Keyboard
In MIR 42 (September 2006), Chris Hakkaart gave an overview of numerical keyboards for calculators and telephones. The ordering of the keys often seemed to be based more on technical considerations than on usability.
Rolf Hofgaard, a Norwegian pioneer of relay calculators, has invented a keyboard that beats any of the layouts Chris discussed... in awkwardness.
In U.S. patent 2261115, from 1941, Hofgaard describes a keyboard with only four keys. The idea is that some numbers can be entered by a combination of keys. This seems a good idea, which is still applied in electronic pocket devices. In the implementation, however, it went wrong, as the image shows.
How did Hofgaard end up with this layout? That is simple: the keys represent powers of two: 1, 2, 4 and 8. The combination of keys needed for a particular number corresponds to the binary representation of that figure.
I've never seen this keyboard in the flesh, which does not surprise me!
Hofgaard's keyboard
Addendum: entering numbers in a binary way was not Hofgaards idea, it was already used in 1907 in a French fire control computing device, the "Pendule Lafrogne" (see Norman Freedman, "Naval firepower: battleship guns and gunnery in the dreadnaught era", Naval Institute Press, 2008, page 308)
Addendum2: yet another strange keyboard is given in U.S. Patent 497058, from 1893. This time the layout is dictated by the machine's construction.
(Note that I reconstructed the keyboard because the patent drawing did not show the 0 and 2 keys)
The keyboard of US497058 (reconstructed)