0809.6: ID Numbers


Puzzle #6: ID Numbers (April 2009)

Until fairly recently, many people thought of logic as the study of the laws of thought. All the basic rules of logic are supposed to be obvious, and they apply to all inferences and implications. In contrast, mathematics is the study of specific domains of objects: numbers, cylinders, differential equations. In the late nineteenth- and early twentieth-centuries, Gottlob Frege, in Germany, and Alfred North Whitehead and Bertrand Russell, in England, attempted to prove that mathematics was really just logic in a complicated disguise. Since 1931, when Kurt Gödel published his two (in)famous incompleteness theorems, the logicist project of Frege and Whitehead and Russell has been considered a failure, though there has been renewed interest in different versions of logicism. This month’s logic puzzle is a bit more mathematical than previous ones, but it requires no more mathematics than simple division.

The Puzzle

Once upon a time, the Hamilton College administration was taken over by a rogue band of number theorists intent on developing a new system of student ID numbers. The number theorists wanted the new ID numbers to have ten digits in which each of the numerals from 0 to 9 appeared exactly once. They also wanted each ID number to be divisible by each of the digits (except 0!).

Questions

1. What would be the smallest possible new ID number?
2. What would be the largest possible new ID number?

Bonus Question

3. How many new ID numbers conforming to the number theorist’s constraints are possible?