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.