Monday, 21 September 2015

Rust remover

When you are trying to solve a problem, sometimes your worldview causes you to unintentionally import prejudices and assumptions that block you from the solution. 

To illustrate, here is an updated version of a puzzle that completely baffled me when I was a kid. In this day and age it won’t fool very many people, and in fact, many people would not see a problem at all.
A father and his son were in a car accident and were taken by ambulance to the hospital. The father was injured, but not seriously, and he was sent to the waiting room. However, the son needed surgery. After he was prepped for the operation, the surgeon came in, but said “I can't operate on him — he's my son!” How is this possible?
When I was young, I subconsciously pictured a doctor as being a man, and this prevented me from seeing the solution, namely that the surgeon was the child’s mother. (Of course, nowadays we recognize that there is a second solution: the surgeon was the son’s other father.)

While I was still teaching I often began one particular course with a small collection of puzzles like this. I used to call them Rust Removers because the students’ thinking always seemed a bit rusty after returning from their summer or Christmas break.

Here’s a dozen puzzles that were carefully posed to entice you into following your preconceptions or somehow cause you to make unwarranted assumptions. You may be able to solve most of them quickly, but very few of my students were able to solve all of them in one sitting. 

If you are absolutely desperate for an answer, a pdf file of the solutions is available here.

1. One night John was reading an exciting book when a power failure threw the room into complete darkness. Nevertheless, he continued reading without a pause. He was not using a laptop, tablet, or e-reader, so how could he do that?
2. I live in a modern high-rise apartment. A lady friend who regularly visits me always gets off the elevator five floors below mine and takes the stairs to my floor. Why?
3. Last night I turned off the light in my bedroom and managed to get into bed before the room was dark. My bed is 3 metres from the light switch. How did I do it?
4. A woman walked up to a counter and handed a book to the cashier. He looked at it and said “Ten dollars.” She paid the man and walked out without the book.  He saw her leave without it but did not call her back.  How come?
5. A man is found shot to death in the front seat of his car. A gun lies out of his reach in the back seat.  All the windows are closed and the doors are locked; there are no bullet holes anywhere in the car. How could this have happened?
6. An escaped prisoner was running along a forest road when he saw a cop car heading towards him. He sprinted into the woods, but before doing this he ran ten metres directly towards the approaching car. Why?
7. A woman had two sons who were born on the same hour of the same day of the same year. But they were not twins, and they were not adopted. How could this be so?
8. Jim Johnson lives in a four story apartment building. He is plant supervisor at an auto factory that is within walking distance of his apartment. Every morning at 0800 h, he walks down a flight of stairs, and when he arrives at his destination he settles back with a cup of Tim Hortons coffee that he purchased along the way and begins to read the newspaper. Halfway through the news his eyelids close and he falls asleep for several hours. Nevertheless, at the end of the month he looks forward to a nice pay raise. How does he get away with this?
9. Take 3 empty paper coffee cups, and put eleven coins in them so that each cup holds an odd number of coins. All the coins must be used. Once you have solved this, put 10 coins in the same cups so that again each cup holds an odd number of coins and all coins are used. (Remember, zero is an even number.) 

The following puzzle is now found throughout the web. I first saw it about 20 years ago. 
10. There is a light in a storage room on the second floor of a building. On the ground floor are three light switches, exactly one of which controls the storage room light, which is a standard incandescent 100 watt bulb. By turning some or all of the switches on or off, it is possible to determine which switch controls the light by making one trip to the storage room. How can this be done?
Outside the storage room, there is no way to determine whether the light is on or off, and disassembling the light switches will not reveal which one controls the light. You are not allowed to be helped by anyone.
11. Two grade six classes were going on a field trip to a museum. They were being transported by two buses each of which had 34 seats.  It so happened that there were 30 boys and 34 girls, and so they put all the boys on one bus and the girls on the other bus. The buses had to stop for a few minutes, and during that time 10 boys snuck onto the girls' bus. But the girls' bus driver noticed that there were too many on the bus, so he sent 10 children (boys and girls) back to the boys' bus. After this was done, were there more boys on the girls' bus than girls on the boys' bus? Or vice versa?

Another older puzzle that has also found its way onto the web:
12. Four people are being pursued by a menacing beast. It is nighttime, and they need to cross a bridge to reach safety. It is pitch black, and only two can cross at once. They need to carry a lamp to light their way. 
Mr. One takes a minimum of 1 minute to cross. Mr Two takes 2 minutes, Mr. Five takes 5 minutes, and Mr. Ten takes 10 minutes. 
If two cross together, the couple is only as fast as the slowest person. For example, if Mr. Ten and Mr. One cross the bridge together, it will take them 10 minutes. A fast person can't carry a slower person to save time. The person or couple crossing the bridge needs the lamp for the entire crossing, and the lamp must be carried back and forth across the bridge (no throwing, etc.). 
 If they don't all get completely across in strictly less than 19 minutes, who ever is on the bridge or left behind will be eaten by the beast. Is it possible for all of them to get across?

