| In some world cultures, the concept of large is
very different to our own. Some people have the rudimentary number system 1, 2,
3, many. There is the story of an African tribesman trying to understand the
number of people killed in the Second World War. He was stunned when told it
was more than 10! Surely, when it comes to truly immense numbers, the number of
electrons, atoms and neutrons in the universe is pretty big; perhaps the
biggest quantity you could think of? Not at all! That number is triflingly
small! Dwarfed by the latent power of the digits 1, 2, 3 and 4.
Not long ago, at my school, a large city comprehensive, I
ran a mathematical competition that ran:
What is the largest number you can make using the digits
1, 2, 3 and 4, the four mathematical signs, +, -, *, ÷, brackets and the
decimal point? You can use each digit once only.
This problem was adapted from one originally published in
Scientific American a few years ago and discussed in detail in a fascinating
book by Clifford A. Pickover ("Computers and the Imagination", Alan Sutton
Publishers, 1991). There the conditions were tighter, in that +, * and ÷
were not permitted. Pickover discusses the largeness of large numbers,
comparing for example, the number of trials necessary for a monkey to type
Shakespeare's Hamlet by random selection of keys with the number of possible
chess games and the number of electrons, atoms and neutrons in the universe. In
what order of size would you put these numbers? In fact, the order, smallest
first, is: the universe number 1079, then the Hamlet number,
approximately 1040000, then the Chess number, 10a, where
a = 1070.5. |
An intriguing point to consider here is that both
the Hamlet and the chess numbers are far larger than we can possibly imagine -
if, that is, we agree that it is surely impossible for us, at our current stage
of evolution, to imagine an entity that is larger than all known matter, let
alone the particles that make up our own brains! Remember that the word for one
million was coined only in the 13th century and billion in the 17th.
As always, I publicised the puzzle on the Maths notice
board, and for the first time in the monthly newsletter that goes home to
parents too. About 40 entries were received. I set the problem to my own middle
ability Year 9 class. Most ideas were disappointingly unimaginative, though
three pupils came up independently with the idea 432 ÷ .1 = 4320. Once
the others had caught wind of the fact that someone had got over 4000 - a huge
number! - there was a scramble to copy the idea and by the end of the lesson
most had assumed they had caught me out and would each now win the prize. No
one mentioned that you could beat this just by putting the digits in the order
4321, and I didn't have the heart to disillusion them. Actually, there were
many entries from other pupils in the school that were smaller. The smallest
was 3 x 421 = 1263. There were also a few of the kind 4 ÷ (1+2-3) =
infinity, "the largest number there is", as one entrant put it. These were
expressly forbidden in Pickover's competition, since he stressed that the
answer must be finite. I decided also to disallow these entries, even though
the rules did not bar them. My argument was that the result of dividing by zero
is not infinity at all but simply undefined, although I knew that many pupils
would not buy that! The top 10 entries in my competition are listed below,
along with their values, as generated by the spreadsheet Excel. |
1, 2, 3, 4 - four digits that DWARFED the
universe