A Conjecture of Apoloniusz Tyszka on the Addition of Rational Numbers

x

1

=1

x

2

=0

2

x

2

-

x

3

=0

{

x

1

=1,

x

2

=0,

x

3

=0}

The absolute value bound is 4.

The maximum value is 1.

The minimum value is 0.

This Demonstration illustrates a conjecture of Apoloniusz Tyszka on the set of solutions of a system of linear equations in

n

variables

x

1

,

x

2

,...,

x

n

; the equations are all of the form

x

i

=1

or

x

j

+

x

k

=

x

l

for

1≤i≤n,1≤j≤k≤n,1≤l≤n

. The conjecture states if such a system has a solution, then there is a solution such that



x

j

≤

n-1

2

for all

1≤j≤n

.

The conjecture is illustrated and tested in several ways depending on the value of the parameters set by the user. For a given number of variables, the user can choose a single system, which is selected at random. The system and its unique solution are then displayed in an explicit (textual) form. If the number of systems is larger than one, then the output is displayed graphically in a form that depends on the specified number of variables. For three and four variables a sample of the actual solutions is displayed inside a rectangle or a rectangular box (we omit the first coordinate as it is always 1). For five and six variables the specified number of systems is created at random and the maximal and minimal values from the solution sets are plotted on two bar charts.