## Numberplays

Many problems requiring numerical solutions can only be solved by setting up some equations and then some detailed analysis. Often the equations have some symmetry, which may involve some manipulation.

Example: Solve the simultaneous equations

(1)

(2)

(3)

Multiply (1) by c, (2) by b and (3) by a to give the three equations

(4)

(5)

(6)

From these we obviously obtainhence

Ifthen from (1)soand thenand

The solutions areor

If(so that) then from (1)

The solutionis found above. The other possibility isand these also satisfy all three equations.

andare symmetrical in the equations (1), (2) and (3) so that if any two are interchanged, the same equations result, in a different order. This means we can swap values ofandto obtain different solution.

Swappingandgives the solution

Swappingandgives the solution

Example: Solve the simultaneous equations

(1)

(2)

(3)

Addto (1), addto (2) and addto (30 to give

andare all equal to a+b+c, they are all equal to each other, so and

The solutions arefor any number

Example: Solve the simultaneous equations:

(1)

(2)

(3)

Adding these equations givesthen

(4)

(5)

(6)

(1)+(4) gives

(2)+(5) gives

(1)+(4) gives