Category Archives: Uncategorized

How many triangles?

It’s been quiet on the blog, but a lot has been happening. University classes in Adelaide have just resumed after a two week mid-semester break. To warm up, I gave my MATH 1070 students the following problem. I found it via Tanya Khovanova who states that it was an entrance problem for the 2016 MIT PRIMES STEP Program. (Read more on Tanya’s blog.)

I drew several triangles on a piece of paper. First I showed the paper to Lev and asked him how many triangles there were. Lev said 5 and he was right. Then I showed the paper to Sasha and asked him how many triangles there were. Sasha said 3 and he was right. How many triangles are there on the paper? Explain.

Here are some solutions from my students, all considered to be correct. The ones in blue originally appeared in Tanya’s blog post (although several students came up with these too). Additional ideas are shown in red below. The black rectangle shows the piece of paper. Two of the rectangles contain instructions instead of diagrams.

Update (8 October 2017): A few new ideas from this week’s classes are shown in purple.

I loved this as an opener to encourage creative problem solving. Thanks Tanya!

How Many Triangles 02.jpg

Another party puzzle

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More party puzzles! This one is from a thoroughly-recommended book, Puzzle Based Learning¹.

Mr and Mrs Smith invited four other couples for a party. When everyone arrived, some of the people in the room shook hands with some of the others. Of course, nobody shook hands with their spouse or themselves, and nobody shook hands with the same person twice.

After that, Mr Smith asked everyone how many times they shook someone’s hand. He received different answers from everybody.

How many times did Mrs Smith shake someone’s hand?

At first glance it seems that there is not enough information to solve the puzzle—which is why I like it! Once we consider each piece of information, we can put the bits together to find a solution.

Warning: mathematical spoilers (but not the solution) ahead. I’ll post the solution in the comments.

Some prompts:

How many people are there at the party?
What is the minimum number of handshakes possible?
What is the maximum number of handshakes possible?
Can you draw a diagram to represent the handshakes made by the person who made the most handshakes?
What can you conclude from this?
What can you add to your diagram?
Can you now solve the puzzle?

Good luck!

[1] Michalewicz, Z., Michalewicz, M., Puzzle Based Learning, Hybrid Publishers, 2008. pp 99-102.

Venn and (the art of) happiness

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I am irresistibly drawn to Venn diagrams. They make me very happy.

I love how accessible they are to emerging mathematicians. We can draw a Venn diagram on the ground and use it to sort objects — even ourselves! — into categories. An animal-sorting example: those that live on land (green hoop), those that live in the water (blue hoop), those that live in both (in the intersection of green and blue), and those that live in neither (outside of both green and blue). Or a shape-sorting example: the green ‘hoop’ contains quadrilaterals, the red ‘hoop’ contains triangles, and the yellow ‘hoop’ contains shapes with right angles.

Source: education.vic.gov.au

Counting the number of regions in a Venn diagram with n sets is a neat argument. Suppose we have a Venn diagram with three sets labelled A, B, C. Then any region is: inside or outside of set A, inside or outside of set B, inside or outside of set C. There are two possibilities for each of three sets, so there are 2×2×2 = 2³ = 8 possible regions. We can generalise this argument to n sets.

Venn diagrams also fascinate experienced mathematicians. See, for example, the survey by Frank Ruskey and Mark Weston in the Electronic Journal of Combinatorics. I am particularly taken with this symmetric Venn diagram¹ for seven sets, and not only because it is named after my home city of Adelaide.

AdelaideRain

Venn diagrams can be clever, comedic or both. Below I share some of my favourites. I’ve tried to attribute authors, where known. If you can fill in the gaps, please let me know. And, do share your favorites with me! (Update: while looking for authors, I found this tumblr account, Fuck Yeah Venn Diagrams, and this twitter account: @venn_diagrams. Ohhhh yeah.)