learn how to play. Just choose "beginner lever" and play until you begin to develop strategies. There are many variations of the game, but stick to the basics that are in Mancala Snails. The rules used there are accurate and true to the real game. It is said that Mancala is the historical precursor to chess. I know for a fact that children love to play this game because they always ask me to get one for the classroom. Now I enjoy playing the computer at Mancala Snails when I am tired and just want to relax.

Possum is still a bit startled by the mess about her and hopes I take the cartons to school in the morning.

Thank you for stopping by.

Visit other participants of the Project at Unplug Your Kids
or clickUnplugged Project

Next week's Project: Pipecleaner (or Twist Tie? Or Wire?)

Technorati Tags: mathematics games, Mancala, Mancala Snails, Unplugged Project, sowing games, combinatorics
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Unplugged Project: Matchsticks Game

Toothpicks (and, not as safely, matchsticks) can be used for a vast number of mathematics games in order to develop geometric knowledge, problem-solving skills and critical thinking, meeting NCTM standards. I have embedded a Flash game in this post so that you and your family can play Matchsticks. It has a scoreboard that will keep your scores forever! So be sure to bookmark this post so that you can come back and play often. The game will open in a new window, so you may have to enable popups for this post only.

Click here to play Matchsticks

Other mathematically excellent sites for toothpick games and puzzles:

Puzzle Corner: Toothpick Games
Toothpick World (with a toothpick applet to
solve the more than forty puzzles available).

Thank you for stopping by.

Visit other participants of the Project at Unplug Your Kids
or clickUnplugged Project

Next week's Project: Egg Cartons

Technorati Tags: mathematics games, Matchsticks, Unplugged Project
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A Fun Arithmetic Game That Sparks Exploration >> Fun Math Blog

Unplugged Project: Puzzles: I missed Puzzle week for the project so I offer my work now.

A fun arithmetic game that sparks exploration >> Fun Math Blog: 

Here’s a game that’s easy and leads to a nice exploration of number theory for those so inclined. Two people play. All you need is a sheet of paper and a pencil or pen. Here’s how to play:

• Each person thinks of a number between 1 and 50 without telling the other person what the number is. Then, each person writes their number on the sheet of paper.
  
• Decide who is going to go first, by tossing a coin or in some other mutually agreeable way.
  
• Players take turns writing down the positive difference between any two numbers on the sheet of paper.
  
• Numbers cannot appear more than once on the paper.
  
• The player who cannot write down a unique positive difference loses.
  

I have used this game in two 5th/6th grade mathematics classes so far. As simple as this game appears on the surface, I suggest that you read the rest the original post to learn its importance to your child's mathematics education. It is a rich and valuable game for them to puzzle over. 

I have posted, as a comment at the original post, what my classes have learned in forty minutes of playing. I won't repeat the observations here in hope that you will play and reach your own conclusions first. But we have not even scratched the surface of the extensions of this game. I need to do some independent study about Euclid's Algorithm before I use it with my older students. I am even considering using it with my college classes. 

If, after learning this game with your child, you find that there is a time when you cannot play with her, you will be interested in this quote from Wild About Math!:

This game is related to Euclid’s algorithm and to the greatest common divisor of two integers. At Cut the Knot there’s a Java version of this game, Euclid’s Game, that you can play alone against the computer. In the computer game the computer picks the two starting number but you can practice determining who should go first.

I use the Cut The Knot site often as a problem solving tool. There are many Java applets there for many classes of problems. We especially use it with the Josephus Flavius problem (which also has extentions and applications to many other problems) but with a far less gruesome problem. 

Here are three screen grabs of Euclid's Game. In the first image, I have chosen to go first because the computer generated numbers are 29 and 18:

In the next image, I purposefully made a subtraction error to demonstrate the feedback that the applet provides:

And finally, I won!

Thank you for visiting. 

To view other projects please click Unplugged Project 

or the links above.

Technorati Tags: Wild About Math!, Fun With Math, Euclid's Algorithm, Euclid's Game, Cut The Knot, GCD, mathematics games

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Futility Closet: Bread Alone

Bread Alone |Futility Closet: 

Andy and Bill are traveling when they meet Carl. Andy has 5 loaves of bread and Bill has 3; Carl has none and asks to share theirs, promising to pay them 8 gold pieces when they reach the next town.