These puzzles are not mine. Some of them came from Martin Gardner’s book “aha! Insight!”  The book has a lot more than just these puzzles, and it can be read in selected short chunks. 

If you like short puzzles that challenge students to overcome fixations in their thinking, you should visit the WODB site. There you will find a collection of puzzles inspired by Christopher Danielson and curated by Mary Bourassa. Each puzzle presents four different things which are such that every three of them have at least one thing in common that is not shared by the fourth one. (WODB = Which One Doesn’t Belong.) 

Friday, 4 September 2015

Magic and Arithmetic Series

Now your high schoolers have learned the two formulas for the sum of an AP. (Do they still call it an Arithmetic Progression?)  So you ask them this:

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Arithmetic sequences and series. I cannot think of a more mind-numbing introduction to them than the way it was done a century ago when I was in school. And a quick googling suggests that the situation may not have improved very much — what I see often begins with a caveat that “You won’t actually need this until you take Calculus.”  Hard on the heels of this are the definitions of the first and last terms, the common difference, and so on. Then comes the formula for the general term, and finally the iconic derivation of the formula for sum of an arithmetic progression.  

Like many math teachers, I also used to tell my students the story of the clever young Gauss. It probably firmed up their belief that you have to be born with a math brain in order to do math.  Raise a glass to Kate Nowak for what she did to introduce AP's. In fact it is her post that prodded me to write this.


If I were able to tardis back a few years, I would probably begin with this: 

Give me the sum of three consecutive numbers.

If the students were to tell me that the sum is 72, I would tell them immediately that the three numbers are 23, 24, and 25. 

And if then I might offer this:

Give me the sum of three consecutive even numbers.

If the sum is 84I would tell them that the three numbers are 26, 28, and 30.

Perhaps even this:
Give me the sum of five consecutive numbers.

For example, if the sum is 45, I would tell them immediately that the five numbers are 7, 8, 9, 10, and 11.


The secret to this is that whenever you have an arithmetic series with an odd number, n, of terms, the sum is always n times the middle term.

It is easy to convince kids that this is the case for the simple cases given above. For example, three consecutive numbers with a middle term m must always be of the form

m - 1,     m ,      m + 1.

Adding, we get 3m. To perform the trick, divide the sum by 3, and subtract 1 to get the first number.

For three consecutive even numbers, the situation is pretty much the same. The three numbers would be 
m - 2,     m,     m + 2,

Adding, the sum is again 3m.

In case you are wondering about doing the trick when you are given the sum of four consecutive numbers: dividing that sum by 2 gives the sum of the middle pair of numbers from which you can easily deduce what the four numbers are.

For example, if the sum is 50, then the sum of the middle pair is 25, so the middle pair is 12 and 13, from which we get the four numbers 11, 12, 13, 14

The sum of an arithmetic series with an even number, n, of terms is always n/2 times the sum of the middle pair. Interestingly, the sum of the middle pair is also the sum of the first term and last term. 

You can pursue this far enough to derive the two formulas for the sum of an arithmetic progression, but I’m not sure that I would push it that far.

Instead I would switch the question around to finding the sum of a longer list of consecutive numbers, à la Kate Nowak.

The possibility of introducing arithmetic sequences and series in this manner grew out of a simple trick from William Simon’s book Mathematical Magic 

A calendar trick

Ask a student to draw a rectangle around three consecutive dates on a calendar month, like so:

This is to be done so that you, the teacher, cannot see what dates have been encircled (for example, have the student stand behind your back while you are facing the rest of the class). Ask the volunteer to show the circled dates to the rest of the students (but not to you) and to tell you the sum of the dates. You can immediately announce the dates that have been circled. 

[A personal aside: 
am notoriously poor at mental arithmetic — a brief description of my troubles is contained here. Using a calendar appeals to me because it forces the three numbers to be small, thus avoiding the floundering that would occur if some cheeky person asks me What are the three numbers if the sum is 14691?] 

After explaining how to do the three-in-a-row trick,  the calendar itself might prompt students to ask questions like this:

How would you do it if we gave you the sum of four consecutive dates?

How would you do it if we gave you the sum of three dates in a vertical line? 

What if we gave you the sum of five consecutive dates? 

The second and third questions the students could answer themselves.

When there are four consecutive dates, there is no middle date and the sum of the four dates is not divisible by 4. I imagine this might be a stumbling block for some students. But, fingers crossed, at least some of them will actually look at what happens when they do divide by 4, and thereby open up other avenues to explore. 

They might, for example, note that dividing by 4 gives the number that is the average of the middle pair of dates.  For the four dates circled above, when I divide the sum by 4, I get 5.5, which is the average of the two middle dates 5 and 6 And not only that, 5.5 is also the average of the first and last dates.

That is: 

Which generalizes to 

But as I said earlier, I probably wouldn't push it this far.

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What is the sum of the following arithmetic progression? Each square represents a term in the progression. 

What about this one? 

Or this one?