Cutting the cake

Bjørn Petter Jelle gave a talk at a school symposium last year with a selection of interesting math problems. I thought his presentation of the cake cutting problem was brilliant.

The classic problem is this: divide a round cake into 8 equal pieces with 3 cuts. He posed the problem and after we discussed in groups, one group offered the classic solution. Quarter the cake with two cuts and then make a third cut horizontally to split the four quarters in half.

Bjørn Petter then produced a cake with lovely sprinkles and big frosting flowers. He offered a knife and asked the group to perform to operation:

 

The result: a very unsatisfying division of the cake. The bottom pieces have almost no frosting:

Back to the drawing board. Another group proposed stacking the four pieces after the second cut and making the third cut vertically. Bjørn Petter produced a second cake and had them try. It worked better, but the beautiful frosting flowers were crushed. Other ideas?

A third group proposed setting the four pieces in a line and cutting with a really big knife. Bjørn Petter produced a third cake... and a sword!

 

Afterwards, we all ate cake of course. I got one of the bad pieces. It didn't matter, his presentation and the series of surprises made for one of the best presentations of a classic problem I'd ever seen. Bravo!

Knight Maze

From the Infinity conference a few years back in Ann Arbor, Michigan. I gave a talk on knight mazes and began the talk with a puzzle. You are a chess knight. Start anywhere you like on the left edge of this board and reach the star by moving only legal knight moves on the white squares. You may not leave the board or land on a blue square.

It's a maze I designed especially to be difficult. Give it a try! The talk focused on interesting maze elements and concluded with an analysis of this maze and the key to why it is so challenging.

Fractal Jugglers

"Nothing but Jugglers"

Making a picture containing "nothing but jugglers" presents a special challenge. In order to properly represent jugglers, the people in the scene must be juggling something, but what can those objects be if the only objects allowed are jugglers? We must conclude that the scene is be recursive with the number of jugglers growing exponentially to infinity as their size decreases to nothing.

Here's the scene as an animated gif. Try to follow one red figure, if you can!

4d juggling visualizer

Click on the window to use arrow keys to rotate this model. Shift-arrow keys scale the model. Flash required.

This is a 3-ball cascade juggling pattern. Imagine filming the juggling pattern then stacking the frames of the film to make a 3d model where depth represents time. That's what this shows. Be sure to click in the window to be able to use the arrow keys. Here is a bigger version.

Tetrahedron teabags

I love these teabags! A nice question: if taken apart what is its shape? Next question, if you want to make a large tetrahedron from the same pattern with a piece of A4 paper (or letter-size paper or whatever), what should the dimensions of your template be? Will you account for the overlap? I haven't tried it yet, but I think there will a nice use of √3 in here somewhere.

Silver ratio windows

I was recently at Jyväskylä University in Finland and saw these windows on the Agora building. They looked like silver rectangles to me, so I snapped a picture to measure. Sure enough, nearly exactly by my measurements on the image.

The silver ratio, by the way, is 1+√2 : 1. It has many connections to the golden ratio. It's equal to 2+1/(2+1/(2+1/(2+1/(2+...


 (Replace all of those 2s in the fraction with 1s and you get the golden ratio).

If you add a unit square to a 1:√2 rectangle (A4 paper, for example) you get a silver rectangle. If you subtract a unit square from a 1:√2 rectangle you also get a silver rectangle.

A4 rectangle plus a unit square is silver:

 A4 rectangle minus a unit square is also silver:

And just like you can remove a series of squares infinitely from a golden rectangle, you can remove a series of 2 squares infinitely from a silver rectangle...

There's lots of other cool properties... I've been fascinated with these "precious metal ratios" lately (gold, silver and yes bronze and others!) so it was exciting to spot these silver ratio windows. I hope this was an instance of intentional mathematics!

Game of Thrones 7-pointed star

I love this motif in the windows in King's Landing in HBO's "Game of Thrones." There are 7 "old gods" in this world which I think are the basis for the pattern of this design. Nice seven-pointed star!

Boggle

I found this nice 10-letter word playing Scramble (Boggle) on my iPhone. The board was one letter shy of making 'tessellation' (you can see the grid in the lower left).

Human Cube

Here's my Human Cube sculpture, painted. I'm quite pleased with the result. It will be exhibited this summer in the College of Fine Arts Gallery at Towson University, Baltimore MD.

If you'd like a copy of the model, it is available (unpainted) for purchase at Shapeways.com.

Lying with statistics

I've talked about how to lie with statistics in some of my university courses. It was with amusement and horror that I saw this at my kids' school. The school scored 2.3 on 5th grade national math test, compared with 2.0 for other schools in the area. This is 17% higher, not 200% higher as the graphic suggests (the base line starts at 1.85 instead of 0). Really, isn't 17% enough?

Carpet puzzle

Waiting in a meeting room at a hotel in Norway after I'd set up for a talk, I became interested in the carpet. I mean, who wouldn't? Carpets are cool. I wanted to spot the repeating unit; which section could be cut out and used as a "stamp" to make the entire pattern? It was a captivating exercise. Maybe you want to try... can you find the repeating unit in this carpet pattern?

 
Here's one solution, click on it to make it larger. I've marked the four corners of one possible repeating unit with small blue blocks. Note that the corners I found make a parallelogram, does this affect producing the carpet and laying it out? This carpet was all over and in some rather large rooms. How do manage to roll this out so the patterns on adjacent pieces match?

There are puzzles to be found everywhere!