## Kiss those Math Headaches GOODBYE!

### Everyday Life Sparks Mathematical Puzzles

So here’s the situation: you’re at the breakfast table, enjoying a bowl of steaming-hot steel-cut oats and maple syrup, and you just poured yourself a mug of black coffee. But then you realize you want to pour some milk in the coffee (sorry, purists). But the milk is in the frig, six feet away. So of course you walk to the frig, grab the milk, bring it to the table, pour some in your coffee, return the milk to the frig and sit back down. Question: could you have done this more efficiently?

Yes, of course. You could have brought your cup of coffee with you as you walked to the frig, poured the milk right there at the frig, returned the milk, and then walked back to the table.

“Morning Joe”

When I realized this this morning, I thought … hmmm. Had I used a bit of forethought, I would save myself an entire round trip from the table to the frig. And while I have no problem making that extra trip (hey, just burned 1.3 calories, right?), the experience made me wonder if anyone has ever developed a mathematics of efficiency for running errands.

I could imagine someone taking initial steps for this. One would create symbols for the various aspects of errands. There would be a general symbol for an errand, and there would be a special ways of denoting: 1) an errand station (like the frig), 2)  an errand that requires transporting an item (like carrying the mug), 3) an errand that requires doing an activity (pouring milk) with two items (mug and milk) at an errand station, 4) an errand that involves picking something up (picking up the mug), and so on. Then one could schematize the process and use it to code various kinds of errands. Eventually, perhaps, one could use such a system to analyze the most efficient way to, say, carry out 15 errands of which 3 involve transporting items, 7 involve picking things up, and 5 involve doing tasks at errand stations. Don’t get me wrong! I have not even begun to try this, but I’ve studied enough math that I can imagine it being done, and that’s one thing I love about math; it allows us to create general systems for analyzing real-world situations and thereby to do those activities more intelligently.

Of course, one reason I’m bringing this up is to encourage people to think more deeply about things that occur in their everyday lives. Activities that appear mundane can become mathematically intriguing when investigated. A wonderful example is the famous problem of the “Bridges of Konigsberg,” explored by the prolific mathematician Leonhard Euler nearly 300 years ago.

Euler in 1736 was living in the town of Konigsberg, now part of Russia. The Pregel River, which flows through Konigsberg, weaves around two islands that are part of the town, and a set of seven lovely bridges connect the islands to each other and to the town’s two river banks. For centuries Konigsberg’s residents wondered if there was a way to take a walk, starting at Point A, crossing each bridge exactly once, and return to Point A. But no one had found a way to do this.

One of the famous Seven Bridges of Konigsberg

Enter Euler. The great mathematician sat down and simplified the problem, turning the bridges into abstract line segments and transforming the bridge entrance and exits into points. Eventually Euler rigorously proved that there is no way to take the walk that people had wondered about. This would be just an interesting little tale, but it has a remarkable offshoot. After Euler published his proof, mathematicians took his way of simplifying the situation and, by exploring it, developed two new branches of math:  topology and graph theory. The graph theory ideas that Euler first explored when thinking about the seven bridges sparked a branch of math that’s used today to determine the most efficient ways of connecting servers that form the backbone of the internet!

Of course, there’s also the classic example of Archimedes shouting “Eureka!” and running through the streets naked after seeing water rise in his bathtub. In that moment, Archimedes, who had been trying to help his king figure out if the crown that was just made for him had been created with pure gold, or with an alloy, saw that the water displacement would help him solve the problem. In the end, Archimedes determined that the crown was not pure gold, and the king rewarded the great thinker for his efforts.

As I write this, I find myself wondering if any of you readers can think of other situations in which everyday life experiences led mathematicians or scientists to major discoveries. It would be enlightening to hear more of these stories.

And, if no such stories spring to mind, check out this site, which lists several such stories.  http://www.sciencechannel.com/famous-scientists-discoveries/10-eureka-moments.htm

In any case, the way that such discoveries occur shows that you never know where a seemingly trivial idea might lead … so it’s good to keep your eyes and mind open.

