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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.

Here is a pendant to Dr Gibson's beautiful dissection of the three squares. I am pretty sure that his proof is new to the world, but I am not sure that mine s so. There are about fifty proofs, differing more or less from each ither, but there are only some half a dozen worth remembering oreaching. The proof I submit (not one of the half dozen) was levised in 1859, when I was a young student in St Andrews, and b may probably have been given long before that date.

Mr J. F. Cameron of Gonville and Caius College, Cambridge, has suggested the use of factorial functions such as or n! to solve Whewell's problem. I had excluded them as beyond the bounds of ordinary arithmetic. I append half a score of my attempts with the factorials; they could be easily increased.