They agree and divide the bread equally among them. When they reach the next town, Carl offers 5 gold pieces to Andy and 3 to Bill.

'Excuse me,' says Andy. 'That's not equitable.' He proposes another arrangement, which, on consideration, Bill and Carl agree is correct and fair.

How do they divide the 8 gold pieces? . . . 

This could be a great portfolio problem. I'll have to play with it to see.

Technorati Tags: Futility Closet, mathematics, puzzle 

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Binary Card Tricks, Part 3

When a student asked why the binary number card trick worked, I decided to help them figure it out. Besides wanting to know why it worked, the kids wanted to know why we could only choose the numbers 1 to 50. These explorations are what makes mathematics exciting. Why do we study mathematics? Not to prepare us for the next course, but to discover and explore why the world works as it does.

We began by making a table of the addends of the numbers from 1 to 50. The only addends we could use were 1, 2, 4, 8, 16, and 32, since we only had one of each:Many students quickly saw patterns in this table: there is only one sum that begins with 1, two with 2, four with 4, etc. Also, 1 was only used once, before it was “changed” to a 2, then a 2 was used twice before it “changed” to a 4, etc.

Some students had difficulty seeing this, so I devised a “tally chart” so that we could see these patterns more clearly:
Very quickly, all students saw the patterns of the zeros and ones.

I then defined some terms for them so that we could speak the same language (if I ever do this again, I may delay this step). I showed them how we had shown that 11(base 2) = 3(base 10), and that this is called the binary number system. It is a way to count using only ones and zeros. We then talked about how the card trick worked because you can write any number with only one of each card that has a power of two on it. Many students saw that in order to choose numbers higher than 63, we would need to add one more card to our deck. We never did discuss why we only choose numbers from 1 to 50 instead of from 1 to 63.

Because of the many Christmas activities in the school, this was as far as I was able to go with one class. One class I never even saw during the whole week so I asked their homeroom teacher to show them the card trick and cut out their cards, which he did. In the future, I would add the NCTM Illumination activity so that students would make their own cards. I would also develop the connections to the base 2 number system and our base 10 system. We might even try other number systems (would a card trick work with base three numbers, even though you would need two cards for each place value?) Would there ever be enough time to do all this?

Related posts:
A Merry Binary Christmas
Binary Card Trick, Part 2

Technorati Tags: binary card trick, base two, base 10, NCTM, place value
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Binary Card Tricks, Part 2

I promised to continue the Merry Binary Christmas post when MacBook was home, and it is, so here are two more important links to that project:

Illuminations: Birthdays and the Binary System
"This lesson is a collection of three activities, all of which revolve around patterns and place value in the binary system. Grades 5‑8 students are drawn into the mathematics by the "magical" ability to guess an unknown number and by the use of birthdays, something they find very relevant. This lesson plan is adapted from the September 1997 edition of Mathematics Teaching in the Middle School."

Mudd Math Fun Facts:Binary Card Trick: "You put a deck of cards in your pocket, and invite anyone in the audience to call out a number between 1 and 15. Then you reach into your pocket, you take out a set of cards whose sum is the number that was called!"

Related posts:
A Merry Binary Christmas
Binary Card Tricks, Part 3

Technorati Tags: NCTM, Mudd Math Fun Facts, binary card trick, Mathematics in the Middle School, Illuminations
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Unplugged Project: Knots And A Cipher Update

Sorry, Mom Unplugged. No string theory today.

I got you-know-who The Dangerous Book for Boysfor Christmas. And

then I got a copy for myself because of the cipher wheel instructions on page sixty-seven. The cipher wheel activity blends perfectly with my Unplugged Project on ciphers. Identical, free instructions can be found here at Scholastic Kid's Spy Academy. (There are great online activities here, at Scholastic, for home and school.)

The connection to this week's Unplugged Project is the section "Five Knots Every Boy Should Know." The Dangerous Book has a web site with a special section for teachers with some free activities to use at recess or at a Parent/Son Night. And one of the activities is this knots section from the book.

