Weird Wipers

It was a rainy day and I was on the bus; not the most pleasant of experiences. I was staring out the side window, hoping that the rain would stop before I got to mine. And then, I noticed something mathematical. What could have been another dull and wet ride took a different turn.

The windshield wipers were moving at different frequencies. Most of the time, the wipers were out of sync. One would move and then the other would follow. But every so often the wipers found their groove and seemed to sync up for a wipe or two. For those of you unfamiliar with this phenomenon, I created a quick visual for you. Insert Desmos.

In-sync:

Out of sync:

I wanted to model the situation mathematically so I could understand what was going on. I pulled out my phone to time the wipers. The left wiper completed 10 strokes in 12.6 seconds. The right wiper made 10 strokes in 12.8s. This meant that the left wiper was operating at a rate of 1.26 strokes per second and the right wiper was operating at a rate of 1.28 strokes per second. Surprising that such a small difference can create such an interesting pattern eh?

Let’s drop the decimals and work with whole numbers; 126 and 128. Since we want the wipers to synchronize, we are looking for a time when both numbers are the ‘same’. This calls for, the least common multiple! After some number crunching, we find that the LCM of 126 and 128 is 8064. Converting back to a decimal, we expect the wipers to synchronize after 80.64 seconds.

This makes intuitive sense to me since most of the bus ride the wipers were out of sync. However, the above calculations tell us that every minute and 20 seconds, the wipers would sync up and go through a couple wipes in tandem.

I wonder why the wipers aren’t perfectly in sync to begin with? Was it a manufacturing defect? Do the wipers have independent motors? Perhaps perfectly in-sync wipers put bus drivers to sleep? Or perhaps the universe just wanted to give me a fun puzzle that rainy day.

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Patterns

Patterns

Patterns are everywhere in mathematics. From the humble pattern of the squares: $1, 4, 9, 16, 25, 36, \ldots$ to the intricate pattern of Fibonacci: $1, 1, 2, 3, 5, 8, 13, \ldots$ One thing I realized recently is that these patterns, and the human mind’s ability to recognize them, can be a powerful teaching technique.

Take for example the age-old question: why is a negative times a negative a positive?

We first start with the fact that $(3) \times (-4) = -12$ which I don’t think is too controversial. If you don’t believe the result, imagine 3 groups of 4 negative signs floating around. There should be a total of 12 negative signs in your mind.

Now look at the following equations. Can you spot the pattern?

$(3) \times (-4) = -12$

$(2) \times (-4) = -8$

$(1) \times (-4) = -4$

$(0) \times (-4) = 0$

$(-1) \times (-4) = ?$

Each time the answer increases by 4. This pattern leads to the logical conclusion that $(-1) \times (-4) = 4$.

Another example I find my students struggle to comprehend is $3^0$. What does raising a number to the power of zero mean? Since exponentiation makes ‘copies’ of the base, $3^4 = 3 \times 3 \times 3 \times 3$ and $3^1 = 3$. But for $3^0$ how can  you make no copies of the base and still get an answer? Again, a pattern can help us to determine the answer:

$3^4 = 81$

$3^3 = 27$

$3^2 = 9$

$3^1 = 3$

$3^0 = ?$

In this case, the pattern is division. Each time the answer is divided by 3. This pattern leads to the logical conclusion that $3^0 = 3 \div 3 = 1$.

As one final example to illustrate the power of this technique, consider $0!$ This concept is especially troublesome for students since factorial is conceptualized as ‘the number of ways of arranging objects.’ As such, a concrete example can be constructed to demonstrate that $3!=6$, the number of ways to arrange 3 objects. But how is the world would someone count the ‘number of ways to arrange no objects? What if we made a pattern?

$4! = 24$

$3! = 6$

$2! = 2$

$1! = 1$

$0! = ?$

As in the previous case, the pattern is division. However, in this case the number that you divide by decreases by 1: $24 \div 4 = 6, \, \, \, 6 \div 3 = 2$. This pattern leads to the logical conclusion that $0!=1$.

I am curious if you have ever used a pattern like the ones above to help explain a tricky concept. Let me know in the comments below!

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Packing Pancakes

Believe it or not, you have been making pancakes wrong your entire life. Now that I have your attention, I want to talk about pancake packing. Imagine your pan as a large circle and each pancake as a smaller circle that you must fit inside the pan. How could you could maximize the amount of space your pancakes use?