### Movie Math: Wake Students Up with Silver Screen Riddles

Last days of the school year … kids getting “antsy.”

Harder and harder to keep their attention … so what’s a teacher to do?

Answer:  Let the media help us with the media generation.

In my May 16 post, I pointed you to a website that showed how math is used in major motion pictures.

In this post I’d like to focus on one such reference to math in the movies, and show how you can turn it into a fun “End-of-Year” lesson.

The clip of Die Hard below has a great scene in which the Bruce Willis character needs to solve a mathematical puzzle in less than five minutes to avoid getting blown up. It’s an exciting scene, and the math is interesting.

I suggest that you first have your class watch this clip.

After watching it, review the solution with your class.
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### Rubik’s Slide: play your way to geometric knowledge

A toy that educates … could it be a dream?

I recently found something that fits that category, educating students in concepts of GEOMETRY.

It’s called the Rubik’s Slide, created by Techno Source. I bought this Rubik’s Slide a few months ago because I needed another puzzle to keep my tutoring clients entertained while I grade their work, which I often do at the start of sessions.

Rubik's Slide Logo

### Fun Math Problem #2

Here is the second in my series of “Fun Math Problems.”

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit! I will post the answer to the problems two days later, after people have had time to respond.

To provide your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.
Please show how you worked the problem. Thanks. I will post the names of the first three people who get this right.

The Problem:  Before you go out to lunch, you glance at the clock above your desk. When you come back from lunch, you glance at the clock again, and you notice something strange. The minute and the hour hand have exchanged places from the positions they had just before you went to lunch.

The question is:  how long were you away?

Image by The Hidaway (Simon) via Flickr

### FUN MATH PROBLEM — Circling the Square & Vice-Versa

From time to time I will post interesting math problems.

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit!

I will post the answer to the problems two days later, after people have had time to respond.

To post your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.

The problem: Which provides the fuller fit? Putting a circular peg in a square hole, or putting a square peg in a circular hole? To get credit, show all work, and justify your answer by expressing each “fit” as a percent.

a) By “fit,” I mean the ratio of the smaller shape to the larger shape, expressed as a percent. For
example, if a ratio is 4 to 5, that would represent a “fit” of 80 percent.

b) For the circular peg in the square hole, assume that the diameter of the circle equals the side of the
square. For the square peg in a circular hole, assume that the diameter of the circle equals the diagonal of the square.

c) By “fuller fit,” I mean the larger of the two ratios.

Have fun!

### Problem of the Week — Answer

Answer to the 10/1/2010 Problem of the Week

The problem:  Certain digits appear the same when reflected across horizontal lines or vertical lines. This week’s problem:  which two-digit numerals appear the same when reflected across a horizontal line? Which two-digit numerals appear the same when reflected across a vertical line? To answer, provide the list for the horizontal line and the list for the vertical line.

Solution, sent in by Jo Ehrlein, of Oklahoma City, OK:

Assuming you write the #1 with no serifs, then here are the single digits that are the same when reflected across a horizontal line:1, 3, 8, 0.  That means that the 2 digit numbers that are the same when reflected across a horizontal line are:  10, 11, 13, 18, 30, 31, 33, 38, 80, 81, 83, 88

2 digit numbers are only the same when reflected across a vertical line if both digits are the same AND the individual digits are the same when reflected across a vertical line. The single digits that meet that criteria are1, 8, 000 isn’t a valid 2 digit number.That means the 2 digit numbers that are the same when reflected across a vertical axis are 11 and 88.

Well done, Jo!

So the winner’s circle this week has one member:

Jo Ehrlein, Oklahoma City, OK

And, in Jo’s honor, here is our ceremonial picture of Oklahoma City, home of Ralph Ellison,author of Invisible Man,  if I recall correctly.

Oklahoma City, OK

Congratulations to everyone who worked on this problem. I had some detailed answers that were partially correct.