Beware of this pdf activity file, though. I have had to use my school PC while my Macintosh is in the shop. And the pdf file for the knots activity crashed this PC twice. Finally, I right-clicked on the link, downloaded the file and opened it locally. It then worked. Be sure to click all of the stickie notes on the book graphic. You can print out certificates, download a section of the book and even print out badges after completion of parts of the book. There are also other valuable links.

I am still planning a math club that will engage kids, especially boys, and I think I will use this Dangerous book. It is stuffed with mathematics activities.

Please note that I am blogging on a PC, which I find inordinately difficult. All of the formatting is differently than on a Mac. I also cannot work with my photos because of my lack of PC software. Therefore, my few posts this week are using graphics linked to their source.

Thank you for stopping by.

Visit other participants of the Project at Unplug Your Kids
or clickUnplugged Project

Next week's Project is in two weeks: COLLAGE

Technorati tags: The Dangerous Book For Boys, Conn Iggulden, Hal Iggulden, math club, cipher wheel, knots
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2008 Old Farmer's Almanac Puzzlers

A farmer goes to market with $100 to buy a total of 100 animals. He has to buy cows, which are $10 each; sheep, which are $3 each; and chickens, which are 50 cents each. How many of each animal did he buy for $100? contributed by Fred Raby, Fenelon Falls, Ontario

A potato and a tomato cost 40 cents. A tomato and an onion cost 50 cents. An onion and a potato cost 60 cents. How much does each single vegetable cost? contributed by Sidney Kravitz, Dover, New Jersey

If a hen-and-a-half lays an egg-and-a-half in a day-and-a-half, how many eggs can six hens lay in six days?

Two men travel by day in the same direction around an island twenty-four miles in circumference, and camp at night. Mr. A starts one mile ahead of Mr. B and goes one mile in the first day, three miles in the second day, and so on, increasing his rate by two miles each day. Mr. B goes five miles every day. When do they camp together?

No answers here. These are elementary and are great warm-ups. The answers are in the 2008 Old Farmer's Almanac by Robert B. Thomas. Leave your solutions in the comments!

Technorati tags: Technorati Tags: Old Farmer's Almanac, math, puzzles
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A Merry Binary Christmas

Every year I give my students a puzzle to make and take home. I choose puzzles that will amaze their families and friends. This year we will make these guess-the-number-cards. After you print and cut out these cards, you give them to a friend. You ask them to think of a secret number between one and fifty. Then have them give you back any card that contains that number.

You can "guess" their magic number by adding the powers of two that are on the bottom left corner of each card that they return to you. For instance: if their magic number is 17, they will return cards 1 and 16, which are in the bottom left corners. You add 1 + 16 and tell your friend that 17 was their magic number. They will be amazed! And best of all, they probably will never figure out how you guessed it.

Wild About Math has another version of this trick here that you should check out.

What's the secret? In base 2, 17 is written as 1001 (base 2) because it contains one 16 (or 2^4), no 8s (or 2^3), no 4s (or 2^2), no 2s (or 2^1) and one 1 (2^0).

2^4 + 2^0 = 16 + 1 = 17

You can see that their are many lessons here. I could connect this game to Egyptian and Russian Peasant multiplication. But I'm leaning towards using the Towers of Hanoi game as an extension.

The Tower of Hanoi (sometimes referred to as the Tower of Brahma or the End of the World Puzzle) was invented by the French mathematician, Edouard Lucas, in 1883. He was inspired by a legend that tells of a Hindu temple where the pyramid puzzle might have been used for the mental discipline of young priests. Legend says that at the beginning of time the priests in the temple were given a stack of 64 gold disks, each one a little smaller than the one beneath it. Their assignment was to transfer the 64 disks from one of the three poles to another, with one important provisoøa large disk could never be placed on top of a smaller one. The priests worked very efficiently, day and night. When they finished their work, the myth said, the temple would crumble into dust and the world would vanish

What does this game have to do with our number guessing cards? Read on and you shall see. The legend mentions that 64 disks are used. You can't possible play with all sixty-four disks. The game usually comes with up to seven disks. There are wooden versions with seven disks that you can buy here (the source of the Legend of the Towers of Hanoi above.) You can also play online at many sites.

