Enumerating all 3-tuples

[Paul Moore]

On 10 March 2018 at 02:18, MRAB <python at mrabarnett.plus.com> wrote:
> On 2018-03-10 01:13, Steven D'Aprano wrote:
>>
>> I am trying to enumerate all the three-tuples (x, y, z) where each of x,
>> y, z can range from 1 to ∞ (infinity).
>>
>> This is clearly unhelpful:
>>
>> for x in itertools.count(1):
>>      for y in itertools.count(1):
>>          for z in itertools.count(1):
>>              print(x, y, z)
>>
>> as it never advances beyond x=1, y=1 since the innermost loop never
>> finishes.
>>
>> Georg Cantor to the rescue! (Well, almost...)
[...]
>> Can anyone help me out here?
>>
> Think about the totals of each triple. You'll get totals of 3, then totals
> of 4, etc.
>
> This might help, although the order they come out might not be what you
> want:
>
> def triples():
>     for total in itertools.count(1):
>         for i in range(1, total):
>             for j in range(1, total - i):
>                 yield i, j, total - (i + j)

Mathematically, that's the usual generalisation of Cantor's diagonal argument.

Paul