Saturday 5 January 2013

Before There Were Four-Fours, There Were Four-Threes

*Eyegate Gallery

EVERYONE has encountered the four-fours problem, using four fours and whatever mathematical operations that were allowed to make a number, or a set of numbers. You may even have read that it originated in the famous book of recreations by W. W. Rouse Ball; Wikipedia still has, "The first printed occurrence of this activity is in 'Mathematical Recreations and Essays' by W. W. Rouse Ball published in 1892. In this book it is described as a 'traditional recreation'. "

I know you've heard it before, but here we go again, "Wikipedia is wrong about that."

The first record I have found of a puzzle like these was in an 1818 edition of The schoolmaster's assistant: being a compendium of arithmetic both practical and theoretical : in five parts, and early American Arithmetic by Thomas Dilworth.
This image is from page 189 and part of a collection of "Short and Diverting Questions". As is typical of many of the early such problems, there were no specifications for the operations that might be employed. I have found the same exact problem in the 1800 edition. 
In the same collection of problems, Dilworth poses a problem requesting the use of four threes...(which should give you a big clue if you are stuck on the previous problem of using four figures to make 12.
Ok, even I can do that one, and the dd+d/d format becomes a regular problem through the years with different digits; the most common being in the form of "use four nines to make 100."

Professor Singmaster says the both the Dilworth problems appear in a 1743 edition of this book.
By 1788 similar problems show up in another classic American Arithmetic by Nicolas Pike,"Said Harry to Edmund, I can place four 1's so that, when added, they shall make precisely 12. Can you do so too?"

The first printed version I can find of a question like this that asks about using three or four of the same number to find a set of integers appears in 1881 in a U.K. magazine called, Knowledge: an Illustrated Magazine of Science. It was founded and edited by Richard A Proctor, the English astronomer who is remembered for his maps of Mars (and has a crater there named for him). It may be that the Cupidus Scientiae who submitted the question is, in fact, the editor.

The next edition (Jan 6, 1882) did indeed carry the solutions, as well as correspondence from an H. Snell who provides that 19 = 4! - 4 - 4/4 (he uses the Jarret symbol for factorial which looks like a right angle symbol with the number on the horizontal line)
The editor felt that factorials were inappropriate for the problem as posed. The following week (Jan 13,1882) there were several solutions for 19 from contributors, including (4+4-.4)/.4.

When W. W. Rouse Ball got into the act, it was in the third edition of his MRE 1896 and it was a long step away from the four-fours as we have come to know it. He repeated a problem previously used by Sam Loyd in 1893 which became popular in the United Kingdom; "Make 82 with the seven digits 9, 8, 7, 6, 5, 4, 0." Loyd offered a prize of 100 pounds for the solution. The solution, involving the use of repeating fractions, was given as 80.5 + .97 + .46 = 82 with all the decimal values repeating. This was indicated in the period by using a single dot above the values which repeated.

It was not until the fifth edition of 1911 of MRE that Ball gives the more common version, and describes it as, "An arithmetical amusement, said to have been first propounded in 1881,...) which seems to refer to the posting in Knowledge. By the sixth edition (1914) he has extended the problem to four nines and four threes. This one is significant because it seems to be the first that discusses what values can be achieved by what set of operations.

After a while it even caught on with higher level mathematicians. In 1991 Clifford A. Pickover asked for good approximations to Phi using four fours.
In 1999 it became popular to ask for integers created using the digits 1, 9, 9, 9.
And it seems I saw a few of those floating around the internet at the beginning of 2012.
But remember, it all started with Jack and Harry, and four-threes.