# Indirect Reasoning and the Chessboard

By Jonathan Osters, The Blake School

I have always been a fan of the movie The Prestige (https://www.youtube.com/watch?v=LV-cXixgrho). The opening scene of the movie shows how to do a good magic trick. I think that sometimes, good teaching is similar to the outline described in the movie. Dan Meyer uses three acts frequently, and this problem uses three variations of the same problem.

In class this week, we have been studying indirect proof, which is a very challenging topic for beginning Geometry students. We show that a situation must be true by eliminating the chance that it is false. To illustrate this idea, we explored a classic problem in mathematics involving a checkerboard. The framework of the problem is very simple: you have a standard 8×8 checkerboard, and dominoes that cover two squares each.

The Pledge

The first question I ask them is, can you tile the checkerboard with dominoes? It does not take long for the first students to show a solution, and I ask if there are other solutions. Below are several such solutions.

You can see that there are a variety of different solutions to this problem. They all come to the conclusion that there are 32 dominoes to cover 64 squares, and that for the dominoes to cover the board, there must be an even number of squares.

The Turn

The second question I ask is: suppose the opposite corners of the checkerboard were cut away. Would it still be possible? Many emphatically express that it is, since there are still an even number of squares (62). I turn them loose to explore.

Some students insist, after failing the first time, that it can’t be done. I encourage them to continue trying; maybe they just haven’t found the solution yet! After attempting and failing numerous times, I call the class back together and ask if anyone came up with a solution. I contrast their early, frequent, and easy success with the current problem, and give them two choices: (1) A room full of intelligent, capable students failed to solve a solvable problem when they were able to solve a similar problem minutes ago, or (2) the problem is not actually solvable. Students become inclined to believe the latter, and I insist that if this is the case, then we should be able to explain why the problem is not solvable.

I have students generate a list of given information. There are only two main givens: first, is that each domino will cover two squares, and the other is that there are 62 squares. I remind them, only at this point, that the checkerboard has different colored squares, and ask if they can refine the givens with this new information. The given information is finally refined to: (1) each domino covers a white square and a black square, and (2) there are 32 white squares and 30 black squares on this board.

I then have them prove this checkerboard cannot be covered with dominoes.

Proof:

Assume that the board can be covered with dominoes. Since dominoes cover one black and one white square, n dominoes will cover n black and n white squares. This statement is a contradiction to the given information that there are more white squares than black. Therefore, this board cannot be covered with dominoes. QED

The Prestige

The key to the follow-up is to get the students to drive it with lots of “what if” questions. In class this week, I got the following “what if” questions as follow-up.

“What if you tried this on a 7×7 board, where the opposite corners are different colors?”

“What if, instead of removing a square from two corners, you added a square to two corners?”

“What if you removed a square from each corner instead of just two?”
“What if the squares you removed were different colors but not corner pieces?”

“Will you be able to cover the board if you remove ANY two squares, so long as they are different colors?”

These are absolutely wonderful questions, and get at the heart of what a true mathematician does. They are able to see that the first question, about the 7×7 board, does not work based on the odd number of squares. They come to realize that the adding of two squares creates a problem that reduces to the one we just solved, which is a mathematical technique that requires a very sharp mind.

The last question is the trickiest. It is the most general, to be sure. For instance, the scenario below is one such instance of removing two squares and being able to tile with dominoes.

As to whether it is always possible to do? Well, I will tell you that my friend and I sat down and we have a proof as to whether or not it is always possible… but a magician never reveals all his secrets. [We will post the proof next week 🙂 ]

# A Profound Respect for 1 Million

By Chris Robinson (@isomorphic2crob), The Blake School

How long will it take for your heart to beat ONE MILLION times?

Before you read on, imagine that the ball just dropped, January 1st, 2015, when will your one millionth heart beat occur? Will it be later that day? Some point in January? In July? In 2016? 2017? How long does it really take to reach ONE MILLION heartbeats? Before doing any figuring, take a guess, then after you have committed to that conjecture, go ahead and calculate it.

This question was posed by Marjory Baruch (Syracuse University) to a group of high school math teachers attending PROMYS for Teachers at Boston University this summer. The initial conjectures for this group of math teachers ranged from a couple months to a couple years. All of us were astounded when we calculated the result to be slightly less than two week’s time.

The questions stuck with me as I began to prepare for my year at Blake, and so I decided to ask my class of Algebra 1B ninth graders, my Honors Geometry ninth graders, and my Honors Algebra II tenth graders. Their answers were a much wider range, including a student or two in each class that guessed less than a month. But not too many predicted it to take more than a couple of years. There was very little difference in the guessing from one class to the next leading me to believe from a very small sample size that students ability to estimate the size of large numbers was not necessarily connected with their current success in mathematics. There was however a difference between the teachers and the students.

