More modular arithmetic, and some logic puzzles

Modular arithmetic

Last week we came back to our problem from the previous two sessions about trying to hit all the points on a circle of n points. We made some progress on it, and I talked about the concept of relatively prime numbers (having no factors in common except 1), but I was sensing some restlessness, so this week we moved on to something different.

Logic puzzles

We started working on some simple logic problems, where you have, for example, three people with three different occupations, but you don't know which person has which occupation. Using the clues, you eliminate certain combinations until only one possibility remains.

The Centre for Innovation in Mathematics Teaching gives some examples of these. I used examples from this book.

We did several variations, where the correspondences were still small; just 3 or 4 things to match, but where the clues were more and more tricky.

For example, in one problem matching A, B, and C to X, Y, an Z, it was revealed that A was older than X. We talked about how this clue looks like it is giving only irrelevant information, but the inference you have to make is that if A is older than X, then A can't be X.

It was a lot of fun, and we will do some harder ones like this next week.

A problem from February's contest

There was an interesting problem on the last contest that required factoring to solve. In what I present here I have modified the problem slightly from the original, but if can solve this, you can solve the other.

We were given a rectangle divided into 4 smaller rectangles. The smaller rectangles all had integer sides, but they were not of equal size.

Three of the inner rectangles had areas given. The problem was to figure out the perimeter of the whole figure.

(The graphic is not necessarily to scale, so don't rely on that to solve it!)

One trick to beware of is that you may become focused on finding the area of the fourth rectangle. Although the process is essentially the same, don't forget at the end of your efforts to answer the question that was asked. In this case, we are looking for perimeter.

If you want to check your answer here, the perimeter you get should be 46.

More Modular Arithmetic

We spent our whole time exploring the points-on-a-circle problem from last time. The circle is just a graphical representation of modular arithmetic (a term I finally introduced after we got familiar with the mechanics). The problem is this:

If you are working in Integers Mod n, and you choose a number to count by, will you eventually hit every number in the set?

So, for example, if n = 6, you can count by 1 or 5, and eventually reach every number between 0 and 5, but 0, 2, 3, and 4 will all get stuck in cycles that don't cover all of the possible points.

Why is that?

As a general approach, I have decided to focus on developing intuition, and encouraging the students to make hypotheses. This worked really well today. They have come up with ideas, and were willing to share them even if they weren't sure they were correct. Then we all test together.

The first thing they noticed was that 1 and -1 (the last number before 0) always work. Then someone noticed that if n is even, no even numbers will ever hit any of the odd numbers. This is a key insight, which we talked through the correctness of.

We left off with the hypothesis that prime numbers always work, but we haven't proven it yet.

Eventually we will tie this back to a problem from December's contest, which ask the question: if you start at zero on a number line, and can only move to the right in steps of 3 or 10, what is the largest integer that you cannot get to?

Next week will be a contest.