44. HOW LARGE? 
Using three nines, it is possible to express several large numbers. For example, 999, or 999 (= 99 x 99 x ... x 99 with nine factors). The largest possible number we can obtain in this way is 999 (*)
This number is rather large to handle: it is more than the number of electrons in the visible universe!
So let us try something smaller.
What is the largest number that can be represented with three twos? with three threes? with three fours?
(*) Note, there is an ambiguity in notation here. The interpretation here is 9(99 ). Another possibility is (99)9 which is considerably smaller.
Hint 1
For each of these problems you should find four possible candidates.
Hint 2
Evaluate each possibility
(a calculator may be useful,
but remember we are not asked to find the exact values), and look for changing patterns.
Solution
Let us begin with the three twos. There are four likely candidates for obtaining the largest number. These are: 222, 222, 222 and 222 .
Evaluating these, and placing them in increasing order, we have:
222 = 16, 222, 222 = 22 x 22 = 484, and 222 = 27 x 27 x 27 x 2
= 128 x 128 x 128 x 2 = 4 194 304. Clearly 222 is the largest.
We now repeat this exercise for the threes and fours. For the threes we obtain the possibilities:
333, 333, 333 and 333 .
Evaluating as before we get in order:
333, 333 = 33 x 33 x 33 = 35 937, 333 = 327 (certainly larger than 333),
and 333 (larger again).
For the fours we obtain the possibilities: 444, 444, 444 and 444 .
Placing these in order gives:
444, 444 = 3 786 816, 444 (much larger than 444),
and 444 = 4256 (larger again).
It is interesting to note how the order of the terms changes as we move from two to three to four.
Extensions
1. On the basis of the evidence you now have, what do you think is the largest number which can be obtained with three fives?
2. Work through the simpler problem which asks for the largest number represented by two twos, two threes, or two fours.
Hint 1
Hint 2
Solution
Extensions
