## 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 obtain hence If then from (1) so and then and The solutions are or If (so that ) then from (1) The solution is found above. The other possibility is and these also satisfy all three equations. and are 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 of and to obtain different solution.

Swapping and gives the solution Swapping and gives the solution Example: Solve the simultaneous equations (1) (2) (3)

Add to (1), add to (2) and add to (30 to give     and are all equal to a+b+c, they are all equal to each other, so and The solutions are for any number Example: Solve the simultaneous equations: (1) (2) (3)

Adding these equations gives then (4) (5) (6)

(1)+(4) gives (2)+(5) gives (1)+(4) gives  