Wythoff Array
John Conway
conway at math.Princeton.EDU
Fri Jun 9 03:30:29 CEST 2000
On Fri, 9 Jun 2000 karttu at megabaud.fi wrote:
>
> Aha! By the way, how does Wythoff relate to that array,
> and who was/is Beatty?
I called it after Wythoff because if one breaks the main body
the table into pairs thus:
---------------------
| 1 2| 3 5| 8 13|...
| 4 7|11 18|29 47|...
| 6 10|...
| .......
then these are precisely the "safe combinations" in Wythoff's
game (whose definition you can find in "Winning Ways", say).
Now Wythoff proved that these are precisely the pairs
[n.tau , n.tau^2] ( [] = integer part),
where tau is the golden number "(1+root5)/2" (I prefer this
traditional notation to the more common "phi", from which it
follows that every positive integer appears just once in a
pair of this form.
Beatty (and others both before and after him - I think
including Wythoff himself!) generalized this to the observation
that for any pair of irrational numbers rho, sigma related
by the condition 1/rho + 1/sigma = 1, every positive integer
appears just once in a pair of the form
[n.rho, n.sigma].
So sequences of the form [n.rho] have become known as
"Beatty sequences" ([n.sigma] being the conjugate one to this).
John Conway
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