Obviously, the easy answer would be to just pour batter on the entire pan. A full pan pancake will use all the space. As a diagram:

However, you may not want to eat a bunch of gigantic pancakes (also called crepes in my family). Instead, what if you wanted to make 2 or 3 pancakes per pan by making them smaller? How you could position them on the pan to make the best use of space?

For 2 pancakes, the setup looks like this:

Not a very good use of space. Most of you have probably never made just 2 pancakes on the pan (unless you were out of batter) because it just looks so inefficient. See all the wasted space at the top and bottom? The pan is practically begging you to add another pancake somewhere. Instead, you probably make 3 pancakes. And I bet you positioned them like this:

Using some careful math, you can show that the 2 pancake setup uses only 50% of the pan. While the 3 pancake setup uses 64.6% of the pan. That is a big improvement in pan coverage and it probably explains why no one makes only 2 pancakes. But what if you made more?

The best way to position 4 pancakes is like so:

And this uses 68.6% of the pan. Unless you are very mathematically minded, it probably never occurred to you to try and fit more pancakes in the pan. The efficiency gained by trying to squeeze in that 4th pancake only added 4% more coverage. This is hardly worth most people’s effort.

However, there is something very interesting about this 4 pancake setup. It actually beats out the 5 pancake setup!

The above 5 pancakes only cover 68.5% of the pan! And it gets worse with 6:

Now we are only covering 66.7%. See all that wasted space in the middle! 6 pancakes is a terrible way to cook. It is only when we hit the lucky number 7 that we finally have an increase in pan coverage:

Here, we get 77.7% coverage. So technically, if you wanted to maximize the space on your pan making small pancakes, you should make 7 pancakes at a time. However, I think that at this point, the pancakes become too small to make sense from a meal perspective, even if it makes sense from a math perspective.

If you really wanted to go for it, you could pack 61 pancakes on the pan. This gets your coverage up to 81.3% and the image looks quite nice:

However, these pancakes are more like flecks of batter than true food.

To sum it up, you should probably make 4 pancakes instead of 3. Your intuition correctly told you that 3 is more efficient than 2. However, you were fooled into stopping at 3. You should really continue on and perfect your 4 pancake skills. If you have to cook a large batch of say 20 pans of pancakes, your 4 pancake method will be able to do it in only 19 pans. Math just saved you 5 minutes. You’re welcome.

Pictures courtesy of Wikipedia: https://en.wikipedia.org/wiki/Circle_packing_in_a_circle

Update:

One of my readers mentioned to me that in Sweden, they often make mini-pancakes. They are called plättar and I am told they are very good. Do you know how many they make per pan?

I guess I shouldn’t have been so quick to write off the 7 configuration as too small to make sense. A very interesting cultural connection!

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Sum of Squares

Here is an interesting problem I have been playing around with. We all remember square numbers from grade school:  22 = 4 or 62 = 36. Now consider the lowly number 5. While it can be squared, it, by itself, is not a square number. That makes 5 sad. However, maybe 5 could be the next best thing. Could we find a way to write 5 as the sum of two squares?

Well, we could try 5 = 2 + 3. But neither 2 or 3 is a square. What about 5 = 4 + 1? Bingo! 4 = 22 and 1 = 12. The interesting thing, is that if you try to do this with 6, you find that is simply doesn’t work. It won’t work with 7 either, but it does with 8. Can you find the two squares that add up to 8?

Of course you can; 8 = 22 + 22.

Now for the fun part. I guarantee you that it will work for 40. Why 40 you might ask? Well, if it worked for 5 and it worked for 8, then it will work for 5 × 8, or 40.

Sure enough, 40 = 36 + 4 = 62 + 22.

Using this insight, we could go even higher. 40 × 8 = 320. Assuming I am correct, it should work for 320. Now I don’t know about you, but I have a hard time spotting two squares that add up to 320. Wouldn’t it be nice if there was a formula that would use what I already know about 8 and 40 to tell us which squares add up to 320?

As I was working through this problem, I stumbled upon just such a formula. It’s called the “sum of squares” because it sums two squares.  The nice thing is that it tells us how to arrive at the two squares that we are going to add together.

