| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
(Stolen from the "Emperor's New Mind" by Roger Penrose, a book that I don't understand entirely yet, to put it mildly.) Is there a quadratic function on x and y (ax�+bxy+cy�+dx+ey+f) which induces a one-to-one correspondence between (x,y) the pairs of natural numbers and the naturals themselves. ie: each (x,y) under this function goes to a *different* natural number, and each natural n is the image of some pair under this function)? If you can find such a function, can you find any others? How many are there? Regards, Andrew.
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1212.1 | 4GL::GILBERT | Ownership Obligates | Tue Mar 27 1990 14:26 | 12 | |
The function (x�+2xy+y�+3x+y)/2 induces the mapping: (0,0) -> 0 (0,1) -> 1 (0,2) -> 3 (0,3) -> 6 (0,4) -> 10 (1,0) -> 2 (1,1) -> 4 (1,2) -> 7 (1,3) -> 11 (2,0) -> 5 (2,1) -> 8 (2,2) -> 12 (3,0) -> 9 (3,1) -> 13 (4,0) -> 14 It may be better understood in the form: (x+y)(x+y+1) ------------ + x 2 | |||||
| 1212.2 | from the same pen as 919.1... | HERON::BUCHANAN | combinatorial bomb squad | Wed Mar 28 1990 12:55 | 9 |
Yes, that's it. Strange: I'd never thought of this packing algorirhm as having a simple quadratic form until it was pointed out to me. Clearly there's another solution with x <-> y. Any others? (I dunno, and haven't looked.) Regards, Andrew. | |||||
| 1212.3 | AITG::DERAMO | Dan D'Eramo, nice person | Wed Mar 28 1990 15:57 | 11 | |
re .2
>> Strange: I'd never thought of this packing algorirhm as
>> having a simple quadratic form until it was pointed out
>> to me.
Yes, same here. I had just assumed it would involve some
kind of "if ... then formula_1 else formula_2"
definition.
Dan
| |||||