Course Diary

Week 1

Monday, January 13th
Today we discussed Gilbreath's First Principle. (This is also known as the Magnetic Color Principle. See Norman Gilbreath, The Linking Ring, 1966, pp. 70-71.) Along the way we described a standard deck of 52 cards, the effect of cutting a deck of cards, and a perfect shuffle.

This magic trick also had a small intro to probability.

Homework: Bring a standard pack of card with you to all classes!

Weekly Writing: (due Wednesday, January 22nd) Time to write your resume! Here is the U Kentucky Career Center resume guide (this used to be better) and the wonderful Princeton University resume guide.

See the Writing Assignment page for further details.

There is also a reading assignment due next Wednesday. See our Reading Assignment page.

Wednesday, January 15th
Gilbreath's 1st Principle, including proof that it works; the symmetric group; perfect outshuffles and modular arithmetic.

Friday, January 17th
Quiz 1 today.

Perfect outshuffles (cont'd); Euler's theorem; the phi function (aka Euler totient function) & properties.

Week 2

Monday, January 20th
NO CLASS -- MARTIN LUTHER KING JR DAY

Wednesday, January 22nd
Proof that Gilbreath's First Principle works; Properties of the Euler phi function;

Friday, January 24th
Grammar review; Magic trick with dealing 4 different suits; Computing smallest number of perfect outer shuffles needed to return deck to original order.

Week 3

Monday, January 27th
T. Ehrenborg's digraph proof of Euler's theorem that a^φ(m) ≣ 1 mod m;

Wednesday, January 29th
Resume writing clinic.

Friday, January 31st
More shuffles today: Backwards perfect shuffles, Perfect inshuffles and outshuffles. Stated and proved Gilbreath's second principle for 3 suits, and more generally, t types. We realized that dealing the cards preserves the relative order of the original deck, and that when we shuffle the deck, the last t cards we "drop" involves the cards in an interval (window) of size t, where t=the number of types.

Week 4

Monday, February 3rd
Today we took a closer look at perfect inner shuffles. We saw that one could understand what is happening to the cards (labeled 1 through 2n) if we add two "fake" cards: 0 at the top and * at the top. Then a perfect inner shuffle of the original deck corresponds to a perfect outer shuffle of this slightly larger deck.

We determined the smallest value k so that after k perfect inner shuffles the deck returns back to its original order. For a deck with 6 cards, it takes 3 shuffles, but for a deck with 52 cards, it takes 52 shuffles! Note that when computing φ(53) = 52, we had to realize that 53 is a prime number.

Homework: Please select your magic trick and partner (if you wish) for the Mini Magic Week. Also, bring a hard copy of your revised resume to class on Wednesday.

Wednesday, February 5th
Continued with Baby Hummer today. Recall this is a trick with four cards. This involves CATO moves, where CA = cut anywhere and TO = two over = turn over the top two cards. After the CATO moves, there is a flip of two cards (the top, and then the third one down).

Today we saw that viewing Baby Hummer on a circle allowed us to see what properties of the cards were being preserved under the moves. We then moved on to 10 card Hummer and CATO = cut and two over moves. More next time...

Homework: You can read about Baby Hummer at the beginning of chapter 1 in Diaconis and Graham.

Friday, February 7th
We first talked about when trying to determine whether or not a number n is prime, it is enough to test all of the positive integers less than or equal to the square root of n. (For example, for 53, we checked 1, 2, 3, 4, 5, 6 and 7, as the square root of 53 is about 7.) I also showed how to find the square root of a positive integer by hand. This follows from Newton's Method, which I may mention on Monday.

We then continued with 10 card Hummer today. The class interrupted me with "A Wave", so then it was time for Quiz 4.

We

Week 5

Monday, February 10th
10 card Hummer (cont'd); Start of the Blind Bartender.

Wednesday, February 12th
No class.

Friday, February 14th
Mini Math Magic Week begins today!!
Day 1:
Don't Bet on the Chiefs (Conrad and Josh), ASK (America, Kynnedi and Shayna), The Mathemagicians (Josh and Chetas).

Week 6

Monday, February 17th
Day 2 of Mini Math Magic Week!!
Obsidian Illusions (Zach), Los Magos (Gael and Michael), the Mystic L, L & M (Lucy, Lucas and Mekha).

Wednesday, February 19th
(To update)

Friday, February 21st
(To update)

Week 7

Monday, February 24th
(To update) Returned to the Blind Bartender's problem today. We considered: 1. 4 cups and no boxing gloves. 2. 4 cups and boxing gloves. When you have n cups, an upper bound for the number of hands assuring a winning strategy is n-1.

Wednesday, February 26th
Finished the Blind Bartender today with 5 cups. Discussed the Ehrenborg-Skinner theorem. Began discussing knots.

Friday, February 28th
Did a number of rope tricks: tying a knot, Snake in a Basket. General discussion about knots. Quiz today.

Week 8

Monday, March 3rd
Returned to the Chefalo knot today. We then discussed Reidemeister moves and (informally) knot congruences. We defined a knot as a simple closed polygonal curve in 3-dimensional space. In this way, we do not end up with infinitely-many knots.