FYI:  Starting this coming week, I’m going to post the Problem of the Week on Monday for teachers who want to use it early in the week. Answers will be posted mid-week.

### Challenge Problem – the ANSWER

Hi everyone,

Here is the answer to yesterday’s challenge problem, the probability problem about catching fish.

First, there are 6 ways to catch the three fish in three casts, getting exactly one trout, one carp and one bass.

You could get the fish in any of these six orders:

TCB / TBC / CTB / CBT / BTC / BCT

Next you find the probability of getting one of these possibilities. Let’s take the first one, TCB.

Keep in mind that after you catch the first fish, the number of fish left goes down by one, to 11; after catching the second fish, the number of fish goes down to 10.

The probability for the TCB possibility is calculated by multiplying:  6/12 x 4/11 x 2/10 = 2/55

When you think about the five other ways to catch the fish, you’ll see that the order of the numerators changes, but the denominators remain 12, 11, 10. So the probability for catching the three fish in any of the six ways is always the same:  2/55.

To get the probability for all six catches, just multiply the probability of one catch by 6:

6/1 x 2/55  =  12/55

And that is the answer: the probability of catching exactly one trout, one carp and one bass is 12/55, which works out to about 21.8%, meaning that this should happen a little more than 1/5 of the time.

I didn’t see anyone submit any answers, but feel free to send them in. Remember that I will post only the correct answers, so no one has to worry about seeing an incorrect answer posted.

Have a great day!

—  Josh

### NOVEMBER PROBLEM OF THE MONTH

The problem:

It’s your friend’s 73rd birthday. You’ve put together a surprise party and baked a special coconut meringue cake. But at the last minute you realize that — golly gee! — you forgot to get candles.

Rummaging through your drawers with just five minutes before your friend’s scheduled arrival, you find that you do have 14 candles. And being a brilliant mathematician, you realize that you can represent the number 73 with these 14 candles, using every candle. How do you do it?

As a hint, here’s a model showing how to do a problem like this, if you are celebrating someone’s 44th birthday, when you have just 13 candles. Notice that each dot on the top row is one candle.

Note that you may use icing to create the symbols: +, –, x, ÷, and you may also put in exponents, using candles to show the value of the exponent.

Have fun!

info@SingingTurtle.com

Make the Subject line: POTM

Please include your full name, where you live, and if you don’t mind, describe your connection to math and math education (for example: teacher, tutor, math enthusiast, etc.).

The first person to send in a correct answer receives a \$20 gift certificate toward the purchase of any Singing Turtle Press products. I’ll fill the winner in on the details by email.

### “Cruel” Sequences Puzzle

Here’s a fun math treat, a great puzzle to pose when you have just a few minutes of class left. It’s guaranteed to drive your students “to distraction.”

As we know, one important area of math involves the study of sequences — those strings of numbers or terms that have some order to them. Often your task in studying sequences is to figure out the next number or term in the sequence — or to figure out the pattern on which the sequence is based.

Here’s a simple example. What would be the next number in this sequence:

1, 4, 9, 16, 25, ?

The answer is 36, since the elements of this sequence are simply the perfect squares of the natural numbers.

Algebra, as we know, involves using letters in place of numbers, so what better time to introduce a sequence involving letters.

Ask someone if s/he can figure out the next letter in this sequence:

O, T, T, F, F, S, S …

Kids will puzzle over this agonizingly. But the answer is remarkably simple. The answer is E.

The reason:  the sequence of letters are the first letters of the first numbers in the Natural Numbers:  One, Two, Three, Four, Five, Six, Seven. The next number is Eight, so the next letter is E. Isn’t that just cruel?

Supposedly, as the story goes, this problem was given both to MIT graduate students and second graders. Guess who got it faster? The second graders.

If there is a moral to the story, it may just be that sometimes we over-think situations. It’s important to keep in mind that the answer may be right in front of our noses.