Is the following trifle worthy of a place in your Mathematical Notes ? I came on the puzzle in a volume of the letters of Dr W. Whewell, the well-known Master of Trinity College, Cambridge. At the end of a letter to Professor A. De Morgan, dated 18th January 1859, he says:

“I can express every whole number from 1 to 15 (I think) by means of four nines. Thus Is it worth while working this out further ?”

A note to the letter states that:

“Two more examples from a fragment in Dr Whewell's handwriting will illustrate the meaning of this:

Here are my solutions of the puzzle for the numbers 0 to 40, excepting 38. Some ingenious young mathematician may be instigated to complement my solutions, and to carry the matter still further.

(a) It is proved in treatises on Algebra that the equations (in three variables for brevity)

have a unique solution given by

provided the determinant

does not vanish.

(b) It is also proved, from (2), that if the “degenerate” homogeneous system

has a non-null solution (i.e. a solution in which the variables are not all zero), then Δ = 0.

It is worthy of remark that there is a method of proving algebraic inequalities which is very generally applicable, and which furnishes proofs of great directness and completeness.

The method consists in expressing the difference A – B in a manifestly positive form, when we have to prove A > B.

The first inequality occurring in school Algebra is usually proved in this way. We have to show that if a, b, and x are positive quantities is nearer to 1 than . We have

The two cases a > b and b > a are then considered separately, and the conclusion drawn.

Theorem.—If the terms of the series Σf(n) never increase as n increases, and if f(n) = 0 then Σf(n) converges or diverges with

has three real roots.—Let x1 be any one root. Then ξ, η, ζ can be chosen uniquely to satisfy simultaneously the three equations

If x1 is complex, so also will be ξ, η, ζ. Let their conjugate complexes be , so that

; etc.