The Cabinet of Curios
Odometers
A number system is an odometer: a row of wheels that roll over and carry into the next wheel. Change how many positions each wheel has and you get base 10, binary, a clock, or the factorial base that Lehmer codes use.
The same number on every odometer
Things to notice
- A wheel is worth what the wheels to its right can count. A turn of the tens wheel comes after the ones wheel shows all 10 readings. A turn of the minutes wheel comes after the seconds wheel shows all 60. Multiply the sizes of the wheels to the right and you get the wheel's place value.
- Same-sized wheels give powers. When every wheel has 10 positions the place values are 1, 10, 10 × 10, 10 × 10 × 10. With 2 positions they're 1, 2, 4, 8, 16 and so on.
- You already use mixed wheels. A clock's seconds and minutes roll over at 60 but its hours roll over at 24. Nobody finds that strange, and it's the same idea as the factorial base.
- The factorial base grows its wheels one at a time. Reading from the right, the wheels have 1, 2, 3, 4 and 5 positions, so the place values are 1, 1, 2 × 1, 3 × 2 × 1, 4 × 3 × 2 × 1: the factorials.
- That's why it fits shuffles. In a Lehmer code the first card is picked from 5, the next from 4, and so on, so each digit is a wheel with one fewer position than the last. Five wheels hold 5 × 4 × 3 × 2 × 1 = 120 readings, one for every order of five things.