Solving this game with even seven disks is daunting, so suggest to your children that they use the problem solving strategy of making a simpler problem. As they solve from one to two to three and greater number of disks, they will begin to notice patterns in how to move the disks.

They may begin to notice that there are a minimum number of moves required to solve the puzzle for different numbers of disks (if not, ask them if there are different numbers of moves possible). I have made a table of the minimum number of moves required for different numbers of disks on the Hanoi game:

Do you see that this game is about the powers of two?

Have your kids make a table like mine (perhaps without the algebra) if it is age appropriate. What are the differences between the minimum number of moves? Hopefully they will recognize that the differences are also powers of two and that will help them write either a recursive or general rule. They can then challenge themselves to solve the seven disk puzzle in 127 moves.

A screen shot of my favorite online implementation of the Towers of Hanoi from NLVM:

Our last activity is to figure out what the value of 2^64 is. If the monks were able to move a disk a second, day and night, without stopping, how long would it be until the world came to an end? I give everybody calculators and let them loose with it. Last night, I found a wonderful tool to help them learn and practice conversions: the unit conversion tool from NLVM. This tool is probably inappropriate for most elementary students, but it's a neat place.

Kids being kids, there is a part of them that will think it marvelous that they are part of predicting the end of the world. The problem is that unless you do it also, you will never know. You will have to provide benchmarks or checks along the way so that the kids don't become lost in these huge numbers. While researching this article, I found two different answers to this problem (I have decided not to link you to them). This doesn't speak too well to our mathematics on the Internet. So be bold and do it yourself! I calculated the correct answer. Is there any relationship between the legend and the actual age of the earth? Of the solar system? Of the universe?

(If you would like a copy of these cards in .pdf or .doc format, please e-mail me.)

Related posts:
Binary Card Tricks, Part 2
Binary Card Tricks, Part 3

Technorati tags: binary+decomposition Towers+of+Hanoi unit+conversion math+games
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MathNotations: Products of Digits: Challenges for Everyone...: Part 2

Remember Dave Marain's Challenge? I posted this challenge a couple of days ago here. He quoted a riddle from a book, The Righteous Men by Sam Bourne:

Just men we are, our number few
Describable in digits two
We're halved if these do multiply
If we few perish then all must die.

Dave then posed a challenge:

(1) Determine a 2-digit positive integer the product of whose digits is one-half the integer. Now, it won't take you long to find such a number (once you get past the elliptical phrasing), but that's just to whet your appetite. The real challenge begins:

(2) Prove that your answer to (1) is unique, i.e., there is only one solution to the problem.

Here is my solution:

Since the solution is a two digit number, we have two "slots" to fill, one for the tens digit and one for the ones digit. The digit in the tens place will be multiplied by 10 and the digit in the ones place will be multiplied by 1.

I will call the first digit, in the tens place, a; and I call the 2nd digit, in the ones place, b.

Because both a and b must be positive integers (we cannot have parts of people or negative amounts of people), we can only consider positive values of b. We are also restricted to the numbers 1 through 9 because we can only use one digit in each place value location.

In order for the expression for the value of a to be positive, 2b must be greater than 10 (or, b must be greater than 5). This leaves us with only the numbers 6, 7, 8 and 9 to consider.

We can now eliminate the odd numbers (7 and 9) because the numerator of the expression for the value of a must be an even number because the denominator will always be even (because the difference of 5 and b is always doubled). If the denominator will always be even, then the numerator must be even in order to get an integer.

We are now left with only 6 and 8 as possible solutions for b:

If b = 6:

If b = 8:

Therefore, 6 is the only possible solution for b. If b = 6, a = 3. Therefore, the integer we are looking for is 36. Half of thirty-six is 18. The product of 3 and 6 is 18. There were 36 people.

My middle school students would undoubtedly solve this problem using a table in Excel. They have had it drilled into them to show three ways to solve a problem: table, graph and equation. In this case, the graph would not be a proof for the solution but it would be confirmation that their solution is correct (assuming that their calculations are correct). I am curious about what equation they would be able to come up with, if any.