Here you can see the mathematician Louis Kauffman performing the rope tricks we have done in class.

Wednesday, March 5th
We then began discussing the Alexander polynomial which is one way to distinguish between knots. We computed A(trefoil) and A(trefoil with extra twist). We computed the Alexander polynomial for the trefoil knot, as well as the trefoil knot with an extra loop (this is the same knot!). We saw these two examples differed by the sign -1.

Here are some knots made out of wood. (Reference: Orderly Tangles, by Alan Holden, 1983.)

The UK Library has the e-book Charles Livingston, Knot Theory available!
Here is Appendix 1 knot table and Appendix 2 Alexander polynomials

Friday, March 7th
Quiz today. Continued with Alexander polynomial today. This is one way to distinguish between knots. We stated (without proof) that if two knots are Reidemeister equivalent then their Alexander polynomial differs by ±t^k for some nonnegative integer k.

I quickly mentioned the theory of 3-colorings. This is another technique for distinguishing knots. To be continued...

Week 9

Monday, March 10th
I began talking about n-colorings of knots today as a way to distinguish between knots. (We did the case n=3.) We also computed a number of examples.

In passing I also mentioned the Four Color Theorem. Since this was the first computer-assisted proof, it was/still is very controversial.

I also stated (without proof) some facts about the Alexander polynomial, including that the coefficients alternate in sign, that a given knot embedding with n crossings has n arcs, and that there are knots having the same Alexander polynomial which are not Reidemeister equivalent.

We gave most of the proof that if a knot K is 3-colorable then it continues to be under any Reidemeister move applied to it. We will return to these ideas algebraically next time.

Wednesday, March 12th
Today we went outside for Conway's game of Rational Tangles. We began by investigating the operations of T = twist and R=rotate. Given a tangle, we computed its rational form, that is, what fraction it represents. To be continued...

Friday, March 14th
Today we had a Zoom lecture on the algebraic coloring of a knot. This involved using colors from Z mod 3 (the congruence classes modulo 3) with the condition 2x = y + z at each crossing. The main argument involved verifying the number of such colorings of the arcs remains constant under Reidemeister moves. We used this to show that the unknot and trefoil knots are different as each has a different number of colorings in this setting.

Lecture

S P R I N G B R E A K

Week 10

Monday, March 24th
Today we showed how to determine the sequence of T's and R's for a given fraction. This comes from Euclid's algorithm for the gcd of two integers. We saw that the equality gcd(a,b) = gcd(a-b,b) corresponds to the twist T and gcd(a,b) = gcd(b,-a) to the rotation R.

Wednesday, March 26th
Finished rational tangles today with covering a knot with a plastic bag, then tangling the knot to undo this, then removing the bag.

We then began talking about magic tricks with the Möbius strip.

Friday, March 28th
Continued with paper magic today. We began analyzing our various paper experiments using a gluing map diagram.

Week 11

Monday, March 31st
Finished exploration of glueing diagrams.

We then began our unit on origami. We learned how to fold hills and valleys from the center of a piece of paper. Our flat fold data supported the theorem that |H-V| = 2.

I learned all of these wonderful ideas from a workshop by the physicist and origami scientist Robert Lang.
Origami handouts

Wednesday, April 2nd
Origami, cont'd. Today we proved two origami theorems using the idea of an ant walking ono the origami crease pattern. We spoke about
Maekawa's theorem: At the vertex of any flat origami crease pattern |H - V| = 2
Kawasaki's theorem: At any vertex the alternating sum of the angles is zero

We then folded the bird foot and the helical Yosimura pattern.

Friday, April 4th
Quiz 11 today. Today we folded the 3-d Miura Ori pattern. This looks like a pile of cubes.

Week 12

Monday, April 7th
Today we folded the Miura Ori pattern. We then watched the Origami TED talk by Robert Lang. Akira Yoshizawa innovated a system to describe how to fold a particular origami form.

The main theorems of a flat origami crease pattern are:
Maekawa's theorem: At the vertex of any flat origami crease pattern |H - V| = 2
Kawasaki's theorem: At any vertex the alternating sum of the angles is zero
Corollary: The sum of the even numbered angles equals the sum of the odd numbered angles equals π.
Corollary: Every crease pattern is two-colorable.

Besides these theorems, he added the origami axiom that a sheet can never penetrate a fold.

Wednesday, April 9th
More folding today... We also watched most of the movie, "Between the Folds".

Friday, April 11th
Quiz 12 today.

Week 13

Monday, April 14th
Began the mathematics of juggling today.

Wednesday, April 16th
Science in the news at Harvard including Kirigami.

Friday, April 18th

Week 14

Monday, April 21st
Day 1 of Final Math Magic Week.

Wednesday, April 23rd
Day 2 of Final Math Magic Week.

Friday, April 25th
Day 3 of Final Math Magic Week.

Week 15

Monday, April 28th
Day 4 of Final Math Magic Week.

Wednesday, April 30th
Final Day of Math Magic Week.
Last Day of Classes

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Last updated: Monday, April 14, 2025.

© 2000 Margaret A. Readdy.