Kiss those Math Headaches GOODBYE!

Archive for the ‘Brain Teasers’ Category

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.

Being Smart?

“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

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.

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Monday the 13th


Today is Monday, the 13th.

So what, right?

Well, maybe not so fast …

If you have a mathematical/logical bent of mind, you might find that interesting.

Friday the 13th is generally considered a bad luck day. So if that is the case, you might wonder if Monday the 13th would be the logical opposite to Friday the 13th, a good luck day. Afterall, Friday is the end of the workweek, and Monday is the beginning of the workweek.

So in that sense, can it be said that Monday and Friday are opposites? And what might that imply.

So here is the challenge. Compose a logical argument as to whether or not Monday the 13th should be considered a lucky day.

That is the challenge for Monday, the 13th of June 2011.

HINT:  You may want to include information about the “truth value” (truthiness, as Steven Colbert likes to say) of statements and their converses.

REWARD:  The first person who presents a compelling logical argument, one way or the other, wins a $10 gift certificate toward the purchase of any Singing Turtle Press products. All comments must be posted by 1 a.m. on Tuesday, the 14th of June, this year.

James Bond Math Challenge


Math in the movies … if there ever was a cool way to explore math, this has to be it. And if you missed my earlier posts on this, check them out here and here.

Math is Cool!

I was looking through the links to movies with math themes, and a question came up.

On the site showing the movies, the text says that there are “mathematical themes and patterns motivated by math” in the introduction scene for the James Bond movie, Casino Royale, this clip:

I’ve watched the clip a few times, and I have my own ideas as to mathematical themes and patterns.

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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.

Math is Cool!

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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Answer to Fun Math Problem #2


ANSWER TO FUN MATH PROBLEM #2

The problem, once again, reads as follows. 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?

I received several answers to this problem, but the first person who got it right was Adrian W. Langman, of Port Angeles, TX.

Here is Adrian’s answer, in his own words:

It’s probably true that the hour hand is near the 12 at the beginning and near the 1 at the end. So it’s about 5 minutes after noon at the beginning, and just a bit after 1 at the end.

(Or, it could be a tad before 11 in the morning at the beginning, and about 5 minutes before noon at the end, if the worker is an early riser.  But this problem is just the geometric mirror image of the one hypothesized above, so the duration of the lunch break will be exactly the same.)

Obviously the lunch break is about 55 minutes. But to find the exact length, let M be the number of minutes past noon at the beginning. I’ll use the obvious coordinate system – the origin at the center of the clock, the clock hands radial lines, the 12 at 0 degrees, and the 3 at 90 degrees.

At M minutes past noon, the minute hand is at M/60 x 360 degrees, i.e. 6M degrees, and the hour hand is at M/60 x 30 degrees (since it’s M/60 of the way from the 12 to the 1, which is at 30 degrees), i.e. M/2 degrees.

So at the end of lunch, since they’ve switched places, the hour hand is at 6M degrees and the minute hand is at M/2 degrees.

Since the minute hand is at M/2 degrees, it is 1/6(M/2) minutes past 1 o’clock, i.e. M/12 minutes past 1 o’clock.

Since the hour hand is at 6M degrees, it’s at 6M-30 degrees past the numeral 1, so it’s 2(6M-30) minutes after 1o’clcock.

Setting M/12 = 2(6M-30) and solving for M yields 720/143 (which is approximately 5.035).

So you left at 720/143 minutes after 12, and returned at 60/143 minutes after 1 o’clock.

So you were gone for 60 + 60/143 – 720/143 minutes,

i.e. 7920/143 minutes, i.e. 55 and 55/143 minutes, which reduces to 55 and 5/13 minutes.

Problem of the Week – 10/18/2010


[Note:  I really am not getting a check from the New Mexico Tourism Department for this post, though I wouldn’t mind if they sent me one!]

Katja and Anthony are on a sightseeing trip in the western United States. Beginning where they land in Santa Fe, NM, they drive 80 miles east to see the historic wild west town of Las Vegas, NM. Then they travel 50 miles north to visit the Kiowa National Grasslands . Next they drive 140 miles west to visit Chaco Canyon National Historical Park. Finally they journey 130 miles south to visit El Malpais National Monument. When they reach El Malpais, how many miles are they from their starting point in Santa Fe?

Please explain how you found your answer, and send answers either as comments to this post, or as emails w/ subject POTW, sent to josh@SingingTurtle.com    I will not post your comments unless and until I determine that it is correct. And then, only on the day when I send out the answer on my blog.

 

Chaco's smaller kivas numbered around 100, eac...

Kivas in Chaco Canyon, New Mexico

 

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

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.