Let’s first return to the case with 5 and 8. The base numbers that produced the squares for 5 were 2 and 1. Let’s call these a and b. For 8, the numbers are 2 and 2. Let’s call these c and d. I’m going to give you the formula and then explain at the end how we obtained it.  For now, just trust me.  At first glance, the formula may not feel intuitive.  But if you hang in there and do the example yourself a couple of times, you’ll get comfortable with it pretty quickly.

First square = a × c + b × d

Second square = a × db × c

Not what you would expect hey? Those are some complicated formulas for such a simple puzzle. However, if we plug in our values, we get:

First square = 2 × 2 + 1 × 2 = 6

Second square = 2 × 21 × 2 = 2

Thus: 40 = 62 + 22

Pretty cool hey! Let’s see if it works for 8 and 40:

8 = 22 + 22

40 = 62 + 22.

First square = 2 × 6 + 2 × 2 = 16

Second square = 2 × 22 × 6 = -8

So, we get: 320 = 162 + (-8)2

Ok, one more fun example.

13 = 22 + 32

17 = 12 + 42

First square = 2 × 1 + 3 × 4 = 14

Second square = 2 × 43 × 1 = 5

And we get: 221 = 142 + 52

That’s all there is to it. A fun exercise is to find out which numbers between 1 and 100 can be written as the sum of two squares. For example, of the three numbers 71, 72, 73, only 2 can be written as the sum of two squares. Can you tell which ones?

Aside:

For those of you wondering why the formula works, a bit of algebra does the trick.

Let n1 and n2 be two positive integers where n1 = a2 + b2 and n2 = c2 + d2. To show that the product, n1n2, can be written as the sum of two squares of integers, observe the following:

n1n2 = (a2 + b2)(c2 + d2)

= a2c2 + a2d2 + b2c2 + b2d2

= (a2c2 + b2d2) + (a2d2 + b2c2)

= (a2c2 + 2abcd + b2d2) + (a2d2 – 2abcd + b2c2)

=(ac + bd)2 + (ad – bc)2

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Fractions and their many names

I have often wondered why some students find fractions so challenging when, for me, the rules of the fraction world are second nature. It just makes sense to me that 1/2 = 2/4. However, new students often struggle with these concepts. What if the answer is related to the fact that the same object can have many different names? In “normal” math, 5 is just 5. It doesn’t go by any other strange or wacky name. But in the fraction world, things are different.

Consider the quantity 15/35. In no way does this resemble the quantity 3/7. Where did the 7 come from? Why is the 3 on the top now instead of the bottom? For no intuitively obvious reason however, the world of fractions declares that these are in fact equivalent! Same object, radically different names. Math, being the logical subject that it is, shouldn’t go around giving the same thing a bunch of different names, right? Or is there a reason for all this relabelling? Consider an analogy.

I usually refer to my father as “Dad.” At work, he is known as “Harv.” His birth certificate declares his name is “Harvey.” My mom usually calls him “Hon.” And if you bumped into him on the street, you might call him “Sir.” While these names are drastically different, they all refer to same person. The reason for the difference is that each name serves a different purpose, depending on the context. Fractions are the same.
Suppose I am baking and the recipe calls for a half cup of flour. That seems simple enough. But what if my half cup measure is dirty. Maybe I would want to use my quarter cup to do the measuring. Then I would use 2 quarter cups since 1/2 = 2/4. The same quantity, but a different name.

Imagine that I am looking for a new job. One job would give me 15 vacation days per year. As a fraction, this is represented as 15/365. A different job offers me 4 days of vacation per 73 calendar days. As a fraction, this is represented as 4/73. To choose the best job, I need to compare these quantities. Using fraction knowledge, I use a different name for 4/73. I use instead, the name 20/365. Now I can clearly see that the second job offer is more appealing with those 20 vacation days per year.

What if your phone battery said, “you have 8/25 of battery remaining.” That would be silly. Of course, the phone should say, “you have 32/100 of battery remaining.” This second name makes it so much easier to understand how many more YouTube’s you can watch until your phone dies.
And personally, I think it is fun for numbers and people to have different names in different contexts. My friend’s grandmother was always affectionately referred to as “Babcia.” Maybe consider that next time you fret over 1/3 being called 2/6.

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Speeding

As I was driving home from my road trip a couple of days ago, a thought occurred to me. Several cars were passing me on the highway, clearly going 10 km/h over the speed limit. Of course, people speed to save time and to get where they are going in a hurry. Sometimes they go 10 km/h over the limit on the highway, and sometimes they do the same thing in the city. I wondered, is there a difference?

