# Penrose tiles and the great icosicosidodecahedron

Last night we completed a fun day of playing around with our Zometool set by building a great icosicosidodecahedron :

A large part of the fun from just playing around with the Zome pieces is that you never know what shapes you’ll end up with! After we finished building this monstrosity something very surprising happened. My younger son was looking at the structure and said that he saw shapes in it that looked like the shapes of Penrose tiles. We have talked about Penrose tiles before, but it was back in March:

Penrose Tiles and some simple 3d Variations

I honestly have no idea why these shapes stuck with him, but kids surprise you sometimes.

Tonight we looked up the angles in the Penrose darts and kites and then did a short calculation of the angles in the great icosicosidodecahedron. The main tool you need is that the angles in a triangle add up to 180 degrees, so that was easy enough to explain to him. To my great delight he was correct – the shapes he saw were indeed exactly the shapes of the Penrose tiles.

Another example of why I love teaching my kids π

# 3 problems from the last week that made me think about math in my work

I’ve seen three interesting problems in the last week that have stuck in my mind. I’m a little embarrassed to admit that I don’t remember where I saw one of them, but here they are with two sources:

(1)

The project here begins with this situation: “Your class needs to raise \$100 to go on a field trip. You decide to sell cups of iced tea and lemonade after school. At the stand, iced tea costs \$0.50 per cup and lemonade costs \$0.80 per cup,” and continues with questions about graphing, lines, slopes, and things like that.

(2) From the puzzle corner section of MIT’s Technology Review:

Link to the November / December Puzzle Corner

βSome men sat in a circle, so that each had two neighbors.
Each had a certain number of coins. The first had one coin more
than the second, who had one coin more than the third, and so
on. The first gave one coin to the second, who gave two coins to
the third, and so on, each giving one coin more than he received,
for as long as possible. There were then two neighbors, one of
whom had four times as much as the other. How many men were there in the circle, and how much money did each one have?β

I worked through this problem with my kids here: Family Math 215

(3) From an unknown source, and perhaps not quite transcribed perfectly (sorry, again, for forgetting who showed me this problem):

“10 couples attend a party. At the party each person shakes the hand of every person that he/she doesn’t already know. Assume, naturally, that each member of a couple knows each other! At some point one of the people at the party asks everyone else how many hands that he/she shook. The answers come back 0, 1, 2, 3, 4, 5, 6, 7, and 8. How many hands did the person who asked this question shake?

Each of these three questions gave me something to think about, though I’ve spent the most time thinking about the first question for reasons that I’m hard pressed to explain.

What’s interesting to me is that I’m starting to think that the process of working through (2) and (3) is more relevant to the math that I do on a day to day basis at work than the process of working through (1) is. That seems a little odd to me since (2) and (3) are pretty contrived, almost math contest-like, problems and (1) is more naturally geared to a business setting.

But I think that it is the process of searching for patterns and ideas on the way to the solution of (2) and (3) that makes them appeal to me. Rarely are the problems that I have to think through at work as clear cut as (1) is. It isn’t that I think (1) is a poor problem or exercise. Quite the opposite, actually, and ways to play around with it have been on my mind for several days now. I think, though, that the mathematical thinking required to solve (2) and (3) are more valuable in business.

Some examples of problems that I’ve had to think through at work:

(A) You are approached by a well-known professional sports team who is playing in three different tournaments. The owners of the team have promised the players a total of \$1,000,000 of bonuses if the team wins all three tournaments – but they have to win all three or the bonuses will not be paid. The friendly odds makers say that the chance of the team winning each tournament is 50%, 50%, and 10%. The owners would like to buy an insurance policy from you that will cover the cost of the bonuses should the team win all three tournaments. How much would you charge for this insurance?

(B) You are approached by a well-known professional sports team who has signed a famous athlete to a contract that will pay a salary of \$10,000,000. The contract is guaranteed even if the player is injured and unable to play. The team would like to buy a policy from you that will pay the \$10,000,000 in the event of an injury preventing the player from playing. You look at the historical data and find (and I’m making this number up) that 5% of the time a player similar to this player is injured and misses the season. You also see several stories in the press about performance enhancing drugs that could be either masking, or perhaps preventing, injuries in the historical data that you’ve reviewed. How much would you charge for this insurance?

(C) You are asked to insure a large prize at a poker tournament. Suppose for simplicity that the prize is \$100,000 and everyone agrees that the mathematical chance of the prize being awarded is exactly 1 in 100. What other things do you need to consider before agreeing to insure this prize for \$5,000?

So, none of these examples really match any of the three original problems exactly. But also none of the work problems really fall into the category of questions where the “right” answers just pop out of formulas. All three of the work questions require you to explore the situation a bit before heading down the path to an answer. I think that’s why questions (2) and (3) above appeal to me a bit more and (1) does even though I recognize that the lessons being taught in (1) are important to getting to the solution of both (2) and (3), and all of the problems I mentioned from work.