Four Fours. Some Arithmetical Puzzles

Extract

An arithmetical amusement, said to have been first propounded in 1881, is the expression in the ordinary arithmetic and algebraic notation of the consecutive integers from 1 upwards, as far as practicable, by the use of four “4”s. I have mentioned the problem in my Mathematical Recreations, but my friend Mr. Oscar Eckenstein has now carried the solutions considerably further. I think a bare statement of our procedure may be of interest, and perhaps some readers of the Gazette may amuse themselves by filling in the details of the analysis or applying similar methods to higher numbers.