math O

• Carroll’s sillylogisms:
  • All pets without inner core have no Chi
  • Any pet with large teeth can easily withstand an attack
  • Pets that are not adorable have no inner core
  • Any pet able to withstand an attack possesses an inner Chi

Games: Hiding on a circle, Second largest, Estimate number of digits

Homework:

• Create a sillylogism a la Carroll’s, and post it here, as a comment. We’ll solve them in class…

• solving hyperbola problem. Vieta theorem, again
• from recursion to generating function.
• Bee sounds…
• Carroll’s sillylogisms.
• is there an \(\mathbb{F}_4\)?

Game: hiding on a circle

Homework:

• Consider the midpoints of the segments formed by intersecting the line \(x+2y=c\) with the hyperbola \(xy=4\). As \(c\) varies, what are the positions of those midpoints?
• Create a sillylogism a la Carroll’s, and post it here, as a comment. We’ll solve them in class…

• properties of parabola
• from recursion to generating function.
• Carroll’s sillylogisms.
• is there an \(\mathbb{F}_4\)?

Game: second best on \(\{0,1,2\}\)…

Homework:

• Consider the midpoints of the segments formed by intersecting the line \(x+2y=c\) with the hyperbola \(xy=4\). As \(c\) varies, what are the positions of those midpoints?
• Create a sillylogism a la Carroll’s, and post it here, as a comment. We’ll solve them in class…

• from recursion to generating function.
• couple of geometry problems: using similarities for fun and profit.
• is there an \(\mathbb{F}_4\)?

Game: second best on \(\{0,1,2\}\)…

• generating functions, – \(1/(1-x)^3\): how the coefficients behave.
  recursion on the coefficients.
• from recursion to generating function.
• couple of geometry problems: using similarities for fun and profit.
• is there an \(\mathbb{F}_4\)?

Game: second best on \(\{0,1,2\}\)…

• generating functions:\[(1+x+x^3)^4=x^{12}+4 x^{10}+4 x^9+6 x^8+12 x^7+10 x^6+12 x^5+13 x^4+8 x^3+6 x^2+4 x+1\]
  what about \[(1+x+\ldots+x^9)^2?\]
• pure formality! work out \(1/(1-x)^2\)
• couple of geometry problems: equipartitions without symmetries; misshaped Magen David.
• is there an \(\mathbb{F}_4\)?
• game of socialism for 2 players

Game: second best on \(\{0,1,2\}\)…

• recalling some combinatorial notions
• arithmetic of the reals: positivity
• is there an \(\mathbb{F}_4\)?
• game of socialism for 2 players

Game: second best

• Cutting the round pie into equitable parts (homework and other variants)
• and of finite fields: solving quadratic equations, – when this is possible? some experiments.
• game of socialism for 2 players

Game: second best

• Putin, Medvedev and generating functions
• finite fields: solving quadratic equations.
• Is there an \(\mathbb{F}_4\)?

Game: Robespierre (ou Danton?)

• Cutting the round pie into equitable parts (homework!)
• arithmetic of reals
• and of finite fields

Game: Robespierre

• Algebra of integer remainders: when does it form a field?
• More on convexity: homework.
• Morse sequences

Game: ?

homework

• Arithmetic of finite fields. Solving linear equations
• Experiment with Morse sequences

• Worked out the two prisoners problem from Jan 16; three prisoners still is homework!
• Inequalities like
  \[\sqrt{x+a}+\sqrt{x-a}\leq 2\sqrt{x}:
  \]
  role of convexity.
• Constructed \(\mathbb{F}_3\) by direct analysis.

Game: wordles

homework

• describe the field \(\mathbb{F}_5\) consisting of five elements. (Key question to answer: how many times you need to add \(1\) to itself to get \(0\)? in that field?)
• using convexity prove that for \(0<a<b<x\), one has \[\sqrt{x+a}+\sqrt{x-a}>\sqrt{x+b}+\sqrt{x-b}.\]
• question from Jan. 16 for three prisoners (now the numbers can be \(\{0,1,2\}\)).

• Guessing your number
• Guessing games: strategies and probabilities

Game: wordles

homework

Recall the definition: a field is a collection of elements (numbers) such that one can add and multiply them. Also, one has two special elements, \(0\) and \(1\) such that \(0+a=0\) and \(1\cdot a=a\) for any number \(a\). Lastly, for any number \(a\) there is a number \(b: a+b=0\) (we call this \(b\) simply \(-a\), and use the shorthand \(c-a:=c+(-a)\)), and for any \(a\neq 0\), there is a number \(b: a\cdot b=1\), – as before, we use the shorthand \(c/a:=c\cdot b\) for such reciprocal element \(b\)).

These operations are subject to rules: commutativity, associativity, distributivity.

• describe the field \(\mathbb{F}_3\) consisting of three elements, \(\{0,1,a\}\). (Key question to answer: what is \(1+1\) in that field?)
• two prisoners have numbers \(0,1\) on their foreheads (can be the same! say, \(0,0\)). They can see the number of the other, but not their own. They are asked (privately) to guess their own number; they are freed if at least one of them has the correct guess. Is there a strategy for them to win?
• same question for three prisoners (now the numbers can be \(\{0,1,2\}\)).

• Can one find a multiple of 1237 consisting only of 1’s and 0s?
• Loose ends: very very skinny polynomials
• a bit of 10-adic arithmetic

Game: hype