In the chart above, I failed to skitch in the point where both graphs intersect. But our students know how to explore the Excel chart to find where they intersect. As expected, the graph reflects the table from which it was made.

I am considering giving them this problem as one of their portfolio problems. I hope I have proof-read the first solution so that it is clear and error-free.

Technorati tags: product+of+digits mathematics+challenge Dave+Marain

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MathNotations: Products of Digits: Challenges for Everyone...

MathNotations: Products of Digits: Challenges for Everyone...: "Just men we are, our number few Describable in digits two We're halved if these do multiply If we few perish then all must die. "

Dave continues:

(1) Determine a 2-digit positive integer the product of whose digits is one-half the integer. Now, it won't take you long to find such a number (once you get past the elliptical phrasing), but that's just to whet your appetite. The real challenge begins:

(2) Prove that your answer to (1) is unique, i.e., there is only one solution to the problem.

Comment: We're looking for more than an exhaustive search through all ninety 2-digit numbers or a programmed solution. The key to this and all of the remaining questions is to find an approach to solving a single equation which has 2 or more variables whose domain is the set of positive integers. Students are usually not introduced to solving such equations but they appear frequently on SATs and Math Contests. Because we are looking only for positive integer solutions, a standard algebraic approach must be supplemented with arithmetic concepts and testing of several possibilities. Number theorists refer to these as Diophantine equations.

I have the solution (or is it "a solution"?). Now I have to see it if it is the only solution. I will update this post as I proceed. Right now I have to get ready for school.

"This investigation was authored by Dave Marain." 

Technorati tags: products+of+digits math+challenge Dave+Marain
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Egyptian geometry and other challenges: Let’s play math!

Egyptian geometry and other challenges: Let’s play math!:
FROM THE MOSCOW PAPYRUS 

(3) The area of a rectangle is 12, and the width is 3/4 of the length. How long are the sides of the rectangle?

The area of a rectangle equals length times width (l•w), which equals 12

(4) One leg of a right triangle is 2 1/2 times the other, and the area is 20. How long are the triangle’s sides?

The area of a triangle equals 1/2 of the base times the height (A=1/2 bh), which equals 20.

Technorati tags: Egyptian+geometry Rhind+papyrus Moscow+papyrus mathematics puzzles
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Egyptian geometry and other challenges: Let’s play math!

Egyptian geometry and other challenges: Let’s play math!:

Here is a math “magic trick” from the Rhind papyrus. (Warning: The Egyptian scribes loved working with fractions.) Your job is to explain why it works. How could Scribe Ahmose know that he would always be able to tell what number his friend had in mind?

• Tell your friend to think of a secret number. [To avoid fractions, pick a multiple of 9.]
  
• Then have him add 2/3 more to his number. [So if he started with 9, he would add 2/3 of 9: 9 6 = 15.]
  
• Finally, tell him to take away 1/3 of this total, and say the answer. [1/3 of 15 is 5, and 15 - 5 = 10.
  
• Your friend would say, “Ten.”]
  
• Now you must subtract 1/10 of that number to find the secret. [1/10 of 10 is 1, so the secret number is 10 - 1 = 9.]
  

OK, here is my algebraic solution: (x is the chosen number):

TA DA! 

I enjoy showing these (but simpler ones) to my college algebra classes. Usually the students enjoy it too and look for other math tricks to solve. Denise has posted the answers. This is my 2nd puzzle to solve. I want to do the triangle one before I peek at the solutions. I hardly ever teach geometry, so I want to look up the triangle inequality thing before I start. 

When clicked, this jpeg will open, full size, in a new window. This would've been more easily understood as a "paper" entry (papyrus / paper) for this week's photohunt instead of my Frost poem, which seemed to confuse people. Too late now.

Technorati tags: Rhind+papyrus Ahmes+papyrus mathematics+trick

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Text Savvy: Three 3s

Text Savvy

Neither of my solutions (my and jonathan's) was submitted by anybody else! I really have to overcome feeling intimidated by these mathematicians because I never submitted our answers. On to the next puzzle:

Your favorite little league baseball team is playing for the championship. The team will play seven games . . .

Technorati tags: Text+Savvy puzzle mathematics probability
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