William A. Webb

Approximating fair division with a limited number of cuts

A large class A of finite algorithms for fairly dividing a cake using k of fewer cuts is described. Assume an algorithm assigns piece Xi to player Pi using associated probability measure μi on measurable subsets of the cake X. If M(n, k) = maxA mini(μi(Xi)) and N(n, k) = maxA(number of i such that μ1(X1⩾ 1n) then for n ⩾ 2, M(n, n − 1) = 1(2n − 2), for n ⩾ 3, M(n, n) ⩾ 1(2n − 3), and for n ⩾ 4, M(n, n + 1) ⩾ 1(2n − 4). Also N(2n − 2, n − 1) = n.

How to cut a cake fairly using a minimal number of cuts

Abstract What is the minimum number of cuts needed to divide a cake among n players so that each player receives at least 1 n of the whole cake? The simple “one cuts - the other chooses” shows that one cut suffices for 2 players. It was previously known that 3 players require 3 cuts and 4 players require 4 cuts with only upper bounds available for n > 4. Algorithms using 6 cuts for 5 players and 8 cuts for 6 players are discussed, which lower the previously known upper bounds. Moreover, it is shown that 6 cuts is the best possible for 5 players.

Polynomial Pell's equation

Consider the polynomial Pells equation X 2 - DY 2 = 1, where D = A 2 + 2C is a monic polynomial in Z[x] and deg C < degA. Then for A,C ∈ Q[x], degC < 2, and B = A/C ∈ Q[x], a necessary and sufficient condition for the polynomial Pells equation to have a nontrivial solution in Z[x] is obtained.

A public key cryptosystem based on complementing sets

A new public key cryptosystem is constructed based on the idea of complementing sets

A combinatorial algorithm to establish a fair border

A_1, A_2, \ldots,A_k of integers. Such sets have the property that all sums

The N -number game for real numbers

a_1 + a_2 + \cdots + a_k where

Sieve methods for polynomial rings over finite fields

a_{i} \in A_{i} , are distinct.

Sums of perturbed sequences of integers

A finite algorithm is given for the following problem: a piece of land bordered by n countries is to be divided equally among these n countries in such a way that each countrys share is connected and adjacent to its original border.

Cake Cutting Algorithms: Be Fair If You Can

If S0 is an n-tuple of real numbers, define a sequence {Sj} by Sj = DSj-1 where D ( a 1 , … , a n ) = ( | a 1 − a 2 | , | a 2 − a 3 | , … , | a n − a 1 | ) . Such a sequence may terminate in all zeros, cycle, or continue indefinitely without repeating. Questions concerning the dimension of the set of S0 which produce cycles are studied.

Lucas' theorem for prime powers

Abstract Basic theorems concerning both the Selberg sieve and the large sieve are shown to hold for polynomial rings over finite fields. Some applications to irreducibles in arithmetic progression and primitive roots are given.