Why do math teachers have such a profound respect for the size of 1 million? Maybe its related to our salaries…. But I digress.

Here’s Hank Green discussing a very similar idea (How long was it a million seconds ago?) on his YouTube channel.

I also had my students place the number 1 million on a number line from 0 to 1 billion, like the one shown below.

Students struggled to see how much larger 1 billion was compared to 1 million, with one student commenting that they always felt like millionaires and billionaires were the essentially the same.

OK, so what’s the big deal (get it? Big.)? I am not sure I have a punch line here, so I will leave you with some observations, and the hope that you find this intriguing as well.

• For most students, and perhaps most teachers, our initial impression of numbers beyond 10,000 take on the same amount of “bigness” until they are compared to other large numbers.
• Asking students to reflect on why they guessed what they did is a useful exercise, and having them write about it for a few minutes can help them develop habits of explaining their mathematical reasoning.

# Guessing: The First Step in Mathematics

by Jonathan Osters, The Blake School

“To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.” — George Polya

Often, some great understandings can be made through the most menial of scenarios. Early in our geometry course, we make sure to tell our students that one of the most important things they can do is form guesses. Hunches. Conjectures. If those conjectures work out in the end, that is wonderful, but if they don’t work out, we get the chance to go back and revise.

Yesterday we started with a seemingly trivial question: If we create a circle and place points along the edge, then connect all the points with line segments, how many regions can be formed in the circle?

We begin with one point on the circle. One point, one region.

(Figure 1)

Add another point and the segment connecting them.

(Figure 2)

Two points, two regions.

Conjecture time! At this early juncture, the conjectures can be varied. The most common are either (a) the number of regions is the same as the number of points, or (b) for each additional point, the number of regions doubles. How, I ask, can we tell if either of these conjectures is correct? Students, now invested in the problem by being forced to make a conjecture about how this will play out, insist that I add a third point.

(Figure 3)

Three points. Four regions.

Students whose conjecture is consistent with the picture holler and playfully taunt those who had conjectured incorrectly. But the game has just begun. I write the following chart on the board:

 Points 1 2 3 4 Regions 1 2 4 ?

I ask if we have any guesses as to what will happen when we add a fourth point. Those who previously had conjectured the “doubling” invariably hold fast to this conjecture, as it has played out so well so far, deciding the regions will again double to 8. Those who had conjectured incorrectly scramble to amend, often eventually claiming that the differences are increasing by 1 (an increase of 1 between one and two points, an increase of two between two and three points) and deciding there will be 7 regions.

(Figure 4)

Four points, eight regions.

More confidence in the “doubling” conjecture, less and less reason to doubt.

What about five points? Now, nearly all students have been converted to the “doubling” conjecture, and the choice of 16 is nearly unanimous. By this time, students are so invested that they clamor to draw this scenario for themselves.

(Figure 5)

And they are correct.

At this point, I ask what the point is in continuing. Don’t we know what will happen for six points? Many students insist we investigate just to confirm.

They work. They count. They pause, then recount. Some re-draw. They are miffed. Sensing this, I ask if they got 32 regions like we “know” we should. Some students say they got 30 regions, while others insist it’s 31, though very few say there are 32 regions*.

As it turns out, both the students who said 30 regions and 31 regions are correct. In certain arrangements of points, three lines become concurrent and cause a region to disappear.

(Figure 6a)                                                 (Figure 6b)

Several students indicate they obtained 30 regions by spacing the points equally around the circle. This scenario is interesting to discuss, as this is not the only scenario that results in concurrence, but the point of concurrence is always the intersection of the lines connecting opposite vertices (in Figure 6a, the region labeled “21” would become a single point).

The purpose of the activity is twofold: first, you can’t get so married to a conjecture that you are unwilling to amend it. While I model such a cavalier attitude during class, I close by making sure they are fully aware that they can’t claim a conjecture as absolute truth. This gets at the second goal: the role of proof in mathematics. To prove the things in math have the properties that they do is absolutely essential, and such proofs establish that we won’t run into unexpected places where our conjecture no longer holds. This lesson remains with the students, and when a student asks if we can assume some geometric fact and “get by” without a proof, the class reminds that student of this activity and how there are no guarantees without a proof.

*Those students who said there are 32 regions did something wrong, like count a region twice, which is why it is always best to number as we count.