More precisely, suppose you had to travel 100 km. In the first situation, you are on the highway clipping along at 100 km/h. The urge to speed wells up inside and you long to shave some time off your trip. If you increase your speed to 110 km/h, how much time do you save?

At your original speed, it will take you 60 minutes to complete the trip. At the increased speed, the trip will only take 54.5 minutes. Thus, you save 5.5 minutes on your trip. Not a ton of time, but still a temptation if you need to get somewhere in a hurry.

Now consider the second situation. You still need to travel 100 km, but you are in the city traveling at a gentle 60 km/h. Suppose you increase your speed to 70 km/h, how much time will save? Take a guess!

My first instinct was that you would still save about 5.5 minutes on the trip since the distance travelled is the same. However, if you crunch the numbers, you find the following:

Time at 60 km/h: 100 minutes

Time at 70 km/h: 85.7 minutes

Time saved: 14.3 minutes

You more than double your time savings! Let’s push this idea to the extreme. Consider a third situation, where you still need to travel 100 km, but the entire time you are stuck in a school zone and must travel 30 km/h! The kind of malicious person who would create a school zone like that escapes me. However, we will continue for the sake of the math. Here are the results:

Time at 30 km/h: 200 minutes

Time at 40 km/h: 150 minutes

Time saved: 50 minutes

You save almost an hour by speeding! Incredible and unexpected! A word of caution. I am in no way advocating speeding, it is a dangerous habit with potentially fatal consequences. However, by investigating the math of the situation, I believe I have uncovered something about human nature. While driving down the highway, speeding feels like a 50-50 choice. You could go a bit faster and cut some time off your trip. Or you could obey the speed limit. In either case, the difference won’t be that noteworthy. Yet, when people are stuck in a school zone, the temptation to speed can feel overwhelming. The vehicle is moving so slowly and it feels like going even 10 km/h over the speed limit would drastically reduce your commute. I think that this temptation, this intuition, is illuminated in the math above. Speeding in a school zone really does save you much more time than speeding on the highway.

Being a teacher, I can not, in good conscious, leave the discussion there. In the final, real life scenario, when the school zone is only 0.5 km long, here are the numbers:

Time at 30 km/h: 1 minute

Time at 40 km/h: 0.75 minutes

Time saved: 15 seconds

While it might be reasonable to try and speed through a 100 km school zone of torture, the 15 seconds saved in a regular school zone is not worth the risk of a child’s life. As the slogan goes, “normal speed meets every need.”

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Puzzling Probability Part 3

This is part 3 of a series in which I am exploring some interesting probability puzzles.

As someone who loves to walk, probability puzzles that involve considering different routes to the same destination are interesting to me. However, this puzzle requires a warm up. Consider the following:

How many ways are there to walk from A to B (assuming no back tracking)?

At first, I tried to draw all the possible paths:

Then I realized that it was going to take far too long and I could never be sure that I didn’t miss a path. I needed another approach. Instead of trying to figure out how many ways there were to get from A to B, I asked a simpler question: How many ways to get from A to C?

Clearly, we can only get to C in only 1 way. We place a number 1 at the intersection. Similarly, we only have 1 way to get to each intersection point along the top and along the left side. Next, we need to determine how many ways to get to point D.

Since there is 1 way to get to the point above D and 1 way to get to the point left of D, we add these and obtain 2 ways to get to D. Continuing like this with each intersection point on the grid, we have:

Thus, there are 20 different ways to walk from A to B.

Now we can answer the actual question. What is the probability that a random walk from A to B (assuming no back tracking) passes through the middle of the grid below?

Using the above method, we find that there are 70 ways of walking from A to B. To count the number of ways through the middle, we simply redraw the gird to force us through the middle as follows:

Again, using the above method, we find that there are 36 ways of walking through middle.

Hence, the probability is 36/70. If you try this exercise for a 6 by 6 grid, you will find that the probability is 4900/12870.

This was counter intuitive to me. In a very large gird, if you were to randomly walk from one end to the other, it seemed to me very unlikely you would pass exactly through the center. The above solution shows that even for a 6 by 6 grid, the probability of walking through the exact center is 38%. As an extension, what do you think happens to this probably as the grid grows larger? Does it approach a specific value? Or does the probability eventually tend to zero?

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