## Problem of the Week 1073## They Said It Couldn't Be Done
A classic puzzle is to write positive integers as a combination of 1, 2, 3, 4 using the operations of addition, subtraction, multiplication, division, and exponentiation (and as many parentheses as you like), using each of the numbers exactly once. For example, 33 = 1 + 4 It had been believed that 791 is the smallest number with no representation in terms of 1, 2, 3, 4, 5, and 6. But this is not so, as discovered recently by Bruce Torrence. Express 791 using the numbers 1, 2, 3, 4, 5, 6 and the five basic operations above. Extra credit: Do the same for 1, 2, 3, ..., 790. But there must have been something subtle about 791, since that was thought impossible.
Note: Bruce observed that it does not matter whether one insists that each of the numbers be used. For one can combine any unused numbers (other than 1) into a sum S and then either multiply an expression by 1
Source: Bruce Torrence, "Arithmetic combinations," © Copyright 2007 Stan Wagon. Reproduced with permission. |